Differential and Integral Equations

Ill-posedness and the nonexistence of standing-waves solutions for the nonlocal nonlinear Schrödinger equation

Jaime Angulo Pava and Roger Peres de Moura

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Abstract

We establish some properties for the Cauchy problem associated with the nonlocal nonlinear Schrödinger equation $\partial_{t}u=-i\alpha\partial^{2}_{x}u+\beta u\partial_{x}(\,|u|^{2})-i\beta u\mathcal{T}_{h}\partial_{x}(\,|u|^{2}) +i\gamma \left| u\right| ^{2}u,$ where $x, t\;\in\mathbb R,$ $\mathcal{T}_{h}$ is the nonlocal operator $$ \mathcal{T}_{h}u(x)=\frac{1}{2h}p.v. \int^{\infty}_{-\infty}\coth \Big (\frac{\pi(y-x)}{2h} \Big ) u(y)dy, $$ with $\alpha>0$, $\beta\geq 0$, $\gamma \geq 0$, and $h\in(0,+\infty)$. Here $\mathcal{T}_{h}\longrightarrow\mathcal{H}$ when $h\longrightarrow +\infty,$ where $\mathcal{H}$ is the Hilbert transform. We prove rigorously that a Picard interaction scheme can not be applied for solving the Cauchy problem associated with that equation in both the cases $0 < h < \infty$ and $h\rightarrow +\infty,$ with initial data in Sobolev spaces of negative index. Elsewhere, we study the asymptotic behavior of the solution in relation to a spatial variable, and we also establish the nonexistence of a standing-waves solution for the above equation in several cases.

Article information

Source
Differential Integral Equations, Volume 20, Number 10 (2007), 1107-1130.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039298

Mathematical Reviews number (MathSciNet)
MR2365204

Zentralblatt MATH identifier
1212.35153

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Citation

Angulo Pava, Jaime; de Moura, Roger Peres. Ill-posedness and the nonexistence of standing-waves solutions for the nonlocal nonlinear Schrödinger equation. Differential Integral Equations 20 (2007), no. 10, 1107--1130. https://projecteuclid.org/euclid.die/1356039298


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