KdV equation beyond standard assumptions on initial data

Rybkin, Alexei

doi:10.1016/j.physd.2017.10.005

Cited by 10 publications

(6 citation statements)

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“…In the case of a highly regular step-like background, existence for the Cauchy problem has been examined in [22,26,27,35,36,59] using the inverse scattering transform. This process has been adapted to classes of one-sided step-like initial data [46,89,90] and even to one-sided step-like elements of H −1 loc (R) [45]. Despite the lack of assumptions at −∞ (the direction in which radiation propagates), these low-regularity arguments require rapid decay at +∞ and global boundedness from below, while our analysis is symmetric in ±x and in ±u.…”

Section: Introductionmentioning

confidence: 99%

Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

Laurens

2022

Commun. Math. Phys.

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Given a suitable solution V(t, x) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data $$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$ u ( 0 , x ) ∈ V ( 0 , x ) + H - 1 ( R ) . Our conditions on V do include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles $$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$ V ( 0 , x ) ∈ H 5 ( R / Z ) satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022. https://doi.org/10.1088/1361-6544/ac37f5) we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019. https://doi.org/10.4007/annals.2019.190.1.4) where $$V\equiv 0$$ V ≡ 0 . In that setting, it is known that $$H^{-1}(\mathbb {R})$$ H - 1 ( R ) is sharp in the class of $$H^s(\mathbb {R})$$ H s ( R ) spaces.

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Section: Introductionmentioning

confidence: 99%

Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

Laurens

2022

Commun. Math. Phys.

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“…The formula (1.3) is also available in this case with an explicit representation of the tau-function in terms of certain scattering data. We refer to our recent [15,37] where (1.3) is extended to essentially arbitrary functions q with a rapid decay only at +∞. The main feature of such initial profiles is infinite sequence of solitons emitted by the initial step.…”

Section: Introductionmentioning

confidence: 99%

The effect of a positive bound state on the KdV solution: a case study ^*

Rybkin

2021

Nonlinearity

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We consider a slowly decaying oscillatory potential such that the corresponding 1D Schrödinger operator has a positive eigenvalue embedded into the absolutely continuous spectrum. This potential does not fall into a known class of initial data for which the Cauchy problem for the Korteweg-de Vries (KdV) equation can be solved by the inverse scattering transform. We nevertheless show that the KdV equation with our potential does admit a closed form classical solution in terms of Hankel operators. Comparing with rapidly decaying initial data our solution gains a new term responsible for the positive eigenvalue. To some extent this term resembles a positon (singular) solution but remains bounded. Our approach is based upon certain limiting arguments and techniques of Hankel operators.

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“…Long-time asymptotic behavior of solutions for (1) with step-like initial data satisfying q l > q r (the case where a DSW is generated) in a region near the wave front was investigated by Hruslov in [18]. Recently, Rybkin presented an inverse scattering theory to solve the initial value problem with bounded initial data that decays rapidly to 0 as x → +∞ but unrestricted otherwise [33]. For a review on DSWs, see [3] and the articles in this special issue (in particular, see [4,10,27,34,36]).…”

Section: Introductionmentioning

confidence: 99%

On numerical inverse scattering for the Korteweg–de Vries equation with discontinuous step-like data

Bilman

Trogdon

2020

Nonlinearity

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We present a method to compute dispersive shock wave solutions of the Korteweg-de Vries equation that emerge from initial data with step-like boundary conditions at infinity. We derive two different Riemann-Hilbert problems associated with the inverse scattering transform for the classical Schrödinger operator with possibly discontinuous, step-like potentials and develop relevant theory to ensure unique solvability of these problems. We then numerically implement the Deift-Zhou method nonlinear steepest descent to compute the solution of the Cauchy problem for small times and in two asymptotic regions. Our method applies to continuous and discontinuous initial data.

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KdV equation beyond standard assumptions on initial data

Cited by 10 publications

References 60 publications

Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

The effect of a positive bound state on the KdV solution: a case study ^*

On numerical inverse scattering for the Korteweg–de Vries equation with discontinuous step-like data

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KdV equation beyond standard assumptions on initial data

Cited by 10 publications

References 60 publications

Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

The effect of a positive bound state on the KdV solution: a case study *

On numerical inverse scattering for the Korteweg–de Vries equation with discontinuous step-like data

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The effect of a positive bound state on the KdV solution: a case study ^*