THE CONFOCAL INVOLUTIVE SYSTEM AND THE INTEGRABILITY OF THE NONLINEARIZED LAX SYSTEMS FOR AKNS HIERARCHY

The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.

Higher-dimensional Chen-Lee-Liu equation and asymmetric peakon soliton

Integrable systems play a crucial role in physics and mathematics.In particular,the traditional(1+1)-dimensional and(2+1)-dimensional integrable systems have received significant attention due to the rarity of integrable systems in higher dimensions.Recent studies have shown that abundant higher-dimensional integrable systems can be constructed from(1+1)-dimensional integrable systems by using a deformation algorithm.Here we establish a new(2+1)-dimensional Chen-Lee-Liu(C-L-L)equation using the deformation algorithm from the(1+1)-dimensional C-L-L equation.The new system is integrable with its Lax pair obtained by applying the deformation algorithm to that of the(1+1)-dimension.It is challenging to obtain the exact solutions for the new integrable system because the new system combines both the original C-L-L equation and its reciprocal transformation.The traveling wave solutions are derived in implicit function expression,and some asymmetry peakon solutions are found.

ON THE COMPLETE INTEGRABILITY OF NONLINEARIZED LAX SYSTEMS FOR THE CLASSICAL BOUSSINESQ HIERARCHY

The N involutive integrals of motion with linearly independent gradients for the nonlin earized eigenvalue problem corresponding to the classical Boussinesq (CB) hierarchy are given. It is shown that when n=1,2,3, the nonlinearized time parts of Lax systems for the CB hierarchy are transformed into three finite-dimensional integrable Hamiltonian systems under the constraint of the nonlinearized spatial part.

Conservation Laws of a Discrete Soliton System

In this paper, we obtain an infinite number of conservation laws for a discrete soliton system by using a solvable generalized Riccati equation.

ON GENERATING EQUATIONS FOR THE KAUP-NEWELL HIERARCHY

It is shown that the Kaup-Newell hierarchy can be derived from the so-called generating equations which are Lax integrable. Positive and negative flows in the hierarchy are derived simultaneously. The generating equations and mutual commutativity of these flows enable us to construct new Lax integrable equations.

Lax Pairs and Integrability Conditions of Higher-Order Nonlinear Schrdinger Equations

We derive the Lax pairs and integrability conditions of the nonlinear Schrdinger equation with higher-order terms, complex potentials, and time-dependent coefficients. Cubic and quintic nonlinearities together with derivative terms are considered. The Lax pairs and integrability conditions for some of the well-known nonlinear Schrdinger equations, including a new equation which was not considered previously in the literature, are then derived as special cases. We show most clearly with a similarity transformation that the higher-order terms restrict the integrability to linear potential in contrast with quadratic potential for the standard nonlinear Schrdinger equation.

A NEW HIERARCHY OF LAX AND LIOUVILLE INTEGRABLE EVOLUTION EQUATIONS ASSOCIATED WITH AN ISOSPECTRAL PROBLEM IN THE LOOP ALGEBRA■

In this paper, an isospectral problem with five potentials is investigated in loop algebra A2 such that a new hierarchy of evolution equations with five arbitrary functions is obtained. And then by fixing the five arbitrary functions to be certain flmctions and using the trace identity, the generalized Hamiltonian structure of the hierarchy of evolution equations is given, it is shown that this hierarchy of equations is Liouville integrable. Finally some special cases of the isospectral problem are also given.

Some New Reductions From a Lax Integrable System

An isospectral problem with four potentials is discussed. The corresponding hierarchy of Lax integrable evolution equations is derived. For the hierarchy, it is shown that there exist other new reductions except those presented by Tu, Meng and Ma. For each reduction case the relevant Hamiltonian structure is established by means of trace identity.

Higher-dimensional integrable deformations of the modified KdV equation

The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical example of an integrable evolution equation in one spatial dimension.Do there exist integrable analogs of the modified Kd V equation in higher spatial dimensions?In what follows,we present a positive answer to this question.In particular,rewriting the(1+1)-dimensional integrable modified Kd V equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations.Further,we illustrate this idea with examples from the modified Kd V hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.

Searching For(2+1)-dimensional nonlinear Boussinesq equation from(1+1)-dimensional nonlinear Boussinesq equation

A novel(2+1)-dimensional nonlinear Boussinesq equation is derived from a(1+1)-dimensional Boussinesq equation in nonlinear Schr?dinger type based on a deformation algorithm.The integrability of the obtained(2+1)-dimensional Boussinesq equation is guaranteed by its Lax pair obtained directly from the Lax pair of the(1+1)-dimensional Boussinesq equation.Because of the effects of the deformation,the(2+1)-dimensional Boussinesq equation admits a special travelling wave solution with a shape that can be deformed to be asymmetric and/or multivalued.