Global existence in the derivative NLS equation with the inverse scattering transform method Dmitry E. Pelinovsky and Yusuke Shimabukuro Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1 May 15, 2015 Abstract We address existence of global solutions of the derivative nonlinear Schr¨odinger (DNLS) equation without the small-norm assumption. By using the inverse scattering transform method without eigenvalues and resonances, we construct a unique global solution in H 2 (R) ∩ H 1,1 (R) which is also Lipschitz continuous with respect to the initial data. Compared to the existing literature on the spectral problem for the DNLS equation, we transform the Riemann–Hilbert problem in the complex plane to the jump on the real line. 1 Introduction We consider the Cauchy problem for the derivative nonlinear Schr¨odinger (DNLS) equation iut + uxx + i(|u|2 u)x = 0, t > 0, u|t=0 = u0 , (1.1) where the subscripts denote partial derivatives and u0 is defined in a suitable function space, e.g., in the standard Sobolev space H m (R) of squared integrable functions with squared integrable derivatives up to the order m. Local existence of solutions for u0 ∈ H s (R) with s > 23 was established by Tsutsumi & Fukuda [32] by using a parabolic regularization. Later, the same authors [33] used the first five conserved quantities of the DNLS equation and established the global existence of solutions for u0 ∈ H 2 (R) provided the initial data is small in the H 1 (R) norm. Using a gauge transformation of the DNLS equation to a system of two semi-linear NLS equations, for which a contraction argument can be used in the space L2 (R) with the help of the Strichartz estimates, Hayashi [14] proved local and global existence of solutions of the DNLS equation for u0 ∈ H 1 (R) provided that the initial data is small in the L2 (R) norm. More specifically, the initial data u0 is required to satisfy the precise inequality: √ ku0 kL2 < 2π. (1.2) The space H 1 (R) is referred to as the energy space for the DNLS equation because its first three conserved quantities having the meaning of the mass, momentum, and energy are well-defined in the 1 space H 1 (R): I0 = Z R |u|2 dx, (1.3) Z Z I1 = i (¯ uux − u¯ ux )dx − |u|4 dx, R Z Z Z R 1 3i 2 2 |u| (u¯ ux − ux u ¯)dx + |u|6 dx. |ux | dx + I2 = 4 R 2 R R 3i Using the gauge transformation u = ve− 4 Rx −∞ kuk6L6 ≤ one can obtain I2 = kvx k2L2 − |v(y)|2 dy (1.5) and the Gagliardo–Nirenberg inequality [35] 4 kuk4L2 kux k2L2 , π2 1 kvk6L6 ≥ 16 (1.4) (1.6) 1 1 − 2 kvk4L2 kvx k2L2 . 4π Under the small-norm assumption (1.2), the H 1 (R) norm of the function v (and hence, the H 1 (R) norm of the solution u of the DNLS equation) is controlled by the conserved quantities I0 and I2 , once the local existence of solutions in H 1 (R) is established. Developing the approach based on the gauge transformation and a priori energy estimates, Hayashi & Ozawa [15, 16, 26] considered global solutions of the DNLS equation in weighted Sobolev spaces under the same small-norm assumption (1.2), e.g., for u0 ∈ H m (R) ∩ L2,m (R), where m ∈ N. Here and in what follows, L2,m (R) denotes the weighted L2 (R) space with the norm kukL2,m := Z 2 m R 2 (1 + x ) |u| dx 1/2 . More recently, local well-posedness of solutions of the DNLS equation was established in spaces of lower regularity, e.g., for u0 ∈ H s (R) with s ≥ 21 by Takaoka [30] who used the Fourier transform restriction method. This result was shown to be sharp in the sense that the flow map fails to be uniformly continuous for s < 12 [4]. Global existence under the constraint (1.2) was established in H s (R) with subsequent generations of the Fourier transform restriction method and the so-called I2 1 method, e.g., for s > 32 33 by Takaoka [31], for s > 3 and s > 2 by Colliander et al. [6] and [7] respectively, 1 and finally for s = 2 by Miao, Wu and Xu [24]. The key question, which goes back to the paper of Hayashi & Ozawa [15], is to find out if the bound (1.2) is optimal for existence of global solutions of the DNLS equation (1.1). By analogy with the quintic nonlinear Schr¨odinger (NLS) and Korteweg–de Vries (KdV) equations, one can ask if solutions with the L2 (R) norm exceeding the threshold value in the inequality (1.2) can blow up in a finite time. √ The threshold value 2π for the L2 (R) norm corresponds to the constant value of the L2 (R) norm of the stationary solitary wave solutions of the DNLS equation. These solutions can be written in the explicit form: R p 3i x 2 2 u(x, t) = φω (x)eiω t− 4 −∞ |φω (y)| dy , φω (x) = 4ω sech(2ωx), (1.7) √ from which we have kφω kL2 = 2π. Although the solitary wave solutions are unstable in the quintic NLS and KdV equations, it was proved by Colin & Ohta [5] that the solitary wave (1.7) of the DNLS equation is orbitally stable with respect to perturbations in H 1 (R). This result indicates that there exist global solutions of the DNLS equation (1.1) in H 1 (R) with the L2 (R) norm exceeding the threshold value in (1.2). 2 Moreover, Colin & Ohta [5] proved that the moving solitary wave solutions of the DNLS equation are also orbitally stable in H 1 (R). Since the L2 (R) norm of the moving solitary wave solutions is √ bounded from above by 2 π, the orbital stability result indicates that there exist global solutions of the DNLS equation (1.1) if the initial data u0 satisfies the inequality √ ku0 kL2 < 2 π. (1.8) These orbital stability results suggest that the inequality (1.2) is not sharp for the global existence in the DNLS equation (1.1). Furthermore, recent numerical explorations of the DNLS equation (1.1) indicate no blow-up phenomenon for initial data with any large L2 (R) norm [22, 23]. The same conclusion is indicated by the asymptotic analysis in the recent work [29]. Towards the same direction, Wu [36] proved that the solution of the DNLS equations with u0 ∈ 1 H (R) does not blow up in a finite time if the L2 (R) norm of the initial data u0 slightly exceed the threshold value in (1.2). The technique used in [36] is a combination of a variational argument together with the mass, momentum and energy conservation in (1.3)–(1.5). On the other hand, the solution of the DNLS equation restricted on the half line R+ blows up in a finite time if the initial data u0 ∈ H 2 (R+ ) ∩ L2,1 (R+ ) yields the negative energy I2 < 0 given by (1.5) [36]. Proceeding further with sharper Gagliardo–Nirenberg-type inequalities, Wu [37] proved very recently that the global solutions of the DNLS equation exists in H 1 (R) if the initial data u0 ∈ H 1 (R) satisfies the inequality (1.8), which is larger than the inequality (1.2). Our approach to address the same question concerning global existence in the Cauchy problem for the DNLS equation (1.1) without the small L2 (R)-norm assumption relies on a different technique involving the inverse scattering transform theory [2, 3]. As was shown by Kaup & Newell [17], the DNLS equation appears to be a compatibility condition for smooth solutions of the linear Lax system given by ∂x ψ = −iλ2 σ3 + λQ(u) ψ (1.9) and ∂t ψ = −2iλ4 σ3 + 2λ3 Q(u) + iλ2 |u|2 σ3 − λ|u|2 Q(u) + iλσ3 Q(ux ) ψ, (1.10) where ψ ∈ C2 is assumed to be a C 2 function of x and t, λ ∈ C is the (x, t)-independent spectral parameter, whereas the matrix potential Q(u) is given by 0 u . (1.11) Q(u) = −u 0 The Pauli matrices that include σ3 are given by 0 1 0 i σ1 = , σ2 = , 1 0 −i 0 1 0 σ3 = . 0 −1 (1.12) A long but standard computation shows that the compatibility condition ∂t ∂x ψ = ∂x ∂t ψ is equivalent to the DNLS equation iut + uxx + i(|u|2 u)x = 0. The linear equation (1.9) is usually referred to as the Kaup–Newell spectral problem. In a similar context of the cubic NLS equation, it is well known that the inverse scattering transform technique applied to the linear Lax system (associated with the so-called Zakharov–Shabat spectral problem) provides a rigorous framework to solve the Cauchy problem in weighted L2 spaces, e.g., for u0 ∈ H 1 (R) ∩ L2,1 (R) [10, 11, 40] or for u0 ∈ H 1 (R) ∩ L2,s (R) with s > 21 [9]. The function spaces are chosen from the fact that the x-derivative Lax operator associated with the cubic NLS equation differs from (1.9) by the absence of the spectral parameter λ in front of the matrix Q(u). As a result, Neumann series solutions for the Jost functions converge and the analytic extensions of the Jost functions are 3 well defined if u is controlled in the space L1 (R), see, e.g., Chapter 2 in [1]. As was shown originally by Deift & Zhou [11, 40], the inverse scattering problem based on the Riemann–Hilbert problem of complex analysis with a jump across the real line can be solved uniquely if u is defined in a subspace of L2,1 (R), which is continuously embedded into the space L1 (R). The time evolution of the scattering data is well defined if u is posed in space H 1 (R) ∩ L2,1 (R) [10, 11]. For the x-derivative Lax operator in (1.9), the key feature is the presence of the spectral parameter λ in front of the matrix Q(u). As a result, Neumann series solutions for the Jost functions do not converge uniformly if u is only defined in the space L1 (R). Although the Lax system (1.9)–(1.10) appeared long ago and was used many times for formal methods, such as construction of soliton solutions [17], temporal asymptotics [18, 34], and long-time asymptotic expansions [38, 39], no rigorous results on the function spaces for the matrix Q(u) have been found so far to ensure bijectivity of the direct and inverse scattering transforms related to the solution of the Cauchy problem for the DNLS equation (1.1) in a suitable function space for u0 . In this connection, we mention the works of Lee [19, 20] on the local solvability of a generalized Lax system with λn dependence for an integer n ≥ 2 and generic small initial data u0 in Schwarz class. In the follow-up paper [21], Lee also claimed existence of a global solution to the Cauchy problem (1.1) for large u0 in Schwarz class, but the analysis of [21] relies on a “Basic Lemma”, where the Jost functions are claimed to be defined for u0 in L2 (R). However, equation (1.9) shows that the condition u0 ∈ L2 (R) is insufficient for construction of the Jost functions uniformly in λ. We address the bijectivity of the direct and inverse scattering transform for the Lax system (1.9)– (1.10) in this work. We show that the direct scattering transform for the Jost functions of the Lax system (1.9)–(1.10) can be developed under the requirement u0 ∈ L1 (R) ∩ L∞ (R) and ∂x u0 ∈ L1 (R). This requirement is satisfied if u0 is defined in the weighted Sobolev space H 1,1 (R) defined by H 1,1 (R) = u ∈ L2,1 (R), ∂x u ∈ L2,1 (R) . (1.13) Note that it is quite common to use notation H 1,1 (R) to denote H 1 (R) ∩ L2,1 (R) [11, 40], which is different from what we use here in (1.13). Furthermore, we show that asymptotic expansions of the Jost functions are well defined if u0 ∈ H 2 (R) ∩ H 1,1 (R), which also provide a rigorous framework to study the inverse scattering transform based on the Riemann–Hilbert problem of complex analysis. Compared to the existing literature on the Kaup–Newell spectral problem [18, 34, 38, 39], we consider the Riemann–Hilbert problem on the real line for the new spectral parameter z := λ2 . Finally, the time evolution of the scattering data is well defined if u0 ∈ H 2 (R) ∩ H 1,1 (R). To present the global existence result for the DNLS equation (1.1), we shall define eigenvalues and resonances for the Kaup–Newell spectral problem (1.9). Definition 1. We say that λ ∈ C is an eigenvalue of the spectral problem (1.9) if the linear equation (1.9) with this λ admits a solution in L2 (R). Definition 2. We say that λ ∈ R ∪ iR is a resonance of the spectral problem (1.9) if the linear equation (1.9) with this λ admits a solution in L∞ (R) with the asymptotic behavior ( 2 a+ e−iλ x e1 , x → −∞, ψ(x) ∼ 2 a− e+iλ x e2 , x → +∞, where a+ and a− are nonzero constant coefficients, whereas e1 = [1, 0]t and e2 = [0, 1]t . Theorem 1. For every u0 ∈ H 2 (R)∩H 1,1 (R) such that the linear equation (1.9) admits no eigenvalues or resonances in the sense of Definitions 1 and 2, there exists a unique global solution u(t, ·) ∈ H 2 (R) ∩ H 1,1 (R) of the Cauchy problem (1.1) for every t ∈ R. Furthermore, the map H 2 (R) ∩ H 1,1 (R) ∋ u0 7→ u ∈ C(R; H 2 (R) ∩ H 1,1 (R)) 4 is Lipschitz continuous. Remark 1. A sufficient condition that the linear equation (1.9) admits no solution in L2 (R) was found in [27]. This condition is satisfied under the small-norm assumption on the H 1,1 (R) norm of the initial data u0 . See Remark 5 below. Although we believe that there exist initial data u0 with large H 1,1 (R) norm that yield no solutions of the linear equation (1.9) in L2 (R), we have no constructive examples of such initial data. Nevertheless, we will show in the forthcoming work that a finite number of eigenvalues λ ∈ C corresponding to the L2 (R) solutions of the linear equation (1.9) can be included by using algebraic methods such as the Backl¨ und, Darboux, or dressing transformations [8, 9]. Remark 2. The condition of no resonance is used to identify the so-called generic initial data u0 . The non-generic initial data u0 violating this condition are at the threshold case: a small perturbation to u0 changes the number of eigenvalues λ in the linear equation (1.9). Remark 3. Compared to the results of Hayashi & Ozawa [14, 15, 16, 26], where local and global wellposedness of the Cauchy problem for the DNLS equation (1.1) was established in H 2 (R)∩H 1,1 (R) under the small L2 (R) norm assumption (1.2), the inverse scattering transform theory is developed without the smallness assumption on the initial data u0 . Remark 4. An alternative proof of Theorem 1 is developed in [28] by using a different version of the inverse scattering transform for the Lax system (1.9)–(1.10). The paper is organized as follows. Section 2 reports the solvability results on the direct scattering transform for the linear equation (1.9). Section 3 is devoted to the solvability results on the inverse scattering transform for the linear equation (1.9). Section 4 incorporates the time evolution of the linear equation (1.10) and contains the proof of Theorem 1. 2 Direct scattering transform The direct scattering transform is developed for the x-derivative Lax equation (1.9), which we rewrite here for convenience: ∂x ψ = −iλ2 σ3 + λQ(u) ψ, (2.1) where ψ ∈ C2 , λ ∈ C, and the matrices Q(u) and σ3 are given by (1.11) and (1.12). The formal construction of the Jost functions is based on the construction of the fundamental solution matrices Ψ± (x; λ) of the linear equation (2.1), which satisfy the same asymptotic behavior at infinity as the linear equation (2.1) with Q(u) ≡ 0: Ψ± (x; λ) → e−iλ 2 xσ 3 as x → ±∞, (2.2) where parameter λ is fixed in an unbounded subset of C. However, the standard fixed point argument for Volterra’s integral equations associated with the linear equation (2.1) is not uniform in λ as |λ| → ∞ if Q(u) ∈ L1 (R). Integrating by parts, it was suggested in [27] that uniform estimates on the Jost functions of the linear equation (2.1) can be obtained under the condition kukL1 (kukL∞ + k∂x ukL1 ) < ∞. Here we explore this idea further and introduce a transformation of the linear equation (2.1) to a spectral problem of the Zakharov–Shabat type. This will allow us to adopt the direct and inverse scattering transform, which was previously used for the cubic NLS equation [11, 40] (see also [9, 10] 5 for review). Note that the pioneer idea of a transformation of the linear equation (2.1) to a spectral problem of the Zakharov–Shabat type can be found already in the formal work of Kaup & Newell [17]. Let us define the transformation matrices for any u ∈ L∞ (R) and λ ∈ C, 2iλ −u(x) 1 0 and T2 (x; λ) = , (2.3) T1 (x; λ) = 0 1 −u(x) 2iλ If the vector ψ ∈ C2 is transformed by ψ1,2 = T1,2 ψ, then straightforward computations show that ψ1,2 satisfy the linear equations 1 |u|2 u 2 ∂x ψ1 = −iλ σ3 + Q1 (u) ψ1 , Q1 (u) = (2.4) 2i −2iux − u|u|2 −|u|2 and 2 ∂x ψ2 = −iλ σ3 + Q2 (u) ψ2 , 1 |u|2 −2iux + u|u|2 Q2 (u) = . −|u|2 2i −u (2.5) Note that Q1,2 (u) ∈ L1 (R) if u ∈ L1 (R) ∩ L3 (R) and ∂x u ∈ L1 (R). The linear equations (2.4) and (2.5) are of the Zakharov–Shabat-type, after we introduce the complex variable z := λ2 . In what follows, we study the Jost functions and the scattering coefficients for the linear equations (2.4) and (2.5). 2.1 Jost functions Let us introduce the normalized Jost functions by considering solutions ψ1,2 of the linear equations (2.4) and (2.5) with z = λ2 in the form m± (x; z) = ψ1 (x; z)eixz , n± (x; z) = ψ2 (x; z)e−ixz , (2.6) according to the asymptotic behavior m± (x; z) → e1 , n± (x; z) → e2 , as x → ±∞, (2.7) where e1 = [1, 0]t and e2 = [0, 1]t . The normalized Jost functions satisfy the following Volterra’s integral equations Z x 1 0 (2.8) m± (x; z) = e1 + 2iz(x−y) Q1 (u(y))m± (y; z)dy ±∞ 0 e and n± (x; z) = e2 + Z x ±∞ e−2iz(x−y) 0 Q (u(y))n± (y; z)dy. 0 1 2 (2.9) The next two lemmas describe properties of the Jost functions, which are analogues to similar properties of the Jost functions in the Zakharov–Shabat spectral problem (see, e.g., Lemma 2.1 in [1]). Lemma 1. Let u ∈ L1 (R) ∩ L3 (R) and ∂x u ∈ L1 (R). For every z ∈ R, there exist unique solutions m± (·; z) ∈ L∞ (R) and n± (·; z) ∈ L∞ (R) satisfying the integral equations (2.8) and (2.9). Moreover, for every x ∈ R, m− (x; ·) and n+ (x; ·) are continued analytically in C+ , whereas m+ (x; ·) and n− (x; ·) are continued analytically in C− . Finally, there exists a positive z-independent constant C such that km∓ (·; z)kL∞ + kn± (·; z)kL∞ ≤ C, 6 z ∈ C± . (2.10) Proof. It suffices to prove the statement for one Jost function, e.g., for m− . The proof for other Jost functions is analogous. Let us define the integral operator K by Z 1 x 1 |u(y)|2 u(y) 0 f (y)dy. (2.11) (Kf )(x; z) := 2i −∞ 0 e2iz(x−y) −2i∂y u(y) − u(y)|u(y)|2 −|u(y)|2 For every z ∈ C+ and every x0 ∈ R, we have # " kuk2L2 (−∞,x0 ) kukL1 (−∞,x0 ) 1 k(Kf )(·; z)kL∞ (−∞,x0 ) ≤ kf (·; z)kL∞ (−∞,x0 ) . 2 2k∂x ukL1 (−∞,x0 ) + kuk3L3 (−∞,x0 ) kuk2L2 (−∞,x0 ) The operator K is a contraction from L∞ (−∞, x0 ) to L∞ (−∞, x0 ) if the two eigenvalues of the matrix # " kuk2L2 (−∞,x0 ) kukL1 (−∞,x0 ) 1 A= 2 2k∂x ukL1 (−∞,x0 ) + kuk3L3 (−∞,x0 ) kuk2L2 (−∞,x0 ) are located inside the unit circle. The two eigenvalues are given by 1q 1 λ± = kuk2L2 (−∞,x0 ) ± kukL1 (−∞,x0 ) (2k∂x ukL1 (−∞,x0 ) + kuk3L3 (−∞,x0 ) ), 2 2 so that 0 < |λ− | < |λ+ |. Hence, the operator K is a contraction if x0 ∈ R is chosen so that 1q 1 kuk2L2 (−∞,x0 ) + kukL1 (−∞,x0 ) (2k∂x ukL1 (−∞,x0 ) + kuk3L3 (−∞,x0 ) ) < 1. 2 2 (2.12) By the Banach Fixed Point Theorem, for this x0 and every z ∈ C+ , there exists a unique solution m− (·; z) ∈ L∞ (−∞, x0 ) of the integral equation (2.8). To extend this result to L∞ (R), we can split R into a finite number of subintervals such that the estimate (2.12) is satisfied in each subinterval. Unique solutions in each subinterval can be glued together to obtain the unique solution m− (·; z) ∈ L∞ (R) for every z ∈ C+ . Analyticity of m− (x; ·) in C+ for every x ∈ R and the bound (2.10) for m− follows from the absolute and uniform convergence of the Neumann series of analytic functions in z. Remark 5. If u is sufficiently small so that the estimate 1 1q 1 kukL1 (2k∂x ukL1 + kuk3L3 ) < kuk2L2 + 2 2 2 (2.13) holds on R, then Banach Fixed Point Theorem yields the existence of the unique solution m− (·; z) ∈ L∞ (R) of the integral equation (2.8) such that km− (·; z) − e1 kL∞ < 1. This is in turn equivalent to the conditions that the linear equation (2.1) has no L2 (R) solutions for every λ ∈ C and no resonances for every λ2 ∈ R in the sense of Definitions 1 and 2. Therefore, the small-norm constraint (2.13) is a sufficient condition that the assumptions of Theorem 1 are satisfied. Lemma 2. Under the conditions of Lemma 1, for every x ∈ R, the Jost functions m± (x; z) and n± (x; z) satisfy the following limits as |Im(z)| → ∞ along a contour in the domains of their analyticity: lim m± (x; z) = m∞ ± (x)e1 , |z|→∞ and 1 2i m∞ ± (x) := e 1 − 2i n∞ ± (x) := e lim n± (x; z) = n∞ ± (x)e2 , |z|→∞ 7 Rx ±∞ Rx ±∞ |u(y)|2 dy |u(y)|2 dy . (2.14) (2.15) If in addition, u ∈ C 1 (R), then for every x ∈ R, the Jost functions m± (x; z) and n± (x; z) satisfy the following limits as |Im(z)| → ∞ along a contour in the domains of their analyticity: (1) (2) lim z m± (x; z) − m∞ ± (x)e1 = q± (x)e1 + q± (x)e2 |z|→∞ and (1) (2) lim z n± (x; z) − n∞ ± (x)e2 = s± (x)e1 + s± (x)e2 , |z|→∞ where (1) q± (x) := (2) q± (x) := (1) s± (x) := (2) s± (x) := (2.16) (2.17) Z x x 2 dy 1 1 1 R±∞ |u(y)| 4 u(y)∂y u ¯(y) + |u(y)| dy, − e 2i 4 2i ±∞ R x 1 2 1 ¯(x)e 2i ±∞ |u(y)| dy , ∂x u 2i Rx 1 2 1 − ∂x u(x)e− 2i ±∞ |u(y)| dy , 2i Z x x 1 1 − 1 R±∞ |u(y)|2 dy 4 2i u ¯(y)∂y u(y) − |u(y)| dy. e 4 2i ±∞ Proof. Again, we prove the statement for the Jost function m− only. The proof for other Jost functions (1) (2) is analogous. Let m− = [m− , m− ]t and rewrite the integral equation (2.8) in the component form: Z h i 1 x (1) (1) (2) m− (x; z) = 1 + u(y) u ¯(y)m− (y; z) + m− (y; z) dy, (2.18) 2i −∞ and (2) m− (x; z) = − 1 2i Z x −∞ h i (1) (2) e2iz(x−y) (2i∂y u ¯(y) + |u(y)|2 u ¯(y))m− (y; z) + |u(y)|2 m− (y; z) dy. (2.19) Recall that for every x ∈ R, m− (x; ·) is analytic in C+ . By bounds (2.10) in Lemma 1, for every u ∈ L1 (R) ∩ L3 (R) and ∂x u ∈ L1 (R), the integrand of the second equation (2.19) is bounded for every z ∈ C+ by an absolutely integrable z-independent function. Also, the integrand converges to zero for every y ∈ (−∞, x) as |z| → ∞ in C+ . By Lebesgue’s Dominated Convergence Theorem, (2) (1) we obtain lim|z|→∞ m− (x; z) = 0, hence m∞ − (x) := lim|z|→∞ m− (x; z) satisfies the inhomogeneous integral equation Z 1 x ∞ |u(y)|2 m∞ (2.20) m− (x) = 1 + − (y)dy, 2i −∞ 1 Rx 2 |u(y)| dy 2i −∞ with the unique solution m∞ . This proves the limit (2.14) for m− . − (x) = e 1 We now add the condition u ∈ C (R) and use the technique behind Watson’s Lemma related to the Laplace method of asymptotic analysis [25]. For every x ∈ R and every small δ > 0, we split integration in the second equation (2.19) for (−∞, x − δ) and (x − δ, x), rewriting it in the equivalent form: Z x Z x−δ (2) e2iz(x−y) dy e2iz(x−y) φ(y; z)dy + φ(x; z) m− (x; z) = x−δ −∞ Z x e2iz(x−y) [φ(y; z) − φ(x; z)] dy ≡ I + II + III, (2.21) + x−δ 8 where i 1 h (1) (2) (2i∂x u ¯(x) + |u(x)|2 u ¯(x))m− (x; z) + |u(x)|2 m− (x; z) . 2i Since φ(·; z) ∈ L1 (R), we have |I| ≤ e−2δIm(z) kφ(·; z)kL1 . φ(x; z) := − Since φ(·; z) ∈ C 0 (R), we have |III| ≤ 1 kφ(x − ·; z) − φ(x; z)kL∞ (x−δ,x) . 2Im(z) On the other hand, we have the exact value II = − i 1 h 1 − e2izδ φ(x; z). 2iz Let us choose δ := [Im(z)]−1/2 such that δ → 0 as Im(z) → ∞. Then, by taking the limit along the contour in C+ such that Im(z) → ∞, we obtain (2) lim zm− (x; z) = − |z|→∞ 1 1 lim φ(x; z) = − (2i∂x u ¯(x) + |u(x)|2 u ¯(x))m∞ − (x), 2i |z|→∞ 4 (2.22) (2) which yields the limit (2.16) for m− . On the other hand, the first equation (2.18) can be rewritten as the differential equation (1) ∂x m− (x; z) = 1 1 (1) (2) |u(x)|2 m− (x; z) + u(x)m− (x; z). 2i 2i Using m ¯∞ − as the integrating factor, (1) ∂x (m∞ − (x)m− (x; z)) = 1 (2) u(x)m∞ − (x)m− (x; z), 2i (1) we obtain another integral equation for m− : (1) m− (x; z) = m∞ − (x) 1 + m∞ (x) 2i − Z x −∞ (2) u(y)m∞ − (y)m− (y; z)dy, Multiplying this equation by z and taking the limit |z| → ∞, we obtain Z x h i 1 ∞ 1 (1) ∞ 4 lim z m− (x; z) − m− (x) = − m− (x) u(y)∂y u ¯(y) + |u(y)| dy, 4 2i |z|→∞ −∞ (2.23) (2.24) (1) which yields the limit (2.16) for m− . We shall now study properties of the Jost functions on the real axis of z. First, we note that following elementary result from the Fourier theory. For notational convenient, we use sometimes kf (z)kL2z instead of kf (·)kL2 . Proposition 1. For every x0 ∈ R− and every w ∈ L2,1 (R), we have Z x √ π 2iz(x−y) p sup kwkL2,1 (−∞,x0 ) . e w(y)dy ≤ 2 1 + x2 x∈(−∞,x0 ) −∞ Moreover, if w ∈ H 1 (R), then Z sup 2iz x∈R Lz (R) x e −∞ 2iz(x−y) w(y)dy + w(x) 9 (2.25) 0 L2z (R) ≤ √ πk∂x wkL2 . (2.26) Proof. Here we give a quick proof of the bounds (2.25) and (2.26) based on Plancherel’s theorem of Fourier analysis. For every x ∈ R and every z ∈ R, we write Z x Z 0 2iz(x−y) f (x; z) := e w(y)dy = e−2izy w(y + x)dy, −∞ −∞ so that kf (x; ·)k2L2 = Z ∞ Z 0 −∞ −∞ Z 0 = π −∞ Z 0 w(y ¯ 1 + x)w(y2 + x)e2i(y1 −y2 )z dy1 dy2 dz −∞ 2 |w(y + x)| dy = π Z x −∞ |w(y)|2 dy. (2.27) If y ≤ x ≤ 0, we have 1 + y 2 ≥ 1 + x2 , so that the above inequality implies Z x π π 2 (1 + y 2 )|w(y)|2 dy ≤ kwk2L2,1 (−∞,x) . kf (x; ·)kL2 ≤ 2 1 + x −∞ 1 + x2 Taking the supremum in x on (−∞, x0 ) for any fixed x0 ∈ R− yields the bound (2.25). To get the bound (2.26), we note that if w ∈ H 1 (R), then w ∈ L∞ (R) and w(x) → 0 as |x| → ∞. As a result, we have Z x e2iz(x−y) ∂y w(y)dy. 2izf (x; z) + w(x) = −∞ The bound (2.26) follows from the computation similar to (2.27). Subtracting the asymptotic limits (2.14) and (2.15) in Lemma 2 from the Jost functions m± and n± in Lemma 1, we prove that for every fixed x ∈ R± , the remainder terms belongs to H 1 (R) with respect to the variable z if u belongs to the space H 1,1 (R) defined in (1.13). Moreover, subtracting also the O(z −1 ) terms as defined by (2.16) and (2.17) and multiplying the result by z, we prove that the remainder term belongs to L2 (R) if u ∈ H 2 (R) ∩ H 1,1 (R). Note that if u ∈ H 1,1 (R), then the conditions of Lemma 1 are satisfied, so that u ∈ L1 (R) ∩ L3 (R) and ∂x u ∈ L1 (R). Also if u ∈ H 2 (R) ∩ H 1,1 (R), then the additional condition u ∈ C 1 (R) of Lemma 2 is also satisfied. Lemma 3. If u ∈ H 1,1 (R), then for every x ∈ R± , we have 1 m± (x; ·) − m∞ ± (x)e1 ∈ H (R), 1 n± (x; ·) − n∞ ± (x)e2 ∈ H (R). (2.28) Moreover, if u ∈ H 2 (R) ∩ H 1,1 (R), then and (1) (2) 2 z m± (x; z) − m∞ ± (x)e1 − (q± (x)e1 + q± (x)e2 ) ∈ Lz (R) (1) (2) 2 z n± (x; z) − n∞ ± (x)e2 − (s± (x)e1 + s± (x)e2 ) ∈ Lz (R). (2.29) (2.30) Proof. Again, we prove the statement for the Jost function m− . The proof for other Jost functions is analogous. We write the integral equation (2.8) for m− in the abstract form m− = e1 + Km− , (2.31) where the operator K is given by (2.11). Although equation (2.31) is convenient for verifying the boundary condition m− (x; z) → e1 as x → −∞, we note that the asymptotic limit as |z| → ∞ is 10 different by the complex exponential factor. Indeed, for every x ∈ R, the asymptotic limit (2.14) is written as m− (x; z) → m∞ − (x)e1 as |z| → ∞, where 1 2i m∞ − (x) := e Rx −∞ |u(y)|2 dy . Therefore, we rewrite equation (2.31) in the equivalent form (I − K)(m− − m∞ − e1 ) = he2 , (2.32) where we have used the integral equation (2.20) that yields e1 − (I − K)m∞ − e1 = he2 with Z x Rx 1 2 e2iz(x−y) w(y)dy, w(x) := −∂x u(x)e 2i −∞ |u(y)| dy . h(x; z) = (2.33) −∞ 2 If u ∈ H 1,1 (R), then w ∈ L2 (R). By the bound (2.25) in Proposition 1, we have h(x; z) ∈ L∞ x (R; Lz (R)) satisfying the following bound for every x0 ∈ R− , √ π 1 3 sup kh(x; z)kL2z (R) ≤ p k∂x ukL2,1 + ku kL2,1 2 1 + x20 x∈(−∞,x0 ) p ≤ C 1 + x20 (kukH 1,1 + kuk3H 1,1 ), (2.34) where C is a positive u-independent constant, whereas the last estimate is due to the Sobolev inequality kukL∞ ≤ √12 kukH 1 . Let us denote the matrix norm of the 2-by-2 matrix Q by kQk := 2 X i,j=1 |Qi,j |. If u ∈ H 1,1 (R), then u ∈ L1 (R) ∩ L3 (R) and ∂x u ∈ L1 (R) so that Q1 (u) ∈ L1 (R), where the matrix Q1 (u) appears in the integral kernel K given by (2.11). The standard method for Volterra’s integral 2 equations yields the bound for every f (x; z) ∈ L∞ x (R; Lz (R)), k(K n f )(x; z)kL∞ 2 ≤ x Lz 1 kQ1 (u)knL1 kf (x; z)kL∞ 2. x Lz n! 2 Therefore, the operator I − K is invertible on the space L∞ x (R; Lz (R)) with a bounded inverse given by ∞ X 1 k(I − K)−1 kL∞ ≤ kQ1 (u)knL1 = ekQ1 (u)kL1 . (2.35) 2 ∞ 2 x Lz →Lx Lz n! n=0 Moreover, the same Neumann series and the same estimate (2.35) can be obtained in the norm 2 L∞ x ((−∞, x0 ); Lz (R)) for every x0 ∈ R. By using (2.32), (2.34), and (2.35), we obtain the following estimate for every x0 ∈ R− : C kQ1 (u)kL1 3 km− (x; z) − m∞ e kuk 1,1 + kuk 1,1 . H − (x)e1 kL2z (R) ≤ p H 1 + x20 x∈(−∞,x0 ) sup (2.36) − 2 Next, we want to show ∂z m− (x; z) ∈ L∞ x ((−∞, x0 ); Lz (R)) for every x0 ∈ R . We differentiate the (1) (2) t integral equation (2.31) in z and introduce the vector v = [v , v ] with the components (1) v (1) (x; z) = ∂z m− (x; z) and (2) (2) v (2) (x; z) = ∂z m− (x; z) − 2ixm− (x; z). 11 Thus, we obtain from (2.31): (I − K)v = h1 e1 + h2 e2 + h3 e2 , where Z h1 (x; z) = x Z−∞ x h2 (x; z) = Z−∞ x h3 (x; z) = −∞ (2.37) (2) yu(y)m− (y; z)dy, (1) ye2iz(x−y) (2iuy (y) + |u(y)|2 u(y))(m− (y; z) − m∞ − (y))dy, ye2iz(x−y) (2iuy (y) + |u(y)|2 u(y))m∞ − (y)dy. For every x0 ∈ R− , each inhomogeneous term of the integral equation (2.37) can be estimated as follows sup x∈(−∞,x0 ) sup x∈(−∞,x0 ) sup x∈(−∞,x0 ) kh1 (x; z)kL2z (R) ≤ kukL2,1 kh2 (x; z)kL2z (R) ≤ kh3 (x; z)kL2z (R) ≤ (2) sup x∈(−∞,x0 ) km− (x; z)kL2z (R) , 2k∂x ukL2,1 + ku3 kL2,1 3 sup x∈(−∞,x0 ) (1) km− (x; z) − m∞ − (x)kL2z (R) , 2k∂x ukL2,1 + ku kL2,1 , where the upper bounds for h1 and h2 are finite due to estimate (2.36). Using the bounds (2.35), (2.36), 2 − and the integral equation (2.37), we conclude that v(x; z) ∈ L∞ x ((−∞, x0 ); Lz (R)) for every x0 ∈ R . (2) 2 Since xm− (x; z) is bounded in L∞ x ((−∞, x0 ); Lz (R)) by the same estimate (2.36), we finally obtain 2 ∞ ∂z m− (x; z) ∈ Lx ((−∞, x0 ); Lz (R)) for every x0 ∈ R− . This completes the proof of (2.28) for m− . To prove (2.29) for m− , we subtract the O(z −1 ) term as defined by (2.16) from the integral equation (2.32) and multiply the result by z. Thus, we obtain i h (1) (2) (1) (2) (2.38) (I − K) z m− − m∞ − e1 − (q− e1 + q− e2 ) = zhe2 − (I − K)(q− e1 + q− e2 ), (1) (2) where the limiting values q− and q− are defined in Lemma 2. Using the integral equation (2.23), we obtain cancelation of the first component of the source term, so that (1) (2) ˜ 2 zhe2 − (I − K)(q− e1 + q− e2 ) = he with ˜ z) = z h(x; Z x e2iz(x−y) w(y)dy + −∞ 1 − 2i Z x −∞ 1 w(x) 2i h i (1) (2) e2iz(x−y) (2i∂y u ¯(y) + u ¯(y)|u(y)|2 )q− (y) + |u(y)|2 q− (y) dy, ˜ z) ∈ where w is the same as in (2.33). By using bound (2.26) in Proposition 1, we have h(x; ∞ 2 1 1,1 2 1,1 Lx (R; Lz (Z)) if w ∈ H (R) in addition to u ∈ H (R), that is, u ∈ H (R) ∩ H (R). Inverting 2 (I − K) on L∞ x (R; Lz (Z)), we finally obtain (2.29) for m− . The following result is deduced from Lemma 3 to show that the mapping ∞ ∞ ± 1 H 1,1 (R) ∋ u → [m± (x; z) − m∞ ± (x)e1 , n± (x; z) − n± (x)] ∈ Lx (R ; Hz (R)) (2.39) is Lipschitz continuous. Moreover, by restricting the potential to H 2 (R) ∩ H 1,1 (R), subtracting O(z −1 ) terms from the Jost functions, and multiplying them by z, we also have Lipschitz continuity of remain2 ders of the Jost functions in function space L∞ x (R; Lz (R)). 12 Corollary 1. Let u, u ˜ ∈ H 1,1 (R) satisfy kukH 1,1 , k˜ ukH 1,1 ≤ U for some U > 0. Denote the corresponding Jost functions by [m± , n± ] and [m ˜ ±, n ˜ ± ] respectively. Then, there is a positive U -dependent constant C(U ) such that for every x ∈ R± , we have km± (x; ·) − m∞ ˜ ± (x; ·) + m ˜∞ ˜kH 1,1 ± (x)e1 − m ± (x)e1 kH 1 ≤ C(U )ku − u (2.40) kn± (x; ·) − n∞ ˜ ± (x; ·) + n ˜∞ ˜kH 1,1 . ± (x)e2 − n ± (x)e2 kH 1 ≤ C(U )ku − u (2.41) ˆ˜ ± (x; ·)kL2 ≤ C(U )ku − u ˆ ˜kH 2 ∩H 1,1 . n± (x; ·) − n km ˆ ± (x; ·) − m ˜ ± (x; ·)kL2 + kˆ (2.42) and Moreover, if u, u ˜ ∈ H 2 (R) ∩ H 1,1 (R) satisfy kukH 2 ∩H 1,1 , k˜ ukH 2 ∩H 1,1 ≤ U , then for every x ∈ R, there is a positive U -dependent constant C(U ) such that where (1) (2) m ˆ ± (x; z) := z m± (x; z) − m∞ ± (x)e1 − (q± (x)e1 + q± (x)e2 ), (1) (2) n ˆ ± (x; z) := z n± (x; z) − n∞ ± (x)e2 − (s± (x)e1 + s± (x)e2 ). Proof. Again, we prove the statement for the Jost function m− . The proof for other Jost functions is analogous. First, let us consider the limiting values of m− and m ˜ − given by 1 2i m∞ − (x) := e Rx −∞ |u(y)|2 dy , 1 2i m ˜∞ − (x) := e Rx −∞ |˜ u(y)|2 dy Then, for every x ∈ R, we have 1 Rx 2i −∞ (|u(y)|2 −|˜u(y)|2 )dy ∞ − 1 e (x) − m ˜ (x)| = |m∞ − − Z x (|u(y)|2 − |˜ u(y)|2 )dy ≤ C1 (U ) −∞ ≤ 2U C1 (U )ku − u ˜kL2 , (2.43) where C1 (U ) is a U -dependent positive constant. Using the integral equation (2.32), we obtain −1 ˜ −1 ˜ (m− − m∞ ˜− −m ˜∞ − e1 ) − ( m − e1 ) = (I − K) he2 − (I − K) he2 ˜ 2 + [(I − K)−1 − (I − K) ˜ 2 ˜ −1 ]he = (I − K)−1 (h − h)e ˜ 2 + (I − K)−1 (K − K)(I ˜ 2, ˜ ˜ −1 he = (I − K)−1 (h − h)e − K) (2.44) ˜ denote the same as K and h but with u being replaced by u ˜ and h where K ˜. To estimate the first term, we write Z x ˜ e2iz(x−y) [w(y) − w(y)] ˜ dy, (2.45) h(x; z) − h(x; z) = −∞ where 1 2 1 2¯ ∞ ¯ u| u ˜ m ˜ − − ∂x u ¯ + |u| u ¯ m∞ w−w ˜ = ∂x u ˜ + |˜ −. 2i 2i ˜kH 1,1 , where C2 (U ) is another U -dependent By using (2.43), we obtain kw − wk ˜ L2,1 ≤ C2 (U )ku − u positive constant. By using (2.45) and Proposition 1, we obtain for every x0 ∈ R− : √ C (U ) π 2 ˜ z)kL2 (R) ≤ p sup kh(x; z) − h(x; ku − u ˜kH 1,1 . (2.46) z 1 + x20 x∈(−∞,x0 ) 13 This gives the estimate for the first term in (2.44). To estimate the second term, we use (2.11) and 2 ∞ 2 observe that K is a Lipschitz continuous operator from L∞ x (R; Lz (R)) to Lx (R; Lz (R)) in the sense 2 that for every f ∈ L∞ x (R; Lz (R)), we have ˜ kL∞ L2 ≤ C3 (U )ku − u k(K − K)f ˜kH 1,1 kf kL∞ 2, x z x Lz (2.47) where C3 (U ) is another U -dependent positive constant that is independent of f . By using (2.34), (2.35), (2.44), (2.46), and (2.47), we obtain for every x0 ∈ R− : C(U ) km− (x; ·) − m∞ ˜ − (x; ·) + m ˜∞ ku − u ˜kH 1,1 , − (x)e1 − m − (x)e1 kL2z (R) ≤ p 1 + x20 x∈(−∞,x0 ) sup which is the first part of the bound (2.40) for m− and m ˜ − . The other part of the bound (2.40) and the bound (2.42) for m− and m ˜ − follow by repeating the same analysis to the integral equations (2.37) and (2.38). 2.2 Scattering coefficients We shall now define the Jost functions of the original linear equation (2.1), which are related to the Jost functions of the systems (2.4) and (2.5) by using the matrix transformations (2.3). To be precise, we define ϕ± (x; λ) = T1−1 (x; λ)m± (x; z), φ± (x; λ) = T2−1 (x; λ)n± (x; z), (2.48) where the inverse matrices are given by 1 2iλ 0 −1 T1 (x; λ) = 2iλ u(x) 1 and T2−1 (x; λ) 1 1 u(x) = . 2iλ 0 2iλ (2.49) The following corollary follows from Lemma 1 and the representations (2.48)–(2.49). Corollary 2. Let u ∈ L1 (R) ∩ L∞ (R) and ∂x u ∈ L1 (R). For every λ2 ∈ R\{0}, there exist unique functions ϕ± (·; λ) ∈ L∞ (R) and φ± (·; λ) ∈ L∞ (R) such that ϕ± (x; λ) → e1 , as x → ±∞. (2.50) φ± (x; λ) → e2 , Proof. To the condition of Lemma 1, we added the condition u ∈ L∞ (R), which ensures the bound on −1 kT1,2 (·; λ)kL∞ for every λ ∈ C\{0}. Then, the assertion of the corollary follows by the representation (2.48)–(2.49) and by the first assertion of Lemma 1. Remark 6. There is no singularity in the definition of Jost functions at the value λ = 0. The linear equation (2.1) with λ = 0 admit unique Jost functions ϕ± (x; 0) = e1 and φ± (x; 0) = e2 , which yield unique definitions for m± (x; 0) and n± (x; 0): −u(x) 1 , , n± (x; 0) = m± (x; 0) = 1 −¯ u(x) which are compatible with the integral equations (2.8) and (2.9) for z = 0. Remark 7. The only purpose in the definition of the original Jost functions (2.48) is to introduce the standard form of the scattering relations, similar to the one used in the literature [17]. After introducing the scattering data for λ2 ∈ R, we will analyze their behavior in the complex z-plane, instead of the complex λ-plane, where z = λ2 . 14 For every x ∈ R and every λ2 ∈ R\{0}, we define the scattering data according to the following transfer matrix # " 2 ϕ− (x; λ) a(λ) b(λ)e2iλ x ϕ+ (x; λ) = . (2.51) 2 φ− (x; λ) φ+ (x; λ) c(λ)e−2iλ x d(λ) By Remark 6, we can also extend the transfer matrix (2.51) to λ = 0 with a(0) = d(0) = 1 and b(0) = c(0) = 0. Since the coefficient matrix in the linear system (2.1) has zero trace, the Wronskian determinant, denoted by W , of two solutions of the linear equation (2.1) for any λ ∈ C is independent of x. As a result, we verify that the scattering coefficients a, b, c, and d are independent of x. Note that the 2 matrix of normalized Jost functions [ϕ± , φ± ] needs to be multiplied by e−iλ xσ3 to yield a solution of the linear equation (2.1), see asymptotic limits (2.2) and (2.50). In this way, we obtain for every x ∈ R and every λ2 ∈ R, 2 2 a(λ) = W (ϕ− e−iλ x , φ+ e+iλ x ) = W (ϕ− , φ+ ), (2.52) b(λ) = W (ϕ+ e (2.53) −iλ2 x , ϕ− e −iλ2 x )=e −2iλ2 x W (ϕ+ , ϕ− ), where we have used the Wronskian relation W (ϕ+ , φ+ ) = 1, which follows from the boundary conditions (2.50) as x → +∞. Now we note the symmetry on solutions of the linear equation (2.1). If ψ is a solution for any λ ∈ C, then σ1 σ3 ψ¯ is also a solution for the same λ ∈ C, where σ1 and σ3 are Pauli matrices in (1.12). As a result, using the boundary conditions for the normalized Jost functions, we obtain the following relations: ψ± (x; λ) = σ1 σ3 ϕ± (x; λ). By applying complex conjugation to the first equation in system (2.51), acting to the resulting equation by σ1 σ3 , and using the relations σ1 σ3 = −σ3 σ1 and σ12 = σ32 = 1, we obtain the second equation in system (2.51) with the correspondence c(λ) = −b(λ), d(λ) = a(λ), λ2 ∈ R. (2.54) From the Wronskian relation W (ϕ− , φ− ) = 1, which can be established from the boundary conditions (2.50) as x → −∞, we verify that the transfer matrix in system (2.51) has the determinant equals to unity. In view of the correspondence (2.54), this yields the result |a|2 (λ) + |b|2 (λ) = 1, λ2 ∈ R. (2.55) We now study properties of the scattering coefficients a and b in suitable function spaces. We prove that a(λ), λb(λ), and λ−1 b(λ) are the H 1 (R) functions with respect to z if u belongs to H 1,1 (R) defined in (1.13). Moreover, we show that λb(λ) and, of course, λ−1 b(λ) are also in L2,1 (R) if u ∈ H 2 (R) ∩ H 1,1 (R). Lemma 4. If u ∈ H 1,1 (R), then the functions a and a ¯ are continued analytically in C+ and C− with respect to z, whereas we have 1 a(λ) − e 2i R R |u|2 dx , λb(λ), λ−1 b(λ) ∈ Hz1 (R). (2.56) Moreover, if u ∈ H 2 (R) ∩ H 1,1 (R), then λb(λ), λ−1 b(λ) ∈ L2,1 z (R). 15 (2.57) Proof. It follows from the integral equations (2.8)–(2.9) and the transformation (2.48) that the original Jost functions ϕ± and φ± satisfy the following Volterra’s integral equations Z x 1 0 (2.58) ϕ± (x; λ) = e1 + λ 2iλ2 (x−y) Q(u(y))ϕ± (y; λ)dy, ±∞ 0 e and Z φ± (x; λ) = e2 + λ x ±∞ 2 e−2iλ (x−y) 0 Q(u(y))φ± (y; λ)dy. 0 1 (2.59) By taking the limit x → +∞, which is justified for every λ2 ∈ R due to Corollary 2 and Remark 6, and using the scattering relation (2.51) and the transformation (2.48)–(2.49), we obtain Z (2) a(λ) = 1 + λ u(x)ϕ− (x; λ)dx ZR h i 1 (1) (2) = 1+ |u(x)|2 m− (x; z) + u(x)m− (x; z) dx (2.60) 2i R and a ¯(λ) = 1 − λ Z 1 = 1− 2i (1) u(x)φ− (x; λ)dx ZR h R i (1) (2) u(x)n− (x; z) + |u(x)|2 n− (x; z) dx. (2.61) By the representations (2.60) and (2.61), as well as the second assertion of Lemma 1, we can see that a is a function of z on R, which is continued analytically in C+ with respect to z. At the same time, a ¯ is continued analytically in C− . By limits (2.14) in Lemma 2, the scattering coefficient a satisfies the following limit as |Im(z)| → ∞ along a contour in C+ : 1 lim a(λ) = a∞ := e 2i R |z|→∞ R |u|2 dx (2.62) . To prove that a(λ) − a∞ is a H 1 (R) function in z, we use the Wronskian representation (2.52) at fixed x = 0. Recall from the transformation (2.48)–(2.49) that (1) (1) ϕ± (x; λ) = m± (x; z) (2) and (2) φ± (x; λ) = n± (x; z). Subtracting the limiting values for a and the normalized Jost functions m± and n± , we rewrite the Wronskian representation (2.52) at x = 0 explicitly (1) (2) ∞ a(λ) − a∞ = (m− (0; z) − m∞ − (0))(n+ (0; z) − n+ (0)) (2) (1) ∞ ∞ ∞ +m∞ − (0)(n+ (0; z) − n+ (0)) + n+ (0)(m− (0; z) − m− (0)) (2) (1) −ϕ− (0; λ)φ+ (0; λ). (2.63) By (2.28) in Lemma 3, all but the last term in (2.63) belong to H 1 (R) with respect to z. Therefore, we only need to look at the last term in (2.63). Although it may be expressed in terms of the Jost functions m− and n+ by using transformations (2.48)–(2.49), this will create a pole at z = 0. Instead, we write explicitly from the integral equation (2.58): Z x Z x (1) 2iz(x−y) −1 (2) ∞ e λ ϕ± (x; λ) = − e2iz(x−y) u(y) m± (y; z) − m∞ u(y)m± (y)dy − (y) dy. (2.64) ± ±∞ ±∞ 16 By using Proposition 1 in the same way as it was used in the proof of Lemma 3, we confirm that (2) λ−1 ϕ± (0; λ) belongs to H 1 (R) with respect to z. On the other hand, the transformation (2.48)–(2.49) shows that h i (1) (1) (2) 2iλφ± (x; λ) = n± (x; z) + u(x) n± (x; z) − n∞ (x) + u(x)n∞ (2.65) ± ± (x), (1) 1 hence 2iλφ± (0; λ) − u(0)n∞ ± (0) also belongs to H (R) with respect to z. Using the representation 1 (2.63) and the Banach algebra property of H (R), we conclude that a(λ) − a∞ ∈ Hz1 (R). Next, we analyze the scattering coefficient b. By using the representation (2.48)–(2.49) and the Wronskian representation (2.53) at fixed x = 0, we write (1) (2) (2) (1) 2iλb(λ) = m+ (0; z)m− (0; z) − m+ (0; z)m− (0; z). (2.66) (1) By (2.28) in Lemma 3 (after the corresponding limiting values are subtracted from m± (0; z)), we establish that λb(λ) ∈ Hz1 (R). On the other hand, the same Wronskian relation can also be written in the form (1) (2) (1) (2) λ−1 b(λ) = m+ (0; z)λ−1 ϕ− (0; λ) − m− (0; z)λ−1 ϕ+ (0; λ). (2.67) (2) Recalling that λ−1 ϕ± (0; λ) belongs to H 1 (R) with respect to z, we obtain λ−1 b(λ) ∈ Hz1 (R). The first assertion (2.56) of the lemma is proved. −1 To prove the last assertion (2.57) of the lemma, we note that λ−1 b(λ) ∈ L2,1 z (R) because zλ b(λ) = 2,1 1 λb(λ) ∈ Hz (R). On the other hand, to show that λb(λ) ∈ Lz (R), we multiply equation (2.66) by z and write the resulting equation in the form (1) (2) (2) (1) (2) (2) 2iλzb(λ) = m+ (0; z) zm− (0; z) − q− (0) − m− (0; z) zm+ (0; z) − q+ (0) (2) (1) (2) (1) ∞ +q− (0) m+ (0; z) − m∞ (0) − q (0) m (0; z) − m (0) , (2.68) + − + − (2) (2) ∞ where we have used the identity q− (0)m∞ + (0) − q+ (0)m− (0) = 0 that follows from limits (2.14) and (2.16). By (2.28) and (2.29) in Lemma 3, all the terms in the representation (2.68) are in L2 (R) with respect to z, hence λb(λ) ∈ Lz2,1 (R). The last assertion (2.57) of the lemma is proved. We show that the mapping H 1,1 (R) ∋ u → a(λ), λb(λ), λ−1 b(λ) ∈ Hz1 (R) (2.69) is Lipschitz continuous. Moreover, we also have Lipschitz continuity of the mapping H 2 (R) ∩ H 1,1 (R) ∋ u → λb(λ), λ−1 b(λ) ∈ L2,1 z (R). (2.70) The corresponding result is deduced from Lemma 4 and Corollary 1. ukH 1,1 ≤ U for some U > 0. Denote the correCorollary 3. Let u, u ˜ ∈ H 1,1 (R) satisfy kukH 1,1 , k˜ sponding scattering coefficients by (a, b) and (˜ a, ˜b) respectively. Then, there is a positive U -dependent constant C(U ) such that ka(λ) − a∞ − a ˜(λ) + a ˜∞ kHz1 + kλb(λ) − λ˜b(λ)kHz1 + kλ−1 b(λ) − λ−1˜b(λ)kHz1 ≤ C(U )ku − u ˜kH 1,1 . (2.71) Moreover, if u, u ˜ ∈ H 2 (R) ∩ H 1,1 (R) satisfy kukH 2 ∩H 1,1 , k˜ ukH 2 ∩H 1,1 ≤ U , then there is a positive U -dependent constant C(U ) such that kλb(λ) − λ˜b(λ)kL2,1 + kλ−1 b(λ) − λ−1˜b(λ)kL2,1 ≤ C(U )ku − u ˜kH 2 ∩H 1,1 . z z (2.72) Proof. The assertion follows from the representations (2.63), (2.64), (2.65), (2.66), (2.67), and (2.68), as well as the Lipschitz continuity of the Jost functions m± and n± established in Corollary 1. 17 2.3 The jump condition Next we define the scattering data and the jump condition for normalized Jost functions. After we introduced the scattering coefficients and the original Jost functions as functions of the spectral parameter λ in the previous subsection, we shall now work with the scattering data and the new Jost functions in terms of the variable z = λ2 only. The jump condition is then formulated on the real axis in the complex z plane. For notational convenience, we replace the previous notation a(λ) by a(z). By Lemma 4, a(z) and a ¯(z) are continued analytically off the real axis for z in C+ and C− . Also a(z) approaches to a finite limit as |z| → ∞. By a standard theorem of complex analysis on zeros of analytic functions, a(z) has at most finite number of zeros for z in C+ . Each zero of a(z0 ) for an eigenvalue z0 with Im(z0 ) > 0 corresponds to the L2 (R) solution of the spectral problem (2.1) for the corresponding λ0 such that z0 = λ20 . According to the assumptions of Theorem 1, no L2 (R) solutions of the spectral problem (2.1) exist, hence a(z) is assumed to have no zeros for z in C+ . On the other hand, a(z) may also have zeros for z on the real axis. Each zero of a(z) on R corresponds to the resonance, according to Definition 2. By the assumptions of Theorem 1, no resonances of the spectral problem (2.1) exist, therefore, a(z) is assumed to have no zeros for z on the real axis. In this case, the scattering relations (2.51) with the symmetry (2.54) can be rewritten in the form: and ϕ− (x; λ) b(λ) 2iλ2 x − ϕ+ (x; λ) = e φ+ (x; λ) a(z) a(z) (2.73) ¯b(λ) φ− (x; λ) 2 − φ+ (x; λ) = − e−2iλ x ϕ+ (x; λ). a ¯(z) a ¯(z) (2.74) Using transformation matrices in (2.48)–(2.49), we can rewrite these scattering relations in terms of the z-dependent normalized Jost functions m± and n± : and where p± (x; z) = m− (x; z) 2iλb(λ) 2izx − m+ (x; z) = e p+ (x; z) a(z) a(z) (2.75) ¯b(λ) p− (x; z) − p+ (x; z) = − e−2izx m+ (x; z). a ¯(z) 2iλ¯ a(z) (2.76) 1 1 1 u(x) T1 (x; λ)T2−1 (x; λ)n± (x; z) = − n (x; z). u(x) −|u(x)|2 − 4z ± 2iλ 4z −¯ (2.77) Properties of the new functions p± are summarized in the following result. Lemma 5. Under the conditions of Lemma 1, for every x ∈ R, the functions p± (x; z) are continued analytically in C± and satisfy the following limits as |Im(z)| → ∞ along a contour in the domains of their analyticity: lim p± (x; z) = n∞ (2.78) ± (x)e2 , where n∞ ± |z|→∞ are the same as in the limits (2.15). Proof. The asymptotic limits (2.78) follow from the representation (2.77) and the asymptotic limits (2.15) for n± (x; z) as |z| → ∞ in Lemma 2. Using the transformation (2.48)–(2.49), functions p± can be written in the equivalent form 1 1 (2) (1) p± (x; z) = n± (x; z)e2 + φ± (x; λ), (2.79) −¯ u (x) 2iλ 18 where the original Jost functions φ± satisfy the Volterra integral equation (2.59) rewritten for the second component as Z x (2) −1 (1) e−2iz(x−y) u(y)n± (y; z)dy. (2.80) λ φ± (x; λ) = ±∞ (2) Since n± (x; z) are well defined at z = 0 by Remark 6, the functions p± in (2.79) are bounded at z = 0, therefore, the representation (2.77) has a removable pole singularity at z = 0. By Lemma 1 and representation (2.77), for every x ∈ R, the analyticity properties of p± (x; z) with respect to z are the same as those of n± (x; z). For every x ∈ R and z ∈ R, we define two matrices P+ (x; z) and P− (x; z) by p− (x; z) m− (x; z) , p+ (x; z) , P− (x; z) = m+ (x; z), . P+ (x; z) = a(z) a(z) (2.81) The scattering relations (2.75) and (2.76) are now rewritten as the jump condition between functions P± (x; z) across the real axis in z for every x ∈ R: r (z)r− (z) r+ (z)e−2izx P+ (x; z) − P− (x; z) = P− (x; z)R(x; z), R(x; z) := + z ∈ R, (2.82) r− (z)e2izx 0 where we have introduced the scattering data: r¯+ (z) = ¯b(λ) , 2iλ¯ a(z) r− (z) = 2iλb(λ) , a(z) z ∈ R. (2.83) The scattering data satisfy the following properties, which are derived from the previous results. Lemma 6. Assume that a does not vanish on R. If u ∈ H 1,1 (R), then r± ∈ H 1 (R) satisfy the relation r− (z) = 4|z|r+ (z), z ∈ R. (2.84) Moreover, if u ∈ H 2 (R) ∩ H 1,1 (R), then r± ∈ L2,1 (R). The mapping H 2 (R) ∩ H 1,1 (R) ∋ u → (r+ , r− ) ∈ H 1 (R) ∩ L2,1 (R) (2.85) is Lipschitz continuous. Proof. The first assertion follows from Lemma 4, whereas the relation (2.84) is derived from the explicit definition (2.83). To prove Lipschitz continuity of the mapping (2.85), we use the following representation for r− and r˜− that correspond to two potentials u and u ˜, r− − re− = 2iλ(b − eb) 2iλeb 2iλeb + [(e a−e a∞ ) − (a − a∞ )] + (e a∞ − a∞ ). a ae a ae a (2.86) Lipschitz continuity of the mapping (2.85) for r− follows from Corollary 3 and the representation (2.86). Lipschitz continuity of the mapping (2.85) for r+ is studied by using a similar representation. By Lemmas 1, 2, 4, and 5, as well as the assumption of no zeros of a in C+ and a ¯ in C− , the ± matrices P± (x; ·) are continued analytically in C for every x ∈ R and satisfy the limiting behavior h 1 Rx i Rx 1 2 2 P∞ (x) := lim P± (x; z) = e 2i +∞ |u(y)| dy e1 , e− 2i +∞ |u(y)| dy e2 . (2.87) |z|→∞ 19 The boundary conditions (2.87) depend on x. This fact represents an obstacle in the inverse scattering transform, where we are supposed to reconstruct the potential u from the behavior of the analytic continuations of the Jost functions in matrices P± . Therefore, we fix the boundary conditions to the identity matrix by defining new matrices M± (x; z) := [P∞ (x)]−1 P± (x; z), x ∈ R, z ∈ C± . As a result, we obtain the Riemann–Hilbert problem for analytic functions M± (x; ·) in C± , M+ (x; z) − M− (x; z) = M− (x; z)R(x; z), z ∈ R, M± (x; z) → I as |z| → ∞, (2.88) (2.89) where the scattering data r± ∈ H 1 (R) ∩ L2,1 (R) are given in Lemma 6. This Riemann–Hilbert problem is the basis for the inverse scattering transform, considered in Section 3. 3 Inverse scattering transform We are now concerned with the solvability of the Riemann-Hilbert problem (2.89) for the given scattering data r+ , r− ∈ H 1 (R) ∩ L2,1 (R) satisfying the constraint (2.84). Under assumptions of no poles of M± (x; ·) in C± , we are looking for analytic matrix functions M± (x; ·) in C± for every x ∈ R. Once the solution M± (x; ·) of the Riemann–Hilbert problem (2.89) is constructed, we shall identify the reconstruction formulas for the potential u of the linear equation (2.1) and study Lipschitz continuity of the potential u defined in H 2 (R) ∩ H 1,1 (R) in terms of the scattering data r+ and r− defined in H 1 (R) ∩ L2,1 (R). With the time evolution of the scattering data r+ (t, ·) and r− (t, ·) to be established in Section 4, the inverse scattering transform generates a unique global solution u(t, ·) of the DNLS equation (1.1) in function space H 2 (R) ∩ H 1,1 (R) for every t ∈ R. This construction yields the proof of Theorem 1. 3.1 Solution to the Riemann–Hilbert problem Let us start with the definition of the Cauchy operator, which can be found in many sources, e.g., in [11]. For any function h ∈ Lp (R) with 1 ≤ p < ∞, the Cauchy operator acting on h defined by Z h(s) 1 ds, z ∈ C \ R (3.1) (Ch) (z) := 2πi R s − z yields an analytic function of z off the real line such that (Ch) (· + iy) is in Lp (R) for each y 6= 0. When z approaches to a point on the real line transversely from the upper and lower half planes, that is, if y → ±0, the Cauchy operator C becomes the Plemelj projection operators, denoted respectively by P ± . These projection operators are given explicitly by Z h(s) 1 ± P h (z) := lim ds, z ∈ R. (3.2) ǫ↓0 2πi R s − (z ± ǫi) The following proposition summarizes the basic properties of the Cauchy and projection operators. Proposition 2. For every h ∈ Lp (R), 1 ≤ p < ∞, the Cauchy operator Ch is analytic off the real line, decays to zero as |z| → ∞, and approaches to P ± h almost everywhere, when a point z ∈ C± approaches to a point on the real axis by any non-tangential contour from C± . If 1 < p < ∞, then there exists a positive constant Cp such that kP ± hkLp ≤ Cp khkLp , (3.3) 20 where C2 = 1 and Cp → +∞ as p → 1 and p → ∞. If h ∈ L1 (R), then the Cauchy operator admits the following asymptotic limit in either C+ or C− : Z 1 lim z (Ch) (z) = − h(s)ds. (3.4) 2πi R |z|→∞ Proof. Analyticity, decay, and boundary values of C on the real axis follow from Theorem 11.2 and Corollary 2 on pp. 190–191 in [13]. By Sokhotski–Plemelj theorem, we have the relations 1 1 (P ± h)(z) = ± h(z) − (Hh) (z), 2 2 where H is the Hilbert transform given by 1 lim (Hh) (z) := πi ǫ↓0 Z z−ǫ + −∞ Z ∞ z+ǫ z ∈ R, h(s) ds, z−s (3.5) z ∈ R. By Riesz’s theorem (Theorem 3.2 in [12]), H is a bounded operator from Lp (R) to Lp (R) for every 1 < p < ∞, so that the bound (3.3) holds. Finally, the asymptotic limit (3.4) is justified by Lebesgue’s dominated convergence theorem if h ∈ L1 (R). Consider the Riemann-Hilbert problem (2.89) for the given scattering data r+ , r− ∈ H 1 (R)∩L2,1 (R) satisfying the constraint (2.84). Under assumptions of no poles of M± (x; ·) in C± , for every x ∈ R, we reduce the Riemann–Hilbert problem in a complex plane to the Fredholm integral equation on the real line. To do so, we recall that R(x; ·) ∈ L1 (R) ∩ L∞ (R) if r+ , r− ∈ H 1 (R) ∩ L2,1 (R). Therefore, we consider the class of solutions such that M± (x; ·) are analytic in C± and M± (x; · + iy) − I ∈ L2 (R) for every ±y > 0, so that Proposition 2 applies with p = 2. Then, for every x ∈ R, the solution to the Riemann-Hilbert problem (2.89) with the boundary condition (2.87) is given by the implicit expression M± (x; z) = I + (CM− (x; ·)R(x; ·)) (z), z ∈ C± . (3.6) By Proposition 2 with p = 2, the solution of the Riemann–Hilbert problem exists in the form (3.6) if and only if there is a solution of the Fredholm integral equation on the real line: M− (x; z) = I + P − M− (x; ·)R(x; ·) (z), z ∈ R, (3.7) in the class of functions M− (x; ·) − I ∈ L2 (R). Once M− (x; ·) on the real line is found from the Fredholm integral equation (3.7), M+ (x; ·) on the real line can also be found by the projection formula M+ (x; z) = I + P + M− (x; ·)R(x; ·) (z), z ∈ R. (3.8) The following lemma ensures solvability of the integral equations (3.7) and (3.8). Lemma 7. Let r+ , r− ∈ L2 (R) ∩ L∞ (R). Then, for every x ∈ R, there exists a unique solution of the integral equations (3.7) and (3.8) such that M± (x; ·) − I belongs to L2 (R) and kM± (x; ·) − IkL2 ≤ (1 + kr+ kL∞ + kr− kL∞ ) (kr+ kL2 + kr− kL2 ) . (3.9) Proof. Since the integral equations (3.7) and (3.8) are linear, writing g± := M± − I yields the proper version of the inhomogeneous Fredholm’s integral equations g− (z) − (P − g− R) (z) = (P − R) (z), z ∈ R. (3.10) g+ (z) − (P + g− R) (z) = (P + R) (z), 21 For notational convenience, the fixed value of x ∈ R is not written in the system (3.10). By using the Sokhotski–Plemelj formulas (3.5), the system (3.10) restores the inhomogeneous jump condition in the form g+ (z) − g− (z)(I + R(z)) = R(z), z ∈ R. (3.11) If r+ , r− ∈ L2 (R) ∩ L∞ (R), then P ± R ∈ L2 (R) by Proposition 2 and solutions of the integral equations (3.10) are well defined for g+ , g− ∈ L2 (R). We now note the factorization I + R = (I + R+ )(I + R− ), (3.12) where 0 r¯+ (z)e−2izx R+ (x; z) = , 0 0 0 0 R− (x; z) = , r− (z)e2izx 0 (3.13) The matrices I + R± are obviously invertible with the inverse (I + R± )−1 = I − R± . Using the transformation h± = g± (I ± R∓ )−1 and the factorization (3.12), we rewrite the jump condition (3.11) in the form h+ (z) − h− (z) = R(z)(I + R− (z))−1 , z ∈ R, (3.14) where R(I + R− ) −1 0 r¯+ (z)e−2izx = . r− (z)e2izx 0 The inhomogeneous Riemann–Hilbert problem with the jump condition (3.14) has a unique solution in L2 (R) such that h± = P ± R(I + R− )−1 . By the bound (3.3) with C2 = 1 for p = 2 in Proposition 2, we have bounds kh± kL2 ≤ kR(I + R− )−1 kL2 ≤ kr+ kL2 + kr− kL2 . (3.15) Invoking the inverse transformation g± = h± (I ± R∓ ) and using (3.13) and (3.15), we arrive to the bound (3.9). The assertion of the lemma is proved. Modifying the same method, we also obtain a useful bound on the inverse operator for an inhomogeneous Fredholm equation defined by the jump matrix R. Corollary 4. Under the conditions of Lemma 7, there is a unique solution g ∈ L2 (R) of the inhomogeneous equation g − P − gR = f, (3.16) where f ∈ L2 (R) is a given matrix-valued function. Moreover, there is a positive constant C(kr+ kL∞ , kr− kL∞ ) (that depends on kr+ kL∞ , kr− kL∞ ) such that kgkL2 ≤ C(kr+ kL∞ , kr− kL∞ )kf kL2 . (3.17) Proof. By writing f = P + f − P − f from relations (3.5), we also split g = h+ − g− , where h+ and g− satisfy the inhomogeneous equations g− − P − g− R = P − f, h+ − P − h+ R = P + f. (3.18) Next, we obtain existence, uniqueness, and bounds on the components g− and h+ in the system (3.18). To deal with g− , we define the auxiliary equation to the first equation in (3.18) g+ − P + g− R = P + f, 22 such that g+ − g− (I + R) = f. Repeating the proof of Lemma 7, we obtain the existence and uniqueness of solutions for g± ∈ L2 (R) satisfying the bound kg− kL2 ≤ kI − R+ kL∞ kI − R− kL∞ kf kL2 . To deal with h+ , we use equation in (3.18) as follows: P+ − P− (3.19) = I following from the relations (3.5) and rewrite the second h+ (I + R) − P + h+ R = P + f. Let us define the auxiliary equation by h− (I + R) − P − h+ R = P − f. We note that I + R is invertible matrix, because the two eigenvalues of this matrix are strictly positive: !2 r r 4 2 c c 1 1 2 1+ = ± c > 0, (3.20) λ± = 1 + c ± c 2 + 2 4 4 2 2 where c2 = r+ r− = ab ≥ 0 is bounded if r+ , r− ∈ L∞ (R). Moreover, the eigenvalues λ± satisfy the bounds 1 (1 + c)2 ≥ λ+ ≥ λ− ≥ . (3.21) (1 + c)2 Denoting w± := h± (I + R), we obtain the jump problem w+ (I − (I + R)−1 R) − w− = f ⇒ w+ (I + R)−1 − w− = f. Using the factorization (3.12) and repeating the proof of Lemma 7, we obtain the existence and uniqueness of solutions for w± ∈ L2 (R) satisfying the bound kh+ kL2 ≤ k(I + R)−1 kL∞ kw+ kL2 ≤ k(I + R)−1 kL∞ kI + R+ kL∞ kI + R− kL∞ kf kL2 . (3.22) Combining the two estimates (3.19) and (3.22), using the bounds (3.21) and the triangle inequality, we obtain (3.17). 3.2 Reconstruction formulas We shall now recover the potential u of the linear equation (2.1) from the solution of the Riemann– Hilbert problem (2.89). This will give us the map H 1 (R) ∩ L2,1 (R) ∋ (r− , r+ ) 7→ u ∈ H 2 (R) ∩ H 1,1 (R), (3.23) where r− and r+ are related by (2.84). Let us recall the connection formulas between the potential u and the Jost functions of the direct scattering transform in Section 2. By Lemma 2, if u ∈ H 2 (R) ∩ H 1,1 (R), then Rx 1 2 (2) ¯(x)e 2i ±∞ |u(y)| dy = 2i lim zm± (x; z). ∂x u (3.24) |z|→∞ On the other hand, by Lemma 2 and the representation (2.77), if u ∈ H 1,1 (R), then 1 u(x)e− 2i Rx ±∞ |u(y)|2 dy (1) = −4 lim zp± (x; z). |z|→∞ (3.25) Next, we study properties of the potential u recovered by equations (3.24) and (3.25) from properties of the scattering coefficients r+ and r− in the solution (3.6) of the Riemann–Hilbert problem (2.89). The two choices in the reconstruction formulas (3.24) and (3.25) are useful for controlling the potential u on the positive and negative line. We shall proceed separately with the estimates on the two half-lines. 23 3.2.1 Estimates on the positive half-line We shall introduce the following notations for the column vectors of the matrices M± as M± (x; z) = [µ± (x; z), η± (x; z)]. (3.26) By comparing (2.81) and (2.87) with (3.26), we rewrite the reconstruction formulas (3.24) and (3.25) for the upper choice of m+ and p+ as follows: Rx Rx 1 1 2 2 (2) (3.27) ¯(x)e 2i +∞ |u(y)| dy = 2ie− 2i +∞ |u(y)| dy lim zµ− (x; z) ∂x u |z|→∞ and 1 u(x)e− 2i Rx +∞ |u(y)|2 dy 1 = −4e 2i Rx +∞ |u(y)|2 dy (1) lim zη+ (x; z) |z|→∞ (3.28) The asymptotic limit (3.4) in Proposition 2 for the solution (3.6) is justified since r± ∈ H 1 (R)∩L2,1 (R), so that R(x; ·) ∈ L1 (R) ∩ L∞ (x; ·) for every x ∈ R. As a result, the reconstruction formulas (3.27) and (3.28) can be written in the explicit form Z h i Rx Rx 1 1 2 2 1 (2) (2) ¯(x)e 2i +∞ |u(y)| dy = − r− (z)e2izx η− (x; z) + r¯+ (z)e−2izx µ− (x; z) dz e 2i +∞ |u(y)| dy ∂x u π R Z 1 (2) r− (z)e2izx η+ (x; z)dz (3.29) = − π R and u(x)e i Rx +∞ |u(y)|2 dy 2 = πi Z (1) R r¯+ (z)e−2izx µ− (x; z)dz. (3.30) where we have used the Riemann–Hilbert problem (2.89) for the second equality in (3.29). Using the integral equations (3.7) and (3.8) with the definitions (3.26), we obtain the system of integral equations for vectors µ− and η+ , which are used in the reconstruction formulas (3.29) and (3.30). The system of integral equations takes the form µ− (x; z) = e1 + P − r− e2izx η+ (x; ·) (z), (3.31) + −2izx η+ (x; z) = e2 + P r¯+ e µ− (x; ·) (z). (3.32) Denoting M = [µ− − e1 , η + − e2 ] , we write the system (3.31) and (3.32) in the matrix form M − P + M R+ − P − M R− = F, where R± are given by (3.13) and F = e2 P − r− e2izx , e1 P + r¯+ e−2izx , (3.33) (3.34) The following result controls the inhomogeneous term F given by (3.34) by using the Fourier theory. Proposition 3. There is a constant C > 0 such that for every x0 ∈ R+ and every r± ∈ H 1 (R), we have C sup P + r¯+ e−2izx L2 (R) ≤ p (3.35) kr+ kH 1 z 1 + x20 x∈(x0 ,∞) 24 and sup x∈(x0 ,∞) Moreover, if r± ∈ L2,1 (R), then − P r− e2izx L2z (R) r+ e−2izx L2 (R) ≤ Ckzr+ kL2z , sup P + z¯ z x∈R and C kr− kH 1 . ≤p 1 + x20 sup P − zr− e2izx L2 (R) ≤ Ckzr− kL2z . (3.37) (3.38) z x∈R (3.36) Proof. Recall the following elementary result from the Fourier theory. For function r ∈ L2 (R), R a given 2 ikz we use the Fourier transform rb ∈ L (R) with the definition rb(k) := R r(z)e dz. Then, we have r ∈ H 1 (R) if and only if rb ∈ L2,1 (R). Similarly, r ∈ L2,1 (R) if and only if rb ∈ H 1 (R). In order to prove (3.35), we write explicitly Z r+ (s)e−2isx 1 + −2izx lim ds P r¯+ e (z) = 2πi ǫ↓0 R s − (z + iǫ) ! Z Z i(k−2x)s e 1 rc = ds dk + (k) lim ǫ↓0 R s − (z + iǫ) 2πi R Z ∞ i(k−2x)z = rc dk, (3.39) + (k)e 2x where the following residue computation has been used: −ǫ(k−2x) Z is(k−2x) e 1 e , ds = lim lim 0, ǫ↓0 ǫ↓0 2πi R s − iǫ if k − 2x > 0 = χ(k − 2x), if k − 2x < 0 (3.40) with χ being the characteristic function. By the bound (2.25) in Proposition 1, we have for every x0 ∈ R+ , Z ∞ √ 2 π ikz ≤p rc dk kb r+ kL2,1 . sup + (k)e 2 1 + x2 x∈(x0 ,∞) 2x Lz (R) 0 Using this estimate, the equivalency of kb r+ kL2,1 and kr+ kH 1 , and the representation (3.39), we obtain the bound (3.35). The proof of the bound (3.36) follows from a similar representation Z −2x Z 1 r− (s)e2isx − 2izx i(k+2x)z P r− e (z) = rc dk lim ds = − − (k)e 2πi ǫ↓0 R s − (z − iǫ) −∞ and the bound (2.25) in Proposition 1. In order to obtain the bound (3.37), we write + −2i P z¯ r+ e −2izx Z ∞ i(k−2x)z (z) = ∂x (z) = ∂x P r¯+ e rc dk + (k)e 2x Z ∞ i(k−2x)z = −2iz rc dk − 2rc + (k)e + (2x) 2x Z ∞ ′ i(k−2x)z dk. = 2 rc + (k)e + −2izx 2x By the representation (2.27) in Proposition 1 and the equivalence of kc r+ ′ kL2 and kzr+ kL2 , we obtain the bound (3.37). The bound (3.38) is proved similarly. 25 We now study solutions of the system of integral equations (3.31) and (3.32) from Proposition 3 and Corollary 4. Lemma 8. For every x0 ∈ R+ and every r± ∈ H 1 (R), the unique solution of the system of integral equations (3.31) and (3.32) satisfies the estimates sup x∈(x0 ,∞) kµ− (x; z) − e1 kL2z (R) ≤ and sup x∈(x0 ,∞) kη+ (x; z) − e2 kL2z (R) ≤ C(kr+ kL∞ , kr− kL∞ ) p (kr+ kH 1 + kr− kH 1 ) 1 + x20 C(kr+ kL∞ , kr− kL∞ ) p (kr+ kH 1 + kr− kH 1 ) , 1 + x20 (3.41) (3.42) where C(kr+ kL∞ , kr− kL∞ ) is a positive constant that depends on kr+ kL∞ and kr− kL∞ . Moreover, if r± ∈ H 1 (R) ∩ L2,1 (R), then sup k∂x µ− (x; z)kL2z (R) ≤ C(kr+ kL∞ , kr− kL∞ ) (kr+ kH 1 ∩L2,1 + kr− kH 1 ∩L2,1 ) (3.43) sup k∂x η+ (x; z)kL2z (R) ≤ C(kr+ kL∞ , kr− kL∞ ) (kr+ kH 1 ∩L2,1 + kr− kH 1 ∩L2,1 ) (3.44) x∈R and x∈R where C(kr+ kL∞ , kr− kL∞ ) is another positive constant that depends on kr+ kL∞ and kr− kL∞ . Proof. Using the identity P + − P − = I following from relations (3.5) and the identity R+ + R− = (I − R+ )R, following from the factorization (3.12), we rewrite the inhomogeneous equation (3.33) in the matrix form G − P − GR = F, G := M (I − R+ ). (3.45) By Corollary 4, for a given F ∈ L2 (R), we obtain a unique G ∈ L2 (R) satisfying the bound (3.17). Since M = G(I −R+ )−1 = G(I +R+ ), a similar bound also holds for kM kL2 . By substituting estimates (3.35) and (3.36) of Proposition 3 to the bound for kM kL2 , we obtain bounds (3.41) and (3.42). In order to obtain bounds (3.43) and (3.44), we take derivative of the inhomogeneous equation (3.33) in x and obtain ∂x M − P + (∂x M ) R+ − P − (∂x M ) R− = F˜ , (3.46) where F˜ = ∂x F + P + M ∂x R+ + P − M ∂x R− . Note that derivatives in x for every term of F˜ reduce to multiplication of r± by z. By using estimates (3.37) and (3.38) of Proposition 3 and the above estimates for M ∈ L2 (R), we have F˜ ∈ L2 (R), hence the bounds (3.43) and (3.44) follow from the bound (3.17) of Corollary 4. We will use now the result of Proposition 3 and Lemma 8 to show that if r+ , r− ∈ H 1 (R), then the reconstruction formulas (3.29) and (3.30) recover u in class H 1,1 (R+ ), whereas if r+ , r− ∈ H 1 (R) ∩ L2,1 (R), then u ∈ H 2 (R+ ) ∩ H 1,1 (R+ ). Lemma 9. Assume that r± ∈ H 1 (R) satisfies (2.84) and M± are obtained from the solution of the Riemann–Hilbert problem (2.89). Then, u ∈ H 1,1 (R+ ) satisfies the bound kukH 1,1 (R+ ) ≤ C(kr+ kH 1 , kr− kH 1 ) (kr+ kH 1 + kr− kH 1 ) , 26 (3.47) where C(kr+ kH 1 , kr− kH 1 ) is a positive constant that depends on kr+ kH 1 and kr− kH 1 . Moreover, if r± ∈ H 1 (R) ∩ L2,1 (R), then u ∈ H 2 (R+ ) satisfies the bound kukH 2 (R+ ) ≤ C(kr+ kH 1 ∩L2,1 , kr− kH 1 ∩L2,1 ) (kr+ kH 1 ∩L2,1 + kr− kH 1 ∩L2,1 ) , (3.48) where C(kr+ kH 1 ∩L2,1 , kr− kH 1 ∩L2,1 ) is another positive constant that depends on kr+ kH 1 ∩L2,1 and kr− kH 1 ∩L2,1 . Proof. We use the reconstruction formula (3.30) rewritten as follows: Z Rx 2 2 r¯+ (z)e−2izx dz u(x)ei +∞ |u(y)| dy = πi R Z h i 2 (1) + r¯+ (z)e−2izx µ− (x; z) − 1 dz. πi R (3.49) 2,1 The first term is controlled in L2,1 (R) because r+ is in H 1 (R) and its Fourier transform rc + is in L (R). To control the second term in L2,1 (R+ ), we denote Z ∞ h i (1) r¯+ (z)e−2izx µ− (x; z) − 1 dz, I(x) := −∞ use the inhomogeneous equation (3.31), and integrate by parts to obtain Z (1) I(x) = − r− (z)η+ (x; z)e2izx P + r+ e−2izx (z)dz. R By bound (3.35) in Proposition 3, bound (3.42) in Lemma 8, and the Cauchy–Schwarz inequality, we have for every x0 ∈ R+ , (1) sup |I(x)| ≤ kr− kL∞ sup kη+ (x; z)kL2z (R) sup P + r+ e−2izx L2 (R) x∈(x0 ,∞) x∈(x0 ,∞) ≤ x∈(x0 ,∞) z C(kr+ kL∞ , kr− kL∞ ) kr+ kH 1 kr− kL∞ (kr+ kH 1 + kr− kH 1 ) . 1 + x20 By using the triangle inequality and combining the estimates for the two terms in (3.49), we obtain the bound kukL2,1 (R+ ) ≤ C(kr+ kL∞ , kr− kL∞ ) (1 + kr− kL∞ (kr+ kH 1 + kr− kH 1 )) kr+ kH 1 , (3.50) where C(kr+ kL∞ , kr− kL∞ ) is a positive constant that depends on kr+ kL∞ and kr− kL∞ . On the other hand, the reconstruction formula (3.29) can be rewritten in the form Z Rx Rx 1 1 1 |u(y)|2 dy |u(y)|2 dy +∞ +∞ 2i 2i r− (z)e2izx dz e ¯(x)e ∂x u = − π R Z h i 1 (2) − r− (z)e2izx η+ (x; z) − 1 dz. π R By using the same analysis as above, we obtain the bound 1 Rx 2 ¯e 2i +∞ |u(y)| dy 2,1 + ≤ C(kr+ kL∞ , kr− kL∞ ) (1 + kr+ kL∞ (kr+ kH 1 + kr− kH 1 )) kr− kH 1 (. 3.51) ∂ x u L (R ) 1 Combining bounds (3.50) and (3.51), we set v(x) := u(x)e− 2i kvkH 1,1 (R+ ) Rx |u(y)|2 dy and obtain ≤ C(kr+ kH 1 , kr− kH 1 ) kr+ kH 1 (R) + kr− kH 1 (R) , 27 +∞ (3.52) where C(kr+ kH 1 , kr− kH 1 ) is a new positive constant that depends on kr+ kH 1 and kr− kH 1 . Since |v(x)| = |u(x)| and H 1 (R) is embedded into L6 (R), the estimate (3.52) implies the bound (3.47), after the constant C(kr+ kH 1 , kr− kH 1 ) is adjusted accordingly. In order to obtain the bound (3.48), we differentiate I in x: Z ∞ Z ∞ h i (1) (1) z¯ r+ (z)e−2izx µ− (x; z) − 1 dz + I ′ (x) = −2i r¯+ (z)e−2izx ∂x µ− (x; z)dz, −∞ −∞ By bound (3.37) in Proposition 3, bounds (3.42) and (3.44) in Lemma 8, and the Cauchy–Schwarz inequality, we have for every x0 ∈ R+ , sup x∈(x0 ,∞) |I ′ (x)| ≤ 2kr− kL∞ sup x∈(x0 ,∞) +2kzr− kL2 +kr− kL∞ ≤ (1) kη+ (x; z)kL2z (R) x∈(x0 ,∞) x∈(x0 ,∞) x∈(x0 ,∞) (1) sup sup sup kη+ (x; z)kL∞ z (R) (1) k∂x η+ (x; z)kL2z (R) + P zr+ e−2izx sup x∈(x0 ,∞) sup x∈(x0 ,∞) L2z (R) + P r+ e−2izx L2z (R) + P r+ e−2izx L2z (R) C(kr+ kL∞ , kr− kL∞ ) p kr+ kH 1 ∩L2,1 kr− kH 1 ∩L2,1 (kr+ kH 1 ∩L2,1 + kr− kH 1 ∩L2,1 ) . 1 + x20 This bound is sufficient to control I ′ in L2 (R+ ) norm. Continuing the same analysis for reconstruction formula (3.29), we obtain the bound (3.48). By Lemma 9, we obtain the existence of the mapping H 1 (R) ∩ L2,1 (R) ∋ (r− , r+ ) 7→ u ∈ H 2 (R+ ) ∩ H 1,1 (R+ ). (3.53) We now show that this map is Lipschitz. Corollary 5. Let r± , r˜± ∈ H 1 (R) ∩ L2,1 (R) satisfy kr± kH 1 ∩L2,1 , k˜ r± kH 1 ∩L2,1 ≤ ρ for some ρ > 0. Denote the corresponding potentials by u and u ˜ respectively. Then, there is a positive ρ-dependent constant C(ρ) such that ku − u ˜kH 2 (R+ )∩H 1,1 (R+ ) ≤ C(ρ) (kr+ − r˜+ kH 1 ∩L2,1 + kr− − r˜− kH 1 ∩L2,1 ) . (3.54) Proof. By the estimates in Lemma 9, if r± ∈ H 1 (R) ∩ L2,1 (R), then the quantities R Rx x 2 2 i v(x) := u(x)ei +∞ |u(y)| dy and w(x) := ∂x u(x) + |u(x)|2 u(x) ei +∞ |u(y)| dy , 2 are defined in function space H 1 (R+ ) ∩ L2,1 (R+ ). Lipschitz continuity of the corresponding mappings follows from the reconstruction formula (3.29) and (3.30) by repeating the same estimates in Lemma 9. Since |v| = |u|, we can write Rx Rx Rx 2 2 2 u−u ˜ = (v − v˜)e−i +∞ |v(y)| dy + v˜ e−i +∞ |v(y)| dy − e−i +∞ |˜v(y)| dy . Therefore, Lipschitz continuity of the mapping (r+ , r− ) 7→ v ∈ H 1 (R+ ) ∩ L2,1 (R+ ) is translated to Lipschitz continuity of the mapping (r+ , r− ) 7→ u ∈ H 1 (R+ ) ∩ L2,1 (R+ ). Using a similar representation for ∂x u in terms of v and w, we obtain Lipschitz continuity of the mapping (3.53) with the bound (3.54). 28 3.2.2 Estimates on the negative half-line By using (2.81), (2.87), and (3.26), we rewrite the reconstruction formulas (3.24) and (3.25) for the lower choice of m− and p− as follows: Rx Rx 1 1 2 2 (2) (3.55) ¯(x)e 2i −∞ |u(y)| dy = 2ie− 2i +∞ |u(y)| dy a∞ lim zµ+ (x; z) ∂x u |z|→∞ and 1 u(x)e− 2i Rx −∞ 1 R |u(y)|2 dy 1 = −4e 2i Rx +∞ |u(y)|2 dy (1) a ¯∞ lim zη− (x; z), |z|→∞ (3.56) 2 where a∞ := lim|z|→∞ a(z) = e 2i R |u(y)| dy . If we now use the same solution representation (3.6) in the reconstruction formulas (3.55) and (3.56), we obtain the same explicit expressions (3.29) and (3.30). On the other hand, if we rewrite the Riemann–Hilbert problem (2.89) in an equivalent form, we will be able to find nontrivial representation formulas for u, which are useful on the negative half-line. We note that ˜ z))F −1 (z), I + R(x; z) = F− (z)(I + R(x; + where 1 + r¯+ (z)r− (z) 0 F− (z) = , 0 1 and ˜ z) = R(x; 1 0 F+ (z) = , 0 1 + r¯+ (z)r− (z) 0 r¯+ (z)e−2ixz . r− (z)e2ixz r¯+ (z)r− (z) ˜ ± (x; z) = M± (x; z)F± (z) for every x ∈ R and z ∈ R, we transform the Riemann–Hilbert Defining M problem (2.89) in the equivalent form: ˜ + (x; z) − M ˜ − (x; z) = M ˜ − (x; z)R(x; ˜ z), z ∈ R, M (3.57) ˜ lim|z|→∞ M± (x; z) = I, ˜ ± (x; ·) remain the same because r± (z) → 0 as where we note that the boundary conditions for M |z| → ∞ on the real line. We again solve the modified Riemann–Hilbert problem (3.57) and obtain ˜ + (x; ·) and M ˜ − (x; ·) extended to C+ and C− . Therefore, we can rewrite the analytic functions M expression (3.6) in the equivalent form ˜ ± (x; z) = I + C M ˜ − (x; ·)R(x; ˜ ·) (z), z ∈ C± . M (3.58) ˜ ± that the column vectors µ+ in M+ and η− in M− remain What is nice in the construction of M unchanged under the transformation matrices F± . The asymptotic limit (3.4) in Proposition 2 for the solution (3.58) is justified since r± ∈ H 1 (R) ∩ L2,1 (R), so that R(x; ·) ∈ L1 (R) ∩ L∞ (R) for every x ∈ R. As a result, the reconstruction formulas (3.55) and (3.56) can be rewritten in the explicit form Z Rx Rx 1 1 2 2 1 (2) e 2i +∞ |u(y)| dy ∂x u ¯(x)e 2i +∞ |u(y)| dy = − r− (z)e2izx η− (x; z)dz (3.59) π R and u(x)e i Rx +∞ |u(y)|2 dy = = Z h i 2 (1) (1) r¯+ (z)e−2izx (1 + r¯+ (z)r− (z))µ− (x; z) + r− (z)e2izx η− (x; z) dz πi R Z 2 (1) r¯+ (z)e−2izx µ+ (x; z)dz, (3.60) πi R 29 where we have used the first equation of the Riemann–Hilbert problem (3.57) for the second equality in (3.60). To study the reconstruction formulas (3.59) and (3.60), we follow the lines of the previous subsection. First, we obtain the system of integral equations for vectors µ+ and η− from projections of the solution representation (3.58) to the real line: µ+ (x; z) = e1 + P + r− e2izx η− (x; ·) (z), (3.61) η− (x; z) = e2 + P − r¯+ e−2izx µ+ (x; ·) (z). (3.62) The estimates of Proposition 3, Lemma 8, Lemma 9, and Corollary 5 apply to the system of integral equations (3.61) and (3.62) with the only change: x0 ∈ R+ is replaced by x0 ∈ R− because the operators P + and P − swap their places in comparison with the system (3.31) and (3.32). As a result, we extend the statements of Lemma 9 and Corollary 5 to the negative half-line. ˜ ± are obtained from the solution of Lemma 10. Assume that r+ , r− ∈ H 1 (R) satisfies (2.84) and M 1,1 − the Riemann–Hilbert problem (3.57). Then, u ∈ H (R ) satisfies the bound kukH 1,1 (R− ) ≤ C(kr+ kH 1 , kr− kH 1 ) kr+ kH 1 (R) + kr− kH 1 (R) , (3.63) where C(kr+ kH 1 , kr− kH 1 ) is a positive constant that depends on kr+ kH 1 and kr− kH 1 . Moreover, if r± ∈ H 1 (R) ∩ L2,1 (R), then u ∈ H 2 (R− ) satisfies the bound kukH 2 (R− ) ≤ C(kr+ kH 1 , kr− kH 1 ) (kr+ kH 1 ∩L2,1 + kr− kH 1 ∩L2,1 ) , (3.64) where C(kr+ kH 1 , kr− kH 1 ) is another positive constant that depends on kr+ kH 1 and kr− kH 1 . Corollary 6. Let r± , r˜± ∈ H 1 (R) ∩ L2,1 (R) satisfy kr± kH 1 ∩L2,1 , k˜ r± kH 1 ∩L2,1 ≤ ρ for some ρ > 0. Denote the corresponding potentials by u and u ˜ respectively. Then, there is a positive ρ-dependent constant C(ρ) such that ku − u ˜kH 2 (R− )∩H 1,1 (R− ) ≤ C(ρ) (kr+ − r˜+ kH 1 ∩L2,1 + kr− − r˜− kH 1 ∩L2,1 ) . 4 (3.65) Proof of Theorem 1 The scattering data r± satisfying the relation (2.84) change in time according to the linear equation (1.10). As is well-known [17], the time evolution of r± is trivially given by the explicit solution 2 r± (t, z) = r± (0, z)e−2iz t , (4.1) where r± (0, ·) are initial spectral data found from the initial condition u(0, ·) and the direct scattering transform in Section 2. By Lemma 4 and Corollary 3, under the condition that u(0, ·) ∈ H 2 (R) ∩ H 1,1 (R) admits no resonances of the linear equation (1.9), the scattering data r± (0, ·) is defined in H 1 (R) ∩ L2,1 (R) and is a Lipschitz continuous function of u(0, ·). Now the time evolution (4.1) implies that r± (t, ·) remains in H 1 (R) ∩ L2,1 (R) for every t ∈ R. Indeed, kr± (t, ·)kL2,1 = kr± (0, ·)kL2,1 and k∂z r± (t, ·) + 4itzr± (t, ·)kL2 = k∂z r(0, ·)kL2 . Hence, r(t, ·) ∈ H 1 (R) ∩ L2,1 (R) for every t ∈ R. Moreover, the relation (2.84) remains valid for every t ∈ R. The potential u(t, ·) is recovered from the scattering data r± (t, ·) with the inverse scattering transform in Section 3, under the assumption that u(0, ·) does not support eigenvalues in the linear equation (1.9). By Lemmas 9, 10 and Corollaries 5, 6, the potential u(t, ·) is defined in H 2 (R) ∩ H 1,1 (R) for every t ∈ R and is a Lipschitz continuous function of r(t, ·). This yields the proof of Theorem 1. 30 Remark 8. We do not claim that the norm ku(t, ·)kH 2 ∩H 1,1 is uniformly bounded for every t ∈ R. Quite on the contrary, the norm kr(t, ·)kH 1 may grow at most linearly in t, therefore, we may expect that the norm ku(t, ·)kH 2 ∩H 1,1 also grows in t. The finite-time blow-up is however excluded by the constructions of the direct and inverse scattering transforms. Acknowledgement. The authors are indebted to C. Sulem for bringing this problem to their attention. D.P. is supported by the NSERC Discovery grant. Y.S. is supported by the McMaster graduate scholarship. 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