Results 41 to 50 of about 275,930 (166)

Justification of the Nonlinear Schrödinger approximation for a quasilinear Klein-Gordon equation [PDF]

Comm. Math. Phys. 355 (2017), no. 3, 1189-1207, 2016
We consider a nonlinear Klein-Gordon equation with a quasilinear quadratic term. The Nonlinear Schr\"odinger (NLS) equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the quasilinear Klein-Gordon equation.
arxiv   +1 more source

Existence of positive solutions for a class of $p$-Laplacian type generalized quasilinear Schrödinger equations with critical growth and potential vanishing at infinity

Electronic Journal of Qualitative Theory of Differential Equations, 2023
In this paper, we study the existence of positive solutions for the following generalized quasilinear Schrödinger equation \begin{equation*} -\operatorname{div}(g^p(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g'(u)|\nabla u|^p+V(x)|u|^{p-2}u =K(x)f(u)+Q(x)g(u)|
Zhen Li
doaj   +1 more source

Modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition [PDF]

, 2020
We prove the existence of the modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition. This is accomplished in three steps. First, we derive a new reduced asymptotic system for the quasilinear wave equation by modifying H\"{o}rmander's method.
arxiv   +1 more source

Existence of infinitely many radial and non-radial solutions for quasilinear Schrödinger equations with general nonlinearity

Electronic Journal of Qualitative Theory of Differential Equations, 2017
In this paper, we prove the existence of many solutions for the following quasilinear Schrödinger equation \begin{equation*} -\Delta u - u\Delta(|u|^2) + V(|x|)u = f(|x|,u),\qquad x \in \mathbb{R}^N. \end{equation*} Under some generalized assumptions on $
Jianhua Chen, Xianhua Tang, Jian Zhang, Huxiao Luo   +3 more
doaj   +1 more source

Existence of nontrivial solutions for a quasilinear Schrödinger–Poisson system in $\mathbb{R}^3$ with periodic potentials

Electronic Journal of Qualitative Theory of Differential Equations, 2023
In this paper, we study the following quasilinear Schrödinger–Poisson system in $\mathbb{R}^3$ \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda \phi u=f(x,u),&x\in{\mathbb{R}^3},\\ -\Delta \phi-\varepsilon^4\Delta_4\phi=\lambda u^2,&x\in{\mathbb{R}
Chongqing Wei, Anran Li, Leiga Zhao
doaj   +1 more source

Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

Studies in Applied Mathematics, Volume 142, Issue 3, Page 241-268, April 2019., 2019
Abstract The nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude‐frequency domains. In this paper, we take advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions in order to ...
T. Congy, G.A. El, M.A. Hoefer, M. Shearer   +3 more
wiley   +1 more source

Normal form of $O(2)$ Hopf bifurcation in a model of a nonlinear optical system with diffraction and delay

Electronic Journal of Qualitative Theory of Differential Equations, 2017
In this paper we construct an $O(2)$-equivarint Hopf bifurcation normal form for a model of a nonlinear optical system with delay and diffraction in the feedback loop whose dynamics is governed by a system of coupled quasilinear diffusion equation and ...
Stanislav Budzinskiy, Alexander Razgulin
doaj   +1 more source

The Calogero–Moser derivative nonlinear Schrödinger equation

Communications on Pure and Applied Mathematics, Volume 77, Issue 10, Page 4008-4062, October 2024.
Abstract We study the Calogero–Moser derivative nonlinear Schrödinger NLS equation i∂tu+∂xxu+(D+|D|)(|u|2)u=0$$\begin{equation*} i\partial _t u +\partial _{xx} u + (D+|D|)(|u|^2) u =0 \end{equation*}$$posed on the Hardy–Sobolev space H+s(R)$H^s_+(\mathbb {R})$ with suitable s>0$s>0$.
Patrick Gérard, Enno Lenzmann
wiley   +1 more source

Multiple small solutions for Schrödinger equations involving positive quasilinear term

Electronic Journal of Qualitative Theory of Differential Equations, 2020
We consider the multiplicity of solutions of a class of quasilinear Schrödinger equations involving the $p$-Laplacian: \begin{equation*} -\Delta_{p} u+V(x)|u|^{p-2}u+\Delta_{p}(u^{2})u=K(x)f(x,u),\qquad x\in \mathbb{R}^{N}, \end{equation*} where $\Delta_{
Dashuang Chong, Xian Zhang, Chen Huang
doaj   +1 more source

Infinitely many solutions for quasilinear Schrödinger equation with general superlinear nonlinearity

Boundary Value Problems, 2023
In this article, we study the quasilinear Schrödinger equation − △ ( u ) + V ( x ) u − △ ( u 2 ) u = g ( x , u ) , x ∈ R N , $$ -\triangle (u)+V(x)u-\triangle \bigl(u^{2}\bigr)u=g(x,u), \quad x\in \mathbb{R}^{N}, $$ where the potential V ( x ) $V(x)$ and
Jiameng Li, Huiwen Chen, Zhimin He, Zigen Ouyang   +3 more
doaj   +1 more source