arXiv 1102.23742011-02-11nlin.PSmath-phmath.APmath.MP

Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D

P. G. Kevrekidis, A. R. Nahmod, C. Zeng

备注：: 19 pages, 1 Figure, to appear in Nonlinearity
DOI：: 10.1088/0951-7715/24/5/007

英文摘要：

In this note we propose a new set of coordinates to study the hyperbolic or non-elliptic cubic nonlinear Schrodinger equation in two dimensions. Based on these coordinates, we study the existence of bounded and continuous hyperbolically radial standing waves, as well as hyperbolically radial self-similar solutions. Many of the arguments can easily be adapted to more general nonlinearities.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Orbital instability of standing waves for NLS equation on Star Graphs

We consider a nonlinear Schrödinger (NLS) equation with any positive power nonlinearity on a star graph $Γ$ ($N$ half-lines glued at the common vertex) with a $δ$ interaction at the vertex. The strength of the interaction is defined by a fixed value $α\in \mathbb{R}$. In the recent works of Adami {\it et al.}, it was shown that for $α\neq 0$ the NLS equation on $Γ$ admits the unique symmetric (with respect to permutation of edges) standing wave and that all other possible standing waves are nonsymmetric. Also, it was proved for $α<0$ that, in the NLS equation with a subcritical power-type nonl…

2018-01-22math-phmath.APmath.DS

Rogue waves on the background of periodic standing waves in the derivative NLS equation

The derivative nonlinear Schrodinger (DNLS) equation is the canonical model for dynamics of nonlinear waves in plasma physics and optics. We study exact solutions describing rogue waves on the background of periodic standing waves in the DNLS equation. We show that the space-time localization of a rogue wave is only possible if the periodic standing wave is modulationally unstable. If the periodic standing wave is modulationally stable, the rogue wave solutions degenerate into algebraic solitons propagating along the background and interacting with the periodic standing waves. Maximal amplitud…

2021-03-16math-phmath.APmath.MP

Modulational instability of periodic standing waves in the derivative NLS equation

We consider the periodic standing waves in the derivative nonlinear Schrodinger (DNLS) equation arising in plasma physics. By using a newly developed algebraic method with two eigenvalues, we classify all periodic standing waves in terms of eight eigenvalues of the Kaup-Newell spectral problem located at the end points of the spectral bands outside the real line. The analytical work is complemented with the numerical approximation of the spectral bands, this enables us to fully characterize the modulational instability of the periodic standing waves in the DNLS equation.

2021-03-16math-phmath.APmath.DS

Orbitally stable standing waves for the asymptotically linear one-dimensional NLS

In this article we study the one-dimensional, asymptotically linear, non-linear Schrödinger equation (NLS). We show the existence of a global smooth curve of standing waves for this problem, and we prove that these standing waves are orbitally stable. As far as we know, this is the first rigorous stability result for the asymptotically linear NLS. We also discuss an application of our results to self-focusing waveguides with a saturable refractive index.

2012-11-20math.APnlin.PS

Standing lattice solitons in the discrete NLS equation with saturation

We consider standing lattice solitons for discrete nonlinear Schrodinger equation with saturation (NLSS), where so-called transparent points were recently discovered. These transparent points are the values of the governing parameter (e.g., the lattice spacing) for which the Peierls-Nabarro barrier vanishes. In order to explain the existence of transparent points, we study a solitary wave solution in the continuous NLSS and analyse the singularities of its analytic continuation in the complex plane. The existence of a quadruplet of logarithmic singularities nearest to the real axis is proven a…

2018-09-28math-phmath.CAmath.MP

NLS approximation for wavepackets in periodic cubically nonlinear wave problems in $\mathbb{R}^d$

The dynamics of single carrier wavepackets in nonlinear wave problems over periodic structures can be often formally approximated by the constant coefficient nonlinear Schrödinger equation (NLS) as an effective model for the wavepacket envelope. We provide a detailed proof of this approximation result for the Gross-Pitaevskii equation (GP) and a semilinear wave equation, both with periodic coefficients in $\mathbb{N}\ni d$ spatial dimensions and with cubic nonlinearities. The proof is carried out in Bloch expansion variables with estimates in an $L^1$-type norm, which translates to an estimate…

2018-09-18math-phmath.APmath.MP

更多相关论文收起更多论文

Numerical study of the transverse stability of NLS soliton solutions in several classes of NLS type equations

Dispersive PDEs are important both in applications (wave phenomena e.g. in hy- drodynamics, nonlinear optics, plasma physics, Bose-Einstein condensates,...) and a mathematically very challenging class of partial differential equations, especially in the time dependent case. An important point with respect to applications is the stability of exact solutions like solitons. Whereas the linear or spectral stability can be addressed analytically in some situations, the proof of full nonlinear (in-)stability remains mostly an open question. In this paper, we numerically investi- gate the transverse …

2014-01-21math-phmath.MPnlin.PS

Standing waves of the quintic NLS equation on the tadpole graph

The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schrödinger equation with quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $ω\in (-\infty,0)$ is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $L^6$. The set of minimizers includes the set of ground states of the system, which are the global minimizers of the energy at constant mass ($L^2$-norm), but it is actually wider.…

2020-09-10math-phmath.APmath.DS

Mapping Between Generalized Nonlinear Schroedinger Equations and Neutral Scalar Field Theories and New Solutions of the Cubic-Quintic NLS Equation

We highlight an interesting mapping between the moving breather solutions of the generalized Nonlinear Schrodinger (NLS) equations and the static solutions of neutral scalar field theories. Using this connection, we then obtain several new moving breather solutions of the cubic-quintic NLS equation both with and without uniform phase in space. The stability of some stationary solutions is investigated numerically and the results confirmed via dynamical evolution.

2011-01-12nlin.PS

Hamiltonian Hopf bifurcations and chaos of NLS/GP standing-wave modes

We examine the dynamics of solutions to nonlinear Schrodinger/Gross-Pitaevskii equations that arise due to Hamiltonian Hopf (HH) bifurcations--the collision of pairs of eigenvalues on the imaginary axis. To this end, we use inverse scattering to construct localized potentials for this model which lead to HH bifurcations in a predictable manner. We perform a formal reduction from the partial differential equations (PDE) to a small system of ordinary differential equations (ODE). We show numerically that the behavior of the PDE is well-approximated by that of the ODE and that both display Hamilt…

2011-01-31math-phmath.MPnlin.CD

Bifurcations of standing localized waves on periodic graphs

The nonlinear Schrodinger (NLS) equation is considered on a periodic metric graph subject to the Kirchhoff boundary conditions. Bifurcations of standing localized waves for frequencies lying below the bottom of the linear spectrum of the associated stationary Schrodinger equation are considered by using analysis of two-dimensional discrete maps near hyperbolic fixed points. We prove existence of two distinct families of small-amplitude standing localized waves, which are symmetric about the two symmetry points of the periodic graphs. We also prove properties of the two families, in particular,…

2016-09-16math.APmath.DSnlin.PS

Stable NLS solitons in a cubic-quintic medium with a delta-function potential

We study the one-dimensional nonlinear Schrödinger equation with the cubic-quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function. We determine all bound states with a positive soliton profile through explicit formulas and, using bifurcation theory, we describe their behavior with respect to the propagation constant. This information is used to prove their stability by means of the rigorous theory of orbital stability of Hamiltonian systems. The presence of the trapping potential gives rise to a regime where two stable bound sta…

2015-11-09math.APnlin.PSnlin.SI

Novel Superposed Kink and Pulse Solutions for $ϕ^4$, MKdV, NLS and Other Nonlinear Equations

We show that a number of nonlinear equations including symmetric as well as asymmetric $ϕ^4$, modified Korteweg de Vries (MKdV), mixed KdV-MKdV, nonlinear Schrödinger (NLS), quadratic-cubic NLS as well as higher order neutral scalar field theories, higher order KdV-MKdV and higher order quadratic-cubic NLS admit superposed periodic kink and pulse solutions and some of them also admit superposed hyperbolic kink solutions.

2022-02-13nlin.PS

Dynamical stabilization of solitons in cubic-quintic nonlinear Schrödinger model

We consider the existence of a dynamically stable soliton in the one-dimensional cubic-quintic nonlinear Schrödinger model with strong cubic nonlinearity management for periodic and random modulations. We show that the predictions of the averaged cubic-quintic NLS equation and modified variational approach for the arrest of collapse coincide. The analytical results are confirmed by numerical simulations of one-dimensional cubic-quintic NLS equation with rapidly and strongly varying cubic nonlinearity coefficient.

2005-07-04cond-mat.othernlin.PS

Some special solutions to the Hyperbolic NLS equation

The Hyperbolic Nonlinear Schrodinger equation (HypNLS) arises as a model for the dynamics of three-dimensional narrowband deep water gravity waves. In this study, the Petviashvili method is exploited to numerically compute bi-periodic time-harmonic solutions of the HypNLS equation. In physical space they represent non-localized standing waves. Non-trivial spatial patterns are revealed and an attempt is made to describe them using symbolic dynamics and the language of substitutions. Finally, the dynamics of a slightly perturbed standing wave is numerically investigated by means a highly acccura…

2017-06-07math.APnlin.PSphysics.class-ph

Experimental evidence of nonlinear focusing in standing water waves

Nonlinear wave focusing originating from the universal modulation instability (MI) is responsible for the formation of strong wave localizations on the water surface and in nonlinear wave guides, such as optical Kerr media and plasma. Such extreme wave dynamics can be described by breather solutions of the nonlinear Schrödinger equation (NLSE) like by way of example the famed doubly-localized Peregrine breathers (PB), which typify particular cases of MI. On the other hand, it has been suggested that the MI relevance weakens when the wave field becomes broadband or directional. Here, we provide…

2022-04-25nlin.PSphysics.flu-dyn

Radiation dynamics in fast soliton collisions in the presence of cubic loss

We study the dynamics of emission of radiation (small-amplitude waves) in fast collisions between two solitons of the nonlinear Schrödinger (NLS) equation in the presence of weak cubic loss. We calculate the radiation dynamics by a perturbation technique with two small parameters: the cubic loss coefficient $ε_{3}$ and the reciprocal of the group velocity difference $1/β$. The agreement between the perturbation theory predictions and the results of numerical simulations with the full coupled-NLS propagation model is very good for large $β$ values, and is good for intermediate $β$ values. Addit…

2019-07-19nlin.PSphysics.optics