Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data

Liu, Jiaqi; Perry, Peter; Sulem, Catherine

doi:10.1016/j.anihpc.2017.04.002

Cited by 71 publications

(55 citation statements)

References 25 publications

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“…where θ(z; z 0 ) and z 0 are as defined in (12). It is worth noticing that this formula is exactly what arises from Riemann-Hilbert Problem 1 via the formula (14) if only the jump matrix V M (z) in (12) is replaced with the triangular form…”

Section: An Unorthodox Approach To the Corresponding Linear Problemmentioning

confidence: 96%

“…If arg(z − z 0 ) = 0, then obviously |δ(z; z 0 )| = 1, so it remains to prove the estimates hold for Im(z) = 0. Following [12], since ln (1…”

Section: Modification Of Diagonal Jumpmentioning

confidence: 99%

“…Now, following [12], let us examine the resulting double integral for u = 0, i.e., for z = u + iv restricted to the imaginary axis. Some simple trigonometry shows that (122) sup (U,V)∈supp(W(·,·;x,t))…”

Section: and Bmentioning

confidence: 99%

“…In [1], the method of [9] was used to confirm the soliton resolution conjecture for the focusing version of the NLS equation under generic conditions on the discrete spectrum. 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.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: An Unorthodox Approach To the Corresponding Linear Problemmentioning

confidence: 96%

“…If arg(z − z 0 ) = 0, then obviously |δ(z; z 0 )| = 1, so it remains to prove the estimates hold for Im(z) = 0. Following [12], since ln (1…”

Section: Modification Of Diagonal Jumpmentioning

confidence: 99%

Section: and Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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show abstract

“…This model is well-known in the integrable systems literature. The solution of this model is expressed in terms of parabolic cylinder functions, D a pzq (see [17, Chapter 12] for its properties); its construction goes back to [20,13], see also [34] for its derivation in the context of DNLS. Here we provide only the necessary formulas Proposition 3.9.…”

Section: 3mentioning

confidence: 99%

The derivative nonlinear Schrödinger equation: Global well-posedness and soliton resolution

et al. 2019

Self Cite

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We review recent results on global wellposedness and long-time behavior of smooth solutions to the derivative nonlinear Schrödinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and the method of Zhou [48] for treating spectral singularities lead to global wellposedness for general initial conditions in the weighted Sobolev space H 2,2 pRq. For generic initial data that can support bright solitons but exclude spectral singularities, we prove the soliton resolution conjecture: the solution is asymptotic, at large times, to a sum of localized solitons and a dispersive component, Our results also show that soliton solutions of DNLS are asymptotically stable.

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On the Maxwell‐Bloch system in the sharp‐line limit without solitons

Li,

Miller

2023

Comm Pure Appl Math

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We study the (characteristic) Cauchy problem for the Maxwell‐Bloch equations of light‐matter interaction via asymptotics, under assumptions that prevent the generation of solitons. Our analysis clarifies some features of the sense in which physically‐motivated initial‐boundary conditions are satisfied. In particular, we present a proper Riemann‐Hilbert problem that generates the unique causal solution to the Cauchy problem, that is, the solution vanishes outside of the light cone. Inside the light cone, we relate the leading‐order asymptotics to self‐similar solutions that satisfy a system of ordinary differential equations related to the Painlevé‐III (PIII) equation. We identify these solutions and show that they are related to a family of PIII solutions recently discovered in connection with several limiting processes involving the focusing nonlinear Schrödinger equation. We fully explain a resulting boundary layer phenomenon in which, even for smooth initial data (an incident pulse), the solution makes a sudden transition over an infinitesimally small propagation distance. At a formal level, this phenomenon has been described by other authors in terms of the PIII self‐similar solutions. We make this observation precise and for the first time we relate the PIII self‐similar solutions to the Cauchy problem. Our analysis of the asymptotic behavior satisfied by the optical field and medium density matrix reveals slow decay of the optical field in one direction that is actually inconsistent with the simplest version of scattering theory. Our results identify a precise generic condition on an optical pulse incident on an initially‐unstable medium sufficient for the pulse to stimulate the decay of the medium to its stable state.

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Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data

Cited by 71 publications

References 25 publications

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

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

The derivative nonlinear Schrödinger equation: Global well-posedness and soliton resolution

On the Maxwell‐Bloch system in the sharp‐line limit without solitons

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