可精确求解与可积系统
Based on the notion of foldon, we introduce a geometric framework for constructing folded parametric wave representations of exact solutions of some shifted nonlocal nonlinear Schr\"{o}dinger and modified Korteweg-de Vries equations. Unlike…
The Gerdjikov-Ivanov (GI) equation is an important model in the derivative nonlinear Schrodinger system, yet its fully discrete integrable analogues remain unexplored. In this paper, we systematically construct discrete versions of both the…
We show that the dispersionless version of the modified DKP hierarchy originally defined as the limit of relations for the tau-function of the Hirota-Miwa type has an equivalent reformulation as the Yang-Baxter equation for Baxter's…
We collect rank two difference-differential Lax pairs for classical Painlev\'e equations in the literature and put each in $2\times 2$ matrix form with the coefficient matrix of the spectral equation a degree two matrix polynomial. We…
This paper investigates the N-elliptic localized solutions of the Foka-Lenells equation. Based on the corresponding Lax pair, the Weierstrass elliptic functions are adopted to construct the elliptic function solutions and the fundamental…
We extend recent work on the relation between classical surface theory and partial differential equations, focusing on equations of pseudo-spherical type in the sense of Chern--Tenenblat and on a non-trivial generalization motivated by the…
We propose a new formulation of the multi-component short pulse (MCSP) equation that includes the coupled complex short pulse (CCSP) equation as a reduction. Using Hirota's bilinear method, we construct its $N$-soliton solutions in Pfaffian…
We establish a sharp upper bound on the densities of finite-gap solutions of the sine-Gordon equation. The bound is derived directly from the finite-dimensional hierarchy, without explicit integration of the finite-gap solutions. The…
In this study, we investigate Lie symmetries of the (2+1)-dimensional Boussinesq equation, which has been proposed to model the propagation of gravity waves on the water surface, with particular emphasis on the head-on collision of oblique…
A Lax pair $(L,P)$ is sometimes thought of as a structural certificate, in that the spatial operator $L$ carries the spectral data of an integrable system, and its isospectral evolution under $\partial_t L = [L,P]$ encodes the nonlinear…
A direct proof based on commuting finite-dimensional flows and local polynomial invariants is given for a sharp upper bound on the amplitudes of finite-gap solutions of the modified Korteweg-de Vries (mKdV) equation. The maximal amplitude…
Off-shell Bethe vectors for a generic $\fg$ invariant integrable model are constructed through the currents of the Yangian doubles of the classical series. These off-shell Bethe vectors are shown to satisfy the defining properties which…
Recently, a family of unconventional integrators for higher order ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for first order…
We study the long-time asymptotics of the focusing nonlinear Schr\"odinger equation with nonzero boundary conditions in the transition region. Biondini and Mantzavinos showed that, away from the transition curves, the \((x,t)\)-plane…
The Tomimatsu--Sato (TS) family, characterized by the rotation parameter $q$ and the TS index $\delta=n,$ provides an important class of exact stationary axisymmetric vacuum solutions of Einstein's equations, whose integrable structure is…
We consider solutions of the sinh-Gordon Painlev\'e III equation \[ u_{xx} + \frac{1}{x} u_x = \sinh u \] that are real on $(0,\infty)$. They are parametrized by the monodromy parameter $p\in\overline{\mathbb{C}}$, $|p|>1$, and an…
In this work we develop an integrable perturbation theory for the defocusing modified Korteweg-de Vries kink solution based on the squared eigenfunction expansion associated with the underlying Zakharov-Shabat scattering problem. We derive…
Using a non-commutative analogue of the isomonodromic problem associated with the discrete first Painlev\'e hierarchy, we construct a non-commutative version of this hierarchy, denoted by $\text{d-PI}_m^{\text{nc}}$. We show that both…
In this paper, we establishes a connection between noncommutative Laurent biorthogonal polynomials (bi-OPs) and matrix discrete Painlev\'e (dP) equations. We first apply nonisospectral deformations to noncommutative Laurent bi-OPs to obtain…
In this paper, by considering two non-isospectral problems with matrices chosen on the color Lie algebra $\mathfrak{sp}_{1}(6)$, we construct (1+1)-dimensional and (2+1)-dimensional super integrable systems on $\mathfrak{sp}_{1}(6)$.…