Related papers: Two-component soliton systems and the Painlev\'e e…
An integrable semi-discretization of complex and multi-component coupled dispersionless systems via Lax pairs is presented. A Lax pair is proposed for the complex sdCD system. We derive the Lax pair for the multi-component sdCD system…
We derive for the first time fundamental equations that describe soliton spatial profiles consisting of two-photon mode fields and macroscopic polarization of medium. Numerical solutions of this basic equation are presented to give single…
We construct a Lax pair for the $E^{(1)}_6 $ $q$-Painlev\'e system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic…
Let $K \subseteq \mathbb{R}^{2 \times 2}$ be a compact set, let $K^{rc}$ be its rank-one convex hull, and let $L(K)$ be its lamination convex hull. It is shown that the mapping $K \to \overline{L(K)}$ is not upper semicontinuous on the…
One constructs the parity-time symmetric solitons in the complex KP Equation using the totally non-negative Grassmannian. We obtain that every element in the totally non-negative orthogonal Grassmannian corresponds to a parity-time…
We show that if the probabilistic logarithmic-space solver or the deterministic nearly logarithmic-space solver for undirected Laplacian matrices can be extended to solve slightly larger subclasses of linear systems, then they can be use to…
For a Hamiltonian system in R^{2n}, its two-system is defined in the phase space R^{2n} x sp(2n,R). In a sense, it is a combination of the original system and its system in variations with feedback. We study the Hamiltonian forms of the…
The paper develops a second-order time-domain moment matching framework for the structure-preserving model reduction of second-order dynamical systems of high dimension, avoiding the first-order double-sized equivalent system. The moments…
The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to…
The $2$-dimensional Toda lattice ($2$D Toda) is a completely integrable semi-discrete wave equation with the KP-II equation in its continuous limit. Using Darboux transformations, we prove the linear stability of $1$-line solitons for $2$D…
General soliton solutions to a nonlocal nonlinear Schr\"odinger (NLS) equation with PT-symmetry for both zero and nonzero boundary conditions {are considered} via the combination of Hirota's bilinear method and the Kadomtsev-Petviashvili…
A system of two operator equations is considered - one of pseudomonotone type and the other of strongly monotone type - both being strongly coupled. Conditions are given that allow to reduce the solvability of this system to a single…
First, starting from two hierarchies of autonomous St\"{a}ckel ODE's, we reconstruct the hierarchy of KdV stationary systems. Next, we deform considered autonomous St\"{a}ckel systems to non-autonomous Painlev\'{e} hierarchies of ODE's.…
In contrast to mono-constrained flows with N degrees of freedom, binary constrained flows of soliton equations, admitting 2x2 Lax matrices, have 2N degrees of freedom. By means of the existing method, Lax matrices only yield the first N…
We propose a method by which to examine all possible partial difference Lax pairs that consist of 'two by two' discrete linear problems, where the matrices contain one separable term in each entry. We thereby derive new, higher-order…
The one-mode and the two-mode multiboson systems with sl(2,R) symmetry are investigated.Hamiltonians of these systems are integrated using the theory of orthogonal polynomials. The coherent state representation for these systems is…
A general form of $N$-dark soliton solutions of the multi-component Mel'nikov system is presented. Taking the coupled Mel'nikov system comprised of two-component short waves and one-component long wave as an example, its general…
Lie's linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing…
We construct algebro-geometric upper triangular solutions of rank two Schlesinger systems. Using these solutions we derive two families of solutions to the sixth Painlev\'e equation with parameters $({1}/{8}, -{1}/{8}, {1}/{8}, {3}/{8})$…
We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be catalogued according to the minimal period and the number of…