Dynamics of three nonisospectral nonlinear Schrdinger equations

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.

Soliton Resonances of the Nonisospectral Modified Kadomtsev-Petviashvili Equation

Many equations possess soliton resonances phenomenon, this paper studies the soliton resonances of the nonisospectral modified Kadomtsev-Petviashvili (mKP) equation by asymptotic analysis.

Nonisospectral Lotka–Volterra Systems as a Candidate Model for Food Chain

In this paper,we derive a generalized nonisospectral semi-infinite Lotka-Volterra equation,which possesses a determinant solution.We also give its a Lax pair expressed in terms of symmetric orthogonal polynomials.In addition,if the simplified case of the moment evolution relation is considered,that is,without the convolution term,we also give a generalized nonisospectral finite Lotka-Volterra equation with an explicit determinant solution.Finally,an application of the generalized nonisospectral continuous-time Lotka-Volterra equation in the food chain is investigated by numerical simulation.Our approach is mainly based on Hirota’s bilinear method and determinant techniques.

Nonisospectral effects on generating localized waves

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 media.The nonisospectral Korteweg–de Vries,modified Korteweg–de Vries and the Hirota equations are employed to demonstrate the idea.Their solutions are presented in terms of Wronskians and double Wronskians and space-time localized dynamics are illustrated.

A Scheme for Generating Nonisospectral Integrable Hierarchies and Its Related Applications

Under the framework of the complex column-vector loop algebra C^(p),we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu scheme.As applications of the scheme,we work out a nonisospectral integrable Schrodinger hierarchy and its expanding integrable model.The latter can be reduced to some nonisospectral generalized integrable Schrodinger systems,including the derivative nonlinear Schrodinger equation once obtained by Kaup and Newell.Specially,we obtain the famous Fokker-Plank equation and its generalized form,which has extensive applications in the stochastic dynamic systems.Finally,we investigate the Lie group symmetries,fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H_(3)and the matrix linear group SL(2,R)for the generalized Fokker-Plank equation(GFPE).

Conserved Gross-Pitaevskii equations with a parabolic potential

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

Schemes for Generating Different Nonlinear Schrodinger Integrable Equations and Their Some Properties

In the paper,we want to derive a few of nonlinear Schrodinger equations with various formats and investigate their properties,such as symmetries,single soliton solutions,multi-soliton solutions,and so on.First of all,we propose an efficient and straightforward scheme for generating nonisospectral integrable hierarchies of evolution equations for which a generalized nonisospectral integrable Schrodinger hierarchy(briefly GNISH)singles out,from which we get a derivative nonlinear Schrodinger equation,a generalized nonlocal Schrodinger integrable system and furthermore we investigate the symmetries and conserved qualities of the GNISH.Next,we apply the dbar method to obtain a generalized nonlinear Schr?dinger-Maxwell-Bloch(GNLS-MB)equation and its hierarchy by introducing a generalized Zakhrov-Shabat spectral problem,whose soliton solutions and gauge transformations are obtained.