相关论文: Analysis of singular solutions for two nonlinear w…
In this paper, we characterize a family of solitary waves for NLS with derivative (DNLS) by the structue analysis and the variational argument. Since (DNLS) doesn't enjoy the Galilean invariance any more, the structure analysis here is…
wave solutions to nonlinear partial differential equations. We simplify the so called (G'/G)-expansion method and apply two of those methods to simple physical problems.
The focussing anisotropic nonlinear Schr\"odinger equation \begin{align*} \mathrm{i} u_t-\partial_{xx} u + (-\partial_{yy})^s u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R} \times \mathbb{R}^2 \end{align*} is considered for $0<s<1$ and $p>2$.…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
In this paper, a nonlinear Schr\"odinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both…
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma…
We study here the existence of solitary wave solutions of a generalized two-component Camassa-Holm system. In addition to those smooth solitary-wave solutions, we show that there are solitary waves with singularities: peaked and cusped…
We study traveling wave solutions of an equation for surface waves of moderate amplitude arising as a shallow water approximation of the Euler equations for inviscid, incompressible and homogenous fluids. We obtain solitary waves of…
We study the existence of solitary-wave solutions and some of their properties for a general multicomponent long-wave-short-wave interaction system. The system considered here describes the nonlinear interaction of multiple short waves with…
We review asymptotic stability of solitary waves for nonlinear dispersive equations set on the line. Our focus is threefold: first, the nonlinear Schrodinger equation; second, the notion of full asymptotic stability (which states that…
New problem is studied that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. Method is discussed to construct nonlinear ordinary differential equations with exact solutions. Main…
In this paper we study a class of nonlinear Schr\"odinger equations which admit families of small solitary wave solutions. We consider solutions which are small in the energy space $H^1$, and decompose them into solitary wave and dispersive…
A 3D singular integral equation is derived for electromagnetic wave scattering by bodies of arbitrary shape. Its numerical solution by a projection method is outlined.
Solutions of the semilinear wave equation are found numerically in three spatial dimensions with no assumed symmetry using distributed adaptive mesh refinement. The threshold of singularity formation is studied for the two cases in which…
This article consists of study of anisotropic double phase problems with singular term and sign changing subcritical as well as critical nonlinearity. Seeking the help of well known Nehari manifold technique, we establish existence of at…
The aim of this Letter is to derive explicitly the secular equation for Scholte waves at the interface between a fluid and an anisotropic crystal cut along a plane containing the normal to a single symmetry plane, that is containing only…
The solvability for infinite dimensional differential algebraic equations possessing a resolvent index and a Weierstra{\ss} form is studied. In particular, the concept of integrated semigroups is used to determine a subset on which…
We prove nonlinear stability of the fundamental self--similar solution of the wave equation with a focusing power nonlinearity $\psi_{tt}-\Delta \psi=\psi^p$ for $p=3,5,7,...$ in the radial case. The proof is based on a semigroup…
In this lecture, we will survey the study of dynamics of the nonlinear wave equation in recent years. We refer to some lecture notes including such as C. Kenig \cite{Kenig01,Kenig02}, C. Kenig and F. Merle \cite{KM1} J. Shatah and M. Struwe…
Spectral method related to Lame equation with finite-gap potential is used to study the optical cascading equations. These equations are known not to be integrable by inverse scattering method. Due to "partial integrability" two-gap…