可精确求解与可积系统
Large-time patterns of general higher-order lump solutions in the KP-I equation are investigated. It is shown that when the index vector of the general lump solution is a sequence of consecutive odd integers starting from one, the…
The structure of extended affine Weyl symmetry group of higher Painlev\'e equations of $N$ periodicity depends on whether $N$ is even or odd. We find that for even $N$, the symmetry group ${\widehat A}^{(1)}_{N-1}$ contains the conventional…
Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$…
It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka-Volterra equations) and that furthermore these dynamics are Hamiltonian in an appropriately defined nonlinear Poisson…
A hierarchy of differential equations on a Banach Lie-Poisson space related to the restricted Grassmannian is studied. Flows on the groupoid of partial isometries and on the restricted Grassmannian are described, and a momentum map picture…
We investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions annihilated by differential operators and explore the properties of the integrability…
A reduction from the self-dual Yang-Mills (SDYM) equation to the unreduced Fokas-Lenells (FL) system is described in this paper. It has been known that the SDYM equation can be formulated from the Cauchy matrix schemes of the matrix…
We investigate the spectral stability of non-degenerate vector soliton solutions and the nonlinear stability of breather solutions for the coupled nonlinear Schrodinger (CNLS) equations. The non-degenerate vector solitons are spectrally…
We study complexity in terms of degree growth of one-component lattice equations defined on a $3\times 3$ stencil. The equations include two in Hirota bilinear form and the Boussinesq equations of regular, modified and Schwarzian type.…
The evolution, as functions of the "ticking time" $\ell =0,1,2,...$, of the solutions of the system of $N$ quadratic recursions \begin{eqnarray*} x_{n}\left( \ell +1\right) =c_{n}+\sum_{m=1}^{N}\left[ C_{nm}x_{m}\left( \ell \right) \right]…
We apply the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear differential equations. We discuss several examples with goal to illustrate the results from the use of derivatives of composite functions in the…
The tau-function for quad-equations from the ABS classification is briefly explained. It is an auxiliary variable that systematically linearises the Backlund chain. Many equations have the same tau function and are unified by…
The aim of this work is multifold. Firstly, it intends to present a complete, quantitative and self-contained description of the periodic traveling wave solutions and Whitham modulation equations for the Toda lattice, combining results from…
The inverse scattering transform is developed to solve the Maxwell-Bloch system of equations that describes two-level systems with inhomogeneous broadening, in the case of optical pulses that do not vanish at infinity in the future. The…
Two-dimensional reductions of the KP-Whitham system, namely the overdetermined Whitham modulation system for five dependent variables that describe the periodic solutions of the Kadomtsev-Petviashvili equation, are studied and…
We discuss some families of integrable and superintegrable systems in $n$-dimensional Euclidean space which are invariant to $m\geq n-2$ rotations. The integrable invariant Hamiltonian $H=\sum p_i^2+V(q)$ commutes with $n-2$ integrals of…
In this paper we propose a geometric approach to study Painlev\'e equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential…
We propose an extension of the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. This extension allows us to fill in the gap in cluster construction of the $q$-difference Painlev\'e equations, showing…
I present a generalization of our joint works with John Harnad (2021) that relates Schur functions, KP tau functions and KP correlation functions to Schur's $Q$-functions, BKP tau functions and BKP correlation functions, respectively.
Motivated by the simplest case of tt*-Toda equations, we study the large and small $x$ asymptotics for $x>0$ of real solutions of the sinh-Godron Painlev\'e III($D_6$) equation. These solutions are parametrized through the monodromy data of…