Related papers: Solitons on a finite-gap background in Bullough-Do…
In this paper, we study the Dirichlet problem for the implicit degen- erate nonlinear elliptic equation with variable exponent in a bounded domain. We obtain sufficient conditions for the existence of a solution with- out regularization and…
In this article we consider the Cauchy problem for the cubic focusing nonlinear Schr\"o\-dinger (NLS) equation on the line with initial datum close to a particular $N$-soliton. Using inverse scattering and the $\bar{\partial}$ method we…
We study a vector generalizations of the lattice KdV equation and one of the simplest Yamilov equations. We use algebraic properties of a certain class of matrices to derive the N-soliton solutions.
Focusing on multi-solitons for the Klein-Gordon equations, in first part of this paper, we establish their conditional asymptotic stability. In the second part of this paper, we classify pure multi-solitons which are solutions converging to…
There is a lack of knowledge about the topological invariants of non-linear $d$-dimensional systems with a periodic potential. We study these systems through a classification of the linearized NLS/GP equation around their soliton solutions.…
We consider equations of the type: \[\partial_t \omega = \omega R(\omega),\] for general linear operators $R$ in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localized…
We study the removability of a singular set for elliptic equations involving weight functions and variable exponents. We consider the case where the singular set satisfies conditions related to some generalization of upper Minkowski content…
In this paper, we consider a class of systems of nonlinear equations, which arise in discretized mixed formulations of problems in solid mechanics by $hp$-finite elements. We introduce a semismooth Newton solver for this specific class and…
New exact solvable elliptic potentials with free constants for the spectral problems of the third order are found. A time dependence of such potentials gives their isospectral deformations and solutions of nonlinear integrable equations.
The Nahm pole boundary condition for certain gauge theory equations in four and five dimensions is defined by requiring that a solution should have a specified singularity along the boundary. In the present paper, we show that this boundary…
Using Newton polyhedra and non-degeneracy of matrices we present conditions which guarantee the Whitney equisingularity of families of isolated determinantal singularities.
In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in…
We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, $\phi_{xt} + \phi_{xxxz}/4 + \phi_x \phi_{xz} + \phi_{xx} \phi_z/2 + \partial_x^{-1} \phi_{zzz}/4 = 0$. This equation is obtained by unifying two directional…
In this paper, we constuct the multi-point blowup solutions of self-similar type for the inviscid Burgers equation. The shape and blowup dynamics are precisely described. Moreover, the solutions we construct are stable under small…
We have proposed, in our previous papers, a method to characterize integrable discrete soliton equations. In this paper we generalize the method further and obtain a $q$-difference Toda equation, from which we can derive various…
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function…
Based on a superposition method recently proposed to obtain 1-solitary wave solutions of the KdV-Burgers equation \cite{Yua2005}, we show that this method can also be used to find a 2-solitary wave solution of the Novikov-Veselov equation.…
n this paper, we prove existence of nodal solutions for singular semilinear elliptic systems without variational structure where its both components are of sign changing. Our approach is based on sub-supersolutions method combined with…
In this paper we develop a monotonicity formula for elliptic systems with Neumann boundary coupling, proving unique continuation and classification of blow-up profiles. As an application, we obtain strong unique continuation for some fourth…
In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths in a bounded domain $\Omega \subset \mathbb{R}^{n}$. We obtain sufficient conditions and show the…