Related papers: Solitons on a finite-gap background in Bullough-Do…
In this article, we focus on investigating the focusing Kundu-Eckhaus equation with nonzero boundary condition. A appropriate two-sheeted Riemann surface is introduced to map the spectral parameter $k$ into a single-valued parameter $z$.…
It is shown that the periodic DNLS, with cubic nonlinearity, possesses gap solutions, i. e. standing waves, with the frequency in a spectral gap, that are exponentially localized in spatial variable. The proof is based on the linking…
Under investigation in this work is the robust inverse scattering transform of the discrete Hirota equation with nonzero boundary conditions, which is applied to solve simultaneously arbitrary-order poles on the branch points and spectral…
The KdV equation with self-consistent sources (KdVES) is used as a model to illustrate the method. A generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for…
In this paper, we studied N-soliton solutions of an integrable equation.
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.,…
We review some recent results concerning Gibbs measures for nonlinear Schroedinger equations (NLS), with implications for the theory of the NLS, including stability and typicality of solitary wave structures. In particular, we discuss the…
In this paper, we study the existence and regularity of solutions for a class of nonlinear singular elliptic equations involving unbounded coefficients and a singular right-hand side. Specifically, we are interested to problem whose…
Static, cylindrically symmetric solutions to nonlinear scalar-Einstein equations are considered. Regularity conditions on the symmetry axis and flat or string asymptotic conditions are formulated in order to select soliton-like solutions.…
We derive a set of bilinear identities for the determinants of the matrices that have been used to construct the dark soliton solutions for various models. To give examples of the application of the obtained identities we present soliton…
We derive an exact solution to the local nonlinear Schr\"odinger equation (NLSE) using the Darboux transformation method. The new solution describes the profile and dynamics of a two-soliton molecule. Using an algebraically-decaying seed…
In this note, we consider blow-up for solutions of the SU(3) Toda system on a compact surface \Sigma. In particular, we give a complete proof of the compactness result stated by Jost, Lin and Wang and we extend it to the case of…
We consider the focusing energy-critical nonlinear wave equation for radially symmetric initial data in space dimensions $D \ge 4$. This equation has a unique (up to sign and scale) nontrivial, finite energy stationary solution $W$, called…
We study the fully discrete elliptic integrable model Q4 and its immediate trigonometric and rational counterparts (Q3, Q2 and Q1). Singular boundary problems for these equations are systematised in the framework of global singularity…
This paper focuses on investigation of the N-coupled Hirota equations arising in an optical fiber. Starting from analyzing the spectral problem, a kind of matrix Riemann-Hilbert problem is formulated strictly on the real axis. Then based on…
In this paper we study Hirota bilinear forms of the type $P(D) \{f\cdot f\}=0$. We prove that for $P(D)=D_x^mD_y^rD_t^n$ the equations have three-soliton solutions if only if two of nonzero $m,n,p$ are odd and the other one even. We…
In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schr\"{o}dinger equation as the number, $N$, of solitons grows to infinity. We discover configurations of $N$-soliton solutions which…
The problem of construction of the boundary conditions for nonlinear equations is considered compatible with their higher symmetries. Boundary conditions for the sine-Gordon, Jiber-Shabat and KdV equations are discussed. New examples are…
We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a…
In the Bogomol'nyi limit of the Calogero-Sutherland collective-field model we find static-soliton solutions. The solutions of the equations of motion are moving solitons, having no static limit for $\l>1$. They describe holes and lumps,…