Ren-integrable and ren-symmetric integrable systems

A new type of symmetry,ren-symmetry,describing anyon physics and corresponding topological physics,is proposed.Ren-symmetry is a generalization of super-symmetry which is widely applied in super-symmetric physics such as super-symmetric quantum mechanics,super-symmetric gravity,super-symmetric string theory,super-symmetric integrable systems and so on.Supersymmetry and Grassmann numbers are,in some sense,dual conceptions,and it turns out that these conceptions coincide for the ren situation,that is,a similar conception of ren-number(R-number)is devised for ren-symmetry.In particular,some basic results of the R-number and ren-symmetry are exposed which allow one to derive,in principle,some new types of integrable systems including ren-integrable models and ren-symmetric integrable systems.Training examples of ren-integrable KdV-type systems and ren-symmetric KdV equations are explicitly given.

Symmetries and symmetry reductions of the combined KP3 and KP4 equation

To find symmetries,symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics.In this paper,the finite point symmetry group of the combined KP3 and KP4(CKP34)equation is found by means of a direct method.The related point symmetries can be obtained simply by taking the infinitesimal form of the finite point symmetry group.The point symmetries of the CKP34 equation constitute an infinite dimensional KacMoody-Virasoro algebra.The point symmetry invariant solutions of the CKP34 equation are obtained via the standard classical Lie point symmetry method.

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.