Notes on solutions in Wronskian form to soliton equations: KdV-type

This paper can be an overview on solutions in Wronskian/Casoratian form to soliton equations with KdV-type bilinear forms. We first investigate properties of matrices commuting with a Jordan block, by which we derive explicit general solutions to equations satisfied by Wronskian/Casoratian entry vectors, which we call condition equations. These solutions are given according to the coefficient matrix in the condition equations taking diagonal or Jordan block form. Limit relations between these different solutions are described. We take the KdV equation and the Toda lattice to serve as two examples for solutions in Wronskian form and Casoratian form, respectively. We also discuss Wronskian solutions for the KP equation. Finally, we formulate the Wronskian technique as four steps.Comment: 45 page

Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy

New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of $(2+1)$-dimensional integrable equations, including the DS-III equation and the $N$-wave problem. Furthermore, we recover a system that contains two types of the KP equation with self-consistent sources as special cases. Darboux and binary Darboux transformations are applied to generate solutions of the proposed hierarchies

New Exact Solutions for the (3+1)-Dimensional Generalized BKP Equation

The Wronskian technique is used to investigate a (3+1)-dimensional generalized BKP equation. Based on Hirota’s bilinear form, new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions are formally derived. Moreover we analyze the strangely mechanical behavior of the Wronskian determinant solutions. The study of these solutions will enrich the variety of the dynamics of the nonlinear evolution equations

Lie symmetry analysis, conservation laws and analytical solutions for chiral nonlinear Schrödinger equation in (2 + 1)-dimensions

In this work, we consider the chiral nonlinear Schrödinger equation in (2 + 1)-dimensions, which describes the envelope of amplitude in many physical media. We employ the Lie symmetry analysis method to study the vector field and the optimal system of the equation. The similarity reductions are analyzed by considering the optimal system. Furthermore, we find the power series solution of the equation with convergence analysis. Based on a new conservation law, we construct the conservation laws of the equation by using the resulting symmetries.&nbsp

A collection of problems in spectral analysis for self-adjoint and non self-adjoint operators

The overall aim of this dissertation is to investigate some problems in spectral anal- ysis for self-adjoint and non self-adjoint operators which arise in different contexts of physics. In the first part of this thesis we study the problem of localisation of complex eigen- values of non Hermitian perturbations of self-adjoint operators realised by means of complex potentials. In particular, we focus our attention on two different operators. The first one is the Laplacian defined on the real half line. The other is a second order two dimensional operator which arises in the context of the physics of materials, in particular from the study of the Hamiltonian of a double layer graphene. For both we provide Keller-type estimates on the localisation of complex eigenvalues. The existence of trapped waves solutions for a set of equations describing the dynamics of a stratified two layers fluid of different densities, confined in a ocean channel of fixed width and varying depth and subject to rotation is studied in the second chapter. The existence of these solutions is then recovered by proving the existence of points in the point spectrum of a two dimensional operator pencil. We prove that, under some smallness assumptions on the difference between the two fluid densities and some geometric assumption of the channel’s shape, the problem has positive solution. The last part of this dissertation focuses on existence of particular Wronskian type of solutions for the KdV equation of the type of complex complexitons. We study these solutions both from a dynamical point of view when seen evolving in time, and also for fixed values of time if regarded as potentials for a spectral problem for the Schr ̈odinger operator.Open Acces

Multiple lump solutions and their interactions for an integrable nonlinear dispersionless PDE in vector fields

In this article, lump solutions, lump with I-kink, lump with II- kink, periodic, multiwaves, rogue waves and several other interactions such as lump interaction with II-kink, interaction between lump, lump with I-kink and periodic, interaction between lump, lump with II-kink and periodic are derived for Pavlov equation by using appropriate transformations. Additionally, we also present 3-dimensional, 2-dimensional and contour graphs for our solutions