Initial‐Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

Miller, Peter D.; Qin, Zhenyun

doi:10.1111/sapm.12075

Cited by 9 publications

(11 citation statements)

References 20 publications

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“…It is the connection, through the two turning points, of oscillatory solutions onto exponential solutions and back again that yields nontrivial reflection in the semiclassical limit. One obtains from this procedure the asymptotic formulae (see [27,Appendix B] for all details of virtually the same calculation)…”

Section: (22)mentioning

confidence: 99%

A Study of the Direct Spectral Transform for the Defocusing Davey‐Stewartson II Equation the Semiclassical Limit

Assainova

Klein

McLaughlin

et al. 2019

Comm Pure Appl Math

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The defocusing Davey‐Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one‐dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space‐time that become well‐defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey‐Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB‐type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ɛ↓ 0 to a limiting function that is supported in the domain of k‐values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one‐dimensional theory. © 2019 Wiley Periodicals, Inc.

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Section: (22)mentioning

confidence: 99%

A Study of the Direct Spectral Transform for the Defocusing Davey‐Stewartson II Equation the Semiclassical Limit

Assainova

Klein

McLaughlin

et al. 2019

Comm Pure Appl Math

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“…In order to motivate the ∂ steepest descent method, we first consider the Cauchy problem for the linear equation corresponding to (1), namely (15) i ∂q ∂t…”

Section: An Unorthodox Approach To the Corresponding Linear Problemmentioning

confidence: 99%

“…In [12], the large-time behavior of solutions of the derivative NLS equation was studied using ∂ methods, and in [11] the same techniques were used to establish a form of the soliton resolution conjecture for this equation. Similar ∂ methods more based on the original approach of [13,14] have also been useful in studying some problems of nonlinear wave theory not necessarily in the realm of large time asymptotics, for instance [15], which deals with boundary-value problems for (1) in the semiclassical limit. Based on this continued interest in ∂ methods, we decided to write this review paper containing all of the results and arguments of [9], some in a new form, as well as some additional expository material which we hope the reader might find helpful.…”

Section: Introductionmentioning

confidence: 99%

Dispersive Asymptotics for Linear and Integrable Equations by the ∂ ¯ $$\overline {\partial }$$ Steepest Descent Method

Dieng

McLaughlin

Miller

2019

Nonlinear Dispersive Partial Differential Equations and Inverse Scattering

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We present a new and relatively elementary method for studying the solution of the initial-value problem for dispersive linear and integrable equations in the large-t limit, based on a generalization of steepest descent techniques for Riemann-Hilbert problems to the setting of ∂-problems. Expanding upon prior work [9] of the first two authors, we develop the method in detail for the linear and defocusing nonlinear Schrödinger equations, and show how in the case of the latter it gives sharper asymptotics than previously known under essentially minimal regularity assumptions on initial data. acting in L 2 (R; C 2 ) as described, for example, in [6].The modulus |α(z 0 )| of the complex amplitude α(z 0 ) as written in (4) was first obtained by Segur and Ablowitz [18] from trace formulae under the assumption that q(x, t) has the form (3) where E (x, t) is small for large t. Zakharov and Manakov [19] took the form (3) as an ansatz to motivate a kind of WKB analysis of the reflection coefficient r(z) and as a consequence were able to also calculate the phase of α(z 0 ), obtaining for the first time the phase as written in (5). Its [10] was the first to observe the key role played in the large-time behavior of q(x, t) by an "isomondromy" problem for parabolic cylinder functions; this problem has been an essential ingredient in all subsequent studies of the large-t limit and as we shall see Given the reflection coefficient r ∈ H 1,1 1 (R) associated with initial data q 0 ∈ H 1,1 (R) via the spectral transform R for the Zakharov-Shabat operator L, the solution of the Cauchy problem for the nonlinear Schrödinger equation (1) may be described as follows. Consider the following Riemann-Hilbert problem:Riemann-Hilbert Problem 1. Given parameters (x, t) ∈ R 2 , find M = M(z) = M(z; x, t), a 2 × 2 matrix, satisfying the following conditions:Analyticity: M is an analytic function of z in the domain C \ R. Moreover, M has a continuous extension to the real axis from the upper (lower) half-plane denoted M + (z) (M − (z)) for z ∈ R. Jump condition:The boundary values satisfy the jump condition

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“…We collect some results on Smirnoff classes and L 2 -RH problems; detailed proofs can be found in [32]. In the context of smooth contours, more information on L 2 -RH problems can be found in [14,17,23,43]. Proof: Let z 0 ∈ C \D and let ϕ(z) = 1 z−z 0 .…”

Section: Appendix A: L 2 -Riemann-hilbert Problemsmentioning

confidence: 99%

“…Following the many successes of the inverse scattering approach, one of the main open problems in the area of integrable systems in the late twentieth century was the extension of the IST formalism to initial-boundary value (IBV) problems, see [1]. Such an extension was introduced by Fokas in [3] (see also [4,5]) and has subsequently been developed and applied by several authors [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In analogy with the IST on the line, the unified transform of [3] relies for the analysis of an IBV problem on the definition of several spectral functions via nonlinear Fourier transforms and on the formulation of a RH problem.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane

Lenells

2016

Stud Appl Math

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The unified transform method introduced by Fokas can be used to analyze initial-boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann-Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of L 2 -RH problems, we consider the construction of quarter plane solutions which are C 1 in time and C 3 in space.

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Initial‐Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

Cited by 9 publications

References 20 publications

A Study of the Direct Spectral Transform for the Defocusing Davey‐Stewartson II Equation the Semiclassical Limit

A Study of the Direct Spectral Transform for the Defocusing Davey‐Stewartson II Equation the Semiclassical Limit

Dispersive Asymptotics for Linear and Integrable Equations by the ∂ ¯ $$\overline {\partial }$$ Steepest Descent Method

Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane

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