Related papers: Solitons on a finite-gap background in Bullough-Do…
We are concerned with the focusing $L^2$-critical nonlinear Schr\"odinger equations in $\mathbb{R}^d$ for $d=1,2$. The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of $K$…
In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of…
Using the framework of Colombeau algebras of generalized functions, we prove the existence and uniqueness results for global generalized solvability of semilinear hyperbolic systems with nonlinear nonlocal boundary conditions. We admit…
In this paper we study the Nonlinear Schr\"odinger-Maxwell equations (NSM). We are interested to analyse the existence of solitons, namely of finite energy solutions which exhibit stability properties. This paper is divided in two parts. In…
We investigate the large-space and large-time asymptotic behavior of a soliton gas for the focusing nonlinear Schr\"odinger equation. The soliton gas is constructed as the continuum limit of pure $N$-soliton solutions as $N\to\infty$, with…
Quaternion-valued solutions to the non-commutative KdV equation are produced using determinants. The solutions produced in this way are (breather) soliton solutions, rational solutions, spatially periodic solutions and hybrids of these…
It is shown that multisoliton solutions of several well known nonlinear PDEs(x, t) can be obtained by certain separation of variables: each n-soliton arises from a mutual solution of a nonlinear ODE(x), common for all NPDEs considered, and…
We report on some recent existence and uniqueness results for elliptic equations subject to Dirichlet boundary condition and involving a singular nonlinearity. We take into account the following types of problems: (i) singular problems with…
We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the…
This article provides a naturel sequel of previous works [6, 4] regarding the stability of travelling waves for a general one-dimensional Schr\"odinger equation (N LS) with non-zero condition at infinity. The aim of this article is twofold.…
In this paper, we consider a large class of nonlinear equations derived from first-order type methods for solving composite optimization problems. Traditional approaches to establishing superlinear convergence rates of semismooth…
Multi-soliton form factors in sine-Gordon theory from the bootstrap are compared to finite volume matrix elements computed using the truncated conformal space approach. We find convincing agreement, and resolve most of the issues raised in…
The matrix-generalized Bogoliubov-de Gennes systems have recently been considered by the present author [arXiv:1509.04242, Phys. Rev. B 93, 024512 (2016)], and time-dependent and self-consistent multi-soliton solutions have been constructed…
We are concerned with solvability of nonlinear systems involving a discrete singular $\phi$-Laplacian operator of type \begin{equation*} u \mapsto \Delta\left[\phi(\Delta u(n-1))\right] \qquad (n\in \{1, \dots, T\}), \end{equation*}…
We prove an existence and uniqueness result for solutions to linear $X$-elliptic equations with $L^1$ data and zero Dirichlet boundary conditions. Such solutions depend continuously on the datum. Moreover, we show that an improvement in the…
Under some conditions we give a blow-up analysis for solutions of an equation with Dirichlet boundary condition.
In this study, we investigate the Klein-Gordon-Zakharov system with a focus on identifying multi-soliton solutions. Specifically, for a given number $N$ of solitons, we demonstrate the existence of a multi-soliton solution that…
Boundary value problems for non-linear parabolic equations with singular potentials are considered. Existence and non-existence results as an application of different Hardy inequalities are proved. Blow-up conditions are investigated too.
The $n$-fold Darboux transformation $T_{n}$ of the focusing real mo\-di\-fied Kor\-te\-weg-de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the $n$-soliton solutions of the mKdV…
In this paper, local and nonlocal complex reduction of a discrete and a semi-discrete negative order Ablowitz-Kaup-Newell- Segur equations is studied. Cauchy matrix type solutions, including soliton solutions and Jordan-block solutions, for…