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.

Lax Pairs of Integrable Systems in Bidifferential Graded Algebras

Lax pairs regarded as foundations of the inverse scattering methods play an important role in integrable systems.In the framework of bidifferential graded algebras,we propose a straightforward approach to constructing the Lax pairs of integrable systems in functional environment.Some continuous equations and discrete equations are presented.

Group training on the improvement of college students'career decision-making self-efficacy

A group training was conducted on 17 college students to improve their career decision-making self-efficacy (CDMSE). The result showed that there was significant difference between the pre-test and the post-test for the experimental group (n = 17), whereas no significant difference was found between the pre-test and the post- test for the control group (n = 17). In the pre-test, there was no significant difference between the experimental group and the control group, and obvious difference between the two groups was found in the post-test. This indicated that the group training was effective on improving the CDMSE of the college students whose scores of CDMSE were below 27% point of the total students.

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

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 equation.This nonsymmetric equation is special in the sense that it contains only one spacing parameter but consists of two consistent cubes with other integrable lattice equations.Using such a multidimensionally consistent property we are able to derive its two Lax pairs and also construct solutions using B?cklund transformations.

Cauchy matrix approach to three non-isospectral nonlinear Schrödinger equations

This paper aims to develop a direct approach,namely,the Cauchy matrix approach,to non-isospectral integrable systems.In the Cauchy matrix approach,the Sylvester equation plays a central role,which defines a dressed Cauchy matrix to provideτfunctions for the investigated equations.In this paper,using the Cauchy matrix approach,we derive three non-isospectral nonlinear Schrödinger equations and their explicit solutions.These equations are generically related to the time-dependent spectral parameter in the Zakharov–Shabat–Ablowitz–Kaup–Newell–Segur spectral problem.Their solutions are obtained from the solutions of unreduced non-isospectral nonlinear Schrödinger equations through complex reduction.These solutions are analyzed and illustrated to show the non-isospectral effects in dynamics of solitons.

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

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 KPSCS.As 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 model.This function brings time variant velocity for the long wave and also governs the short-wave packet.The variety of interactions of waves resulting from the time-freedom in the dispersion relation is illustrated.

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

The(2+1)-dimensional nonlocal breaking solitons AKNS hierarchy and the nonlocal negative order AKNS hierarchy are presented.Solutions in double Wronskian form of these two hierarchies are derived by means of a reduction technique from those of the unreduced hierarchies.The 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 reductions.Dynamical behaviors of some obtained solutions are illustrated.It 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.

Integrability of auto-B?cklund transformations and solutions of a torqued ABS equation

An auto-B?cklund transformation for the quad equation Q1_(1) is considered as a discrete equation,called H2^(a),which is a so called torqued version of H2.The equations H2^(a) and Q1_(1) compose a consistent cube,from which an auto-B?cklund transformation and a Lax pair for H2^(a) are obtained.More generally it is shown that auto-B?cklund transformations admit auto-Backlund transformations.Using the auto-Backlund transformation for H2^(a)we derive a seed solution and a one-soliton solution.From this solution it is seen that H2^(a) is a semi-autonomous lattice equation,as the spacing parameter q depends on m but it disappears from the plane wave factor.

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 so-called dressing the boundary,which generates soliton solutions by 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.Connection of our construction to the inverse scattering transform is also provided.

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.

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

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 number of multidimensionally consistent affine linear and multiquadratic quadrilateral equations.

Spectrum transformation and conservation laws of lattice potential KdV equation

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.

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.