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Addition formulae, Backlund transformations, periodic solutions, and quadrilateral equations

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Frontiers of Mathematics in China 2019年第1期14卷 203-223页

作者： danda zhang da-jun zhang School of Mathematics and Statistics Ningbo University Ningbo 315211 China Department of Mathematics Shanghai University Shanghai 200444 China

Addition formulae of trigonometric and elliptic functions are used to generate Backlund transformations together with their connecting quadrilateral equations. As a result, we obtain the periodic solutions for a numbe... 详细信息

关键词： Addition formulae trigonometric functions elliptic functions Backlund transformation quadrilateral equations

Conserved Gross-Pitaevskii equations with a parabolic potential

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Communications in Theoretical Physics 2022年第10期74卷 30-36页

作者： Shi-min Liu da-jun zhang Department of Mathematics Shanghai UniversityShanghai 200444People’s Republic of China

An integrable Gross–Pitaevskii equation with a parabolic potential is presented where particle density|u|^(2)is *** also present an integrable vector Gross–Pitaevskii system with a parabolic potential,where the total particle density∑^(n)_(j)=_(1)∣u_(j)∣^(2)is *** equations are related to nonisospectral scalar and vector nonlinear Schrödinger *** many conservation laws are *** transformations between the standard isospectral nonlinear Schrödinger equations and the conserved Gross–Pitaevskii equations,both scalar and vector cases are *** and dynamics are analyzed and *** solutions exhibit features of localized-like waves.

关键词： Gross-Pitaevskii equation gauge transformation nonisospectral conserved particle density

Cauchy matrix structure of the Mel’nikov model of long-short wave interaction

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Communications in Theoretical Physics 2020年第12期72卷 83-93页

作者： Hong-Juan Tian da-jun zhang Department of Mathematics Shanghai UniversityShanghai 200444China

We propose a systematic method to construct the Mel’nikov model of long–short wave interactions,which is a special case of the Kadomtsev–Petviashvili(KP)equation with self-consistent sources(KPSCS).We show details how the Cauchy matrix approach applies to Mel’nikov’s model which is derived as a complex reduction of the *** a new result wefind that in the dispersion relation of a 1-soliton there is an arbitrary time-dependent function that has previously not reported in the literature about the Mel’nikov *** function brings time variant velocity for the long wave and also governs the short-wave *** variety of interactions of waves resulting from the time-freedom in the dispersion relation is illustrated.

关键词： Mel’nikov model of long–short wave interaction Cauchy matrix approach self-consistent sources KP equation

Notes on Canonical Forms of Integrable Vector Nonlinear Schr?dinger Systems

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Chinese Physics Letters 2017年第10期34卷 5-7页

作者： Kui Chen da-jun zhang Department of Mathematics Shanghai University Shanghai 200444

We present canonical forms of integrable vector nonlinear Sehr6dinger systems. Mathematically, it is enough to focus on these canonical forms.

关键词： NLS Notes on Canonical Forms of Integrable Vector Nonlinear Schr?dinger Systems

Nonisospectral effects on generating localized waves

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Communications in Theoretical Physics 2021年第11期73卷 18-30页

作者： Abdselam Silem Hua Wu da-jun zhang Department of Mathematics Shanghai UniversityShanghai 200444China

In this paper we explain how space-time localized waves can be generated by introducing nonisospectral effects which are usually related to non-uniformity of *** nonisospectral Korteweg–de Vries,modified Korteweg–de Vries and the Hirota equations are employed to demonstrate the *** solutions are presented in terms of Wronskians and double Wronskians and space-time localized dynamics are illustrated.

关键词： nonisospectral effects space-time localized wave integrable system bilinear

Solutions of the nonlocal(2+1)-D breaking solitons hierarchy and the negative order AKNS hierarchy

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Communications in Theoretical Physics 2020年第4期72卷 11-22页

作者： Jing Wang Hua Wu da-jun zhang Department of Mathematics Shanghai UniversityShanghai 200444China

The(2+1)-dimensional nonlocal breaking solitons AKNS hierarchy and the nonlocal negative order AKNS hierarchy are *** in double Wronskian form of these two hierarchies are derived by means of a reduction technique from those of the unreduced *** advantage of our method is that we start from the known solutions of the unreduced bilinear equations,and obtain solitons and multiple-pole solutions for the variety of classical and nonlocal *** behaviors of some obtained solutions are *** is remarkable that for some real nonlocal equations,amplitudes of solutions are related to the independent variables that are reversed in the real nonlocal reductions.

关键词： nonlocal (2+1)-dimensional breaking solitons AKNS hierarchy negative order AKNS hierarchy double Wronskian solutions reduction

Dynamics of three nonisospectral nonlinear Schrdinger equations

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Chinese Physics B 2019年第2期28卷 82-93页

作者： Abdselam Silem Cheng zhang da-jun zhang Department of Mathematics Shanghai UniversityShanghai 200444China

Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrdinger equation(NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover,we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes,velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.

关键词： nonisospectral nonlinear Schrodinger equations gauge transformations bilinear forms solitons rogue waves

Spectrum transformation and conservation laws of lattice potential KdV equation

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Frontiers of Mathematics in China 2017年第2期12卷 403-416页

作者： Senyue LOU Ying SHI da-jun zhang Faculty of Science Ningbo University Ningbo 315211 China School of Science Zhejiang University of Science and Technology Hangzhou 310023 China Department of Mathematics Shanghai University Shanghai 200444 China

Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this paper, we derive infinitely many conserved quantities for the lattice potential Korteweg-de Vries equation whose solutions have nonzero backgrounds. The derivation is based on the fact that the scattering data a（z） is independent of discrete space and time and the analytic property of Jost solutions of the discrete Schr5dinger spectral problem. The obtained conserved densities are asymptotic to zero when ｜n｜（or ｜m｜） tends to infinity. To obtain these results, we reconstruct a discrete Riccati equation by using a conformal map which transforms the upper complex plane to the inside of unit circle. Series solution to the Riccati equation is constructed based on the analytic and asymptotic properties of Jost solutions.

关键词： Conserved quantities analytic property conformal map inversescattering transform lattice potential KdV equation

Integrability and solutions of a nonsymmetric discrete Korteweg-de Vries equation

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Communications in Theoretical Physics 2024年第2期76卷 45-50页

作者： Maebel Mesfun da-jun zhang Song-lin Zhao Department of Mathematics Shanghai UniversityShanghai 200444China Newtouch Center for Mathematics of Shanghai University Shanghai 200444China Department of Applied Mathematics Zhejiang University of TechnologyHangzhou 310023China

In this paper,we present Lax pairs and solutions for a nonsymmetric lattice equation,which is a torqued version of the lattice potential Korteweg-de Vries *** nonsymmetric equation is special in the sense that it contains only one spacing parameter but consists of two consistent cubes with other integrable lattice *** such a multidimensionally consistent property we are able to derive its two Lax pairs and also construct solutions using B?cklund transformations.

关键词： discrete Korteweg-de Vries equation nonsymmetric multi-dimensional consistency Backlund transformation solution