Gauge Miura and Bäcklund transformations for generalized A _n -KdV hierarchies

Abstract

The construction of Miura and Bäcklund transformations for A n mKdV and KdV hierarchies are presented in terms of gauge transformations acting upon the zero curvature representation. As in the well known sl(2) case, we derive and relate the equations of motion for the two hierarchies. Moreover, the Miura-gauge transformation is not unique, instead, it is shown to be connected to a set of generators labeled by the exponents of A n . The construction of generalized gauge-Bäcklund transformation for the A n -KdV hierarchy is obtained as a composition of Miura and Bäcklund-gauge transformations for A n -mKdV hierarchy. The zero curvature representation provide a framework which is universal within all flows and generate systematically Bäcklund transformations for the entirely hierarchy.

Comments on the negative grade KdV hierarchy

“…In order to map the mKdV and KdV hierarchies let us consider the Miura-gauge transformation generated by (see [4], [5] )…”

“…In ref. [4], [5] we have related the two hierarchies by a gauge transformation that maps one Lax operator into the other. Such Miura-gauge transformation acting upon the zero curvature maps the flows from one hierarchy into the other.…”

Breather, lump, and interaction solutions to a nonlocal KP system

“…For example, the inverse scattering transform method, the Hirota bilinear method, the Riemann-Hilbert approach, the variable separation method, the Bäcklund transformation method,and the symmetry reduction method. These methods have been widely sharpened in the last few years, e.g., see [18][19][20][21][22][23][24][25][26][27][28][29][30] and the references therein. Since most of the nonlinear PDEs of physically significant are written in high dimensions, whether these equations can be integrated is a question of fundamental importance.…”

Geometrical correspondence of the Miura transformation induced from affine Kac–Moody algebras

“…32,33,42 Several ways have been developed to construct Miura transformations such as symmetry groups 41 and gauge transformation. 22 In a series of papers by Fordy et al, 4,5,29 Miura transformations and its multicomponent extensions can be constructed by the factorization of energy-dependent operators. Such a construction can be utilized to obtain the Miura maps between super integrable systems 51 .…”