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Da-jun Zhang

Publications and source records attributed to Da-jun Zhang.

At least 55 records · Page 3Linked to original sources
arXiv2020

Discrete rogue waves and blow-up from solitons of a nonisospectral semi-discrete nonlinear Schrödinger equation

We investigate the nonisospectral effects of a semi-discrete nonlinear Schrödinger equation, which is a direct integrable discretisation of its continuous counterpart. Bilinear form and double casoratian solution of the equation are presented. Dynamics of solutions are analyzed. Both solitons and multiple pole solutions admit space-time localized rogue wave behavior. And more interestingly, the solutions allow blow-up at finite time $t$.

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arXiv2020

Symmetric discrete AKP and BKP equations

We show that when KP (Kadomtsev-Petviashvili) $τ$ functions allow special symmetries, the discrete BKP equation can be expressed as a linear combination of the discrete AKP equation and its reflected symmetric forms. Thus the discrete AKP and BKP equations can share the same $τ$ functions with these symmetries. Such a connection is extended to 4 dimensional (i.e. higher order) discrete AKP and BKP equations in the corresponding discrete hierarchies. Various explicit forms of such $τ$ functions, including Hirota's form, Gramian, Casoratian and polynomial, are given. Symmetric $τ$ functions of Cauchy matrix form that are composed of Weierstrass $σ$ functions are investigated. As a result we obtain a discrete BKP equation with elliptic coefficients.

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arXiv2020

On the Lattice Potential KP Equation

The paper presents an approach to derive finite genus solutions to the lattice potential Kadomtsev-Petviashvili (lpKP) equation introduced by F.W. Nijhoff, et al. This equation is rederived from compatible conditions of three replicas of the discrete ZS-AKNS spectral problem, which is a Darboux transformation of the continuous ZS-AKNS spectral problem. With the help of these links and by means of the so called nonlinearization technique and Liouville platform, finite genus solutions of the lpKP equation are derived. Semi-discrete potential KP equations with one and two discrete arguments, respectively, are also discussed.

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arXiv2020

New dynamics of the classical and nonlocal Gross-Pitaevskii equation with a parabolic potential

Solutions of the classical and nonlocal Gross-Pitaevskii (GP) equation with a parabolic potential and a gain term are derived by using a second order nonisospectral Ablowitz-Kaup-Newell-Segur system and reduction technique of double Wronskians. Solutions of the classical GP equation show typical space-time localized characteristics. An interesting dynamics, solitons carrying an oscillating wave, are found with mathematical analysis and illustrations. Solutions of some nonlocal cases are also illustrated.

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arXiv2020

Eigenfunction equations of lattice KdV equations and connections to ABS lattice equations with a $δ$ term

We develop lattice eigenfunction equations of lattice KdV equation, which are equations obeyed by the auxiliary functions, or eigenfunctions, of the Lax pair of the lattice KdV equation. This leads to three-dimensionally consistent quad-equations that are closely related to lattice equations in the Adler-Bobenko-Suris (ABS) classification. In particular, we show how the H3($δ$), Q1($δ$) and Q3($δ$) equations in the ABS list arise from the lattice eigenfunction equations by providing a natural interpretation of the $δ$ term as interactions between the eigenfunctions. By construction, exact solution structures of these equations are obtained. The approach presented in this paper can be used as a systematic means to search for integrable lattice equations.

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arXiv2019

Elliptic solutions of Boussinesq type lattice equations and the elliptic Nth root of unity

We establish an infinite family of solutions in terms of elliptic functions of the lattice Boussinesq systems by setting up a direct linearisation scheme, which provides the solution structure for those equations in the elliptic case. The latter, which contains as main structural element a Cauchy kernel on the torus, is obtained from a dimensional reduction of the elliptic direct linearisation scheme of the lattice Kadomtsev-Petviashvili equation, which requires the introduction of a novel technical concept, namely the "elliptic cube root of unity". Thus, in order to implement the reduction we define, more generally, the notion of {\em elliptic $N^{\rm th}$ root of unity}, and discuss some of its properties in connection with a special class of elliptic addition formulae. As a particular concrete application we present the class of elliptic $N$-soliton solutions of the lattice Boussinesq systems.

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arXiv2019

Wronskian solutions of integrable systems

Wronski determinant (Wronskian) provides a compact form for $τ$-functions that play roles in a large range of mathematical physics. In 1979 Matveev and Satsuma, independently, obtained solutions in Wronskian form for the Kadomtsev-Petviashvili equation. Later, in 1981 these solutions were constructed from Sato's approach. Then in 1983, Freeman and Nimmo invented the so-called Wronskian technique, which allows directly verifying bilinear equations when their solutions are given in terms of Wronskians. In this technique the considered bilinear equation is usually reduced to the Plücker relation on Grassmannians, and finding solutions of the bilinear equation is transferred to find a Wronskian vector that is defined by a linear differential equation system. General solutions of such differential equation systems can be constructed by means of triangular Toeplitz matrices. In this monograph we review the Wronskian technique and solutions in Wronskian form, with supporting instructive examples, including the Korteweg-de Vries (KdV) equation, the modified KdV equation, the Ablowitz-Kaup-Newell-Segur hierarchy and reductions, and the lattice potential KdV equation. (Dedicated to Jonathan J C Nimmo).

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arXiv2019

Vector NLS solitons interacting with a boundary

We construct multi-soliton solutions of the n-component vector nonlinear Schrödinger equation on the half-line subject to two classes of integrable boundary conditions (BCs): the homogeneous Robin BCs and the mixed Neumann/Dirichlet BCs. The construction is based on the approach of dressing the integrable BCs: soliton solutions are generated in preserving the integrable BCs at each step of the Darboux-dressing process. Under the Robin BCs, examples, including boundary-bound solitons, are explicitly derived; under the mixed Neumann/Dirichlet BCs, the boundary can act as a polarizer that tunes different components of the vector solitons.

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arXiv2018

Duality for discrete integrable systems II

We generalise the concept of duality to lattice equations. We derive a novel 3 dimensional lattice equation, which is dual to the lattice AKP equation. Reductions of this equation include Rutishauser's quotient-difference (QD) algorithm, the higher analogue of the discrete time Toda (HADT) equation and its corresponding quotient-quotient-difference (QQD) system, the discrete hungry Lotka-Volterra system, discrete hungry QD, as well as the hungry forms of HADT and QQD. We provide three conservation laws, we conjecture the equation admits N-soliton solutions and that reductions have the Laurent property and vanishing algebraic entropy.

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arXiv2018

Discrete Crum's Theorems and Integrable Lattice Equations

In this paper, we develop discrete versions of Darboux transformations and Crum's theorems for two second order difference equations. The difference equations are discretised versions (using Darboux transformations) of the spectral problems of the KdV quation, and of the modified KdV equation or sine-Gordon equation. Considering the discrete dynamics created by Darboux transformations for the difference equations, one obtains the lattice potential KdV equation, the lattice potential modified KdV equation and the lattice Schwarzian KdV equation, that are prototypes of integrable lattice equations. It turns out that, along the discretisation processes using Darboux transformations, two families of integrable systems (the KdV family, and the modified KdV or sine-Gordon family), including their continuous, semi-discrete and lattice versions, are explicitly constructed. As direct applications of the discrete Crum's theorems, multi-soliton solutions of the lattice equations are obtained.

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arXiv2017

Solutions of local and nonlocal equations reduced from the AKNS hierarchy

In the paper possible local and nonlocal reductions of the Ablowitz-Kaup-Newell-Suger (AKNS) hierarchy are collected, including the Korteweg-de Vries (KdV) hierarchy, modified KdV hierarchy and their nonlocal versions, nonlinear Schrödinger hierarchy and their nonlocal versions, sine-Gordon equation in nonpotential form and its nonlocal forms. A reduction technique for solutions is employed, by which exact solutions in double Wronskian form are obtained for these reduced equations from those double Wronskian solutions of the AKNS hierarchy. As examples of dynamics we illustrate new interaction of two-soliton solutions of the reverse-$t$ nonlinear Schrödinger equation. Although as a single soliton it is always stationary, two solitons travel along completely symmetric trajectories in $\{x,t\}$ plane and their amplitudes are affected by phase parameters. Asymptotic analysis is given as demonstration. The approach and relation described in this paper are systematic and general and can be used to other nonlocal equations.

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arXiv2017

Bilinearisation-reduction approach to the nonlocal discrete nonlinear Schrödinger equations

A bilinearisation-reduction approach is described for finding solutions for nonlocal integrable systems and is illustrated with nonlocal discrete nonlinear Schrödinger equations. In this approach we first bilinearise the coupled system before reduction and derive its double Casoratian solutions, then we impose reduction on double Casoratians so that they coincide with the nonlocal reduction on potentials. Double Caosratian solutions of the classical and nonlocal (reverse space, reverse time and reverse space-time) discrete nonlinear Schrödinger equations are presented.

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arXiv2017

On decomposition of the ABS lattice equations and related Bäcklund transformations

The Adler-Bobenko-Suris (ABS) list contains all scalar quadrilateral equations which are consistent around the cube. Each equation in the ABS list admits a beautiful decomposition. In this paper, we first revisit these decomposition formulas, by which we construct Bäcklund transformations (BTs) and consistent triplets. Some BTs are used to construct new solutions, lattice equations and weak Lax pairs.

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arXiv2017

Solutions of the nonlocal nonlinear Schrödinger hierarchy via reduction

In this letter we propose an approach to obtain solutions for the nonlocal nonlinear Schrödinger hierarchy from the known ones of the Ablowitz-Kaup-Newell-Segur hierarchy by reduction. These solutions are presented in terms of double Wronskian and some of them are new.The approach is general and can be used for other systems with double Wronskian solutions which admit local and nonlocal reductions.

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arXiv2017

Rational solutions to the ABS list: Degenerating approach

In the paper we first construct rational solutions for the Nijhoff-Quispel-Capel (NQC) equation by means of bilinear method. These solutions can be transferred to those of Q3$_δ$ equation in the Adler-Bobenko-Suris (ABS) list. Then making use of degeneration relation we obtain rational solutions for Q2, Q1$_δ$, H3$_δ$, H2 and H1. These rational solutions are in Casoratian form and the basic column vector satisfies an extended condition equation set.

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arXiv2017

On a Second Discretization of the ZS-AKNS Spectral Problem: Revisit

In this paper we revisit a discrete spectral problem which was proposed by Ragnisco and Tu in 1989, as a second discretization of the ZS-AKNS spectral problem. We show that the spectral problem corresponds to a bidirectional discretization of the derivative of two wave functions $ϕ_{1,x}$ and $ϕ_{2,x}$. As a connection with higher dimensional systems, the spectral problem and a related hierarchy can be derived from Lax triads of the differential-difference KP hierarchy via a symmetry constraint. Isospectral and nonisospectral flows derived from the spectral problem compose a Lie algebra. By considering its infinite dimensional subalgebras and continuum limit of recursion operator, three semi-discrete AKNS hierarchies are constructed.

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