Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation

Abstract

A Kac-Moody algebra construction for the integrable hierarchy containing the Gardner equation is proposed. Solutions are systematically constructed employing the dressing method and deformed vertex operators which takes into account the nonvanishing boundary value problem for the mKdV hierarchy. Explicit examples are given and besides usual KdV like solitons, our solutions contemplate the large amplitude table-top solitons, kinks, dark solitons, breathers and wobbles.

Nonlinear Stability of MKdV Breathers

“…Namely, we are going to focus on the focusing 5th-order mKdV equation and which we will denote them as 5th, 7th and 9th-mKdV equations hereafter. All these higher order mKdV equations are members of an infinite family of equations, the so call focusing mKdV hierarchy of equations, as it was shown by Alejo-Cardoso [2] (see [9] for a defocusing mKdV version of this hierarchy). Note that we are only interested in focusing mKdV versions since these models are the only mKdV equations bearing regular (not singular) and real breather solutions.…”

Dressing approach to the nonvanishing boundary value problem for the AKNS hierarchy

“…The previously discussed approach for a nontrivial vacuum solution was considered for the mKdV hierarchy in [5,6]. Now we present another example by considering the non-abelian AKNS hierarchy.…”

“…When taking the fields to vanish, only the operator with highest (and/or lowest) grade that is a constant operator purely in K remains, see (13b). Clearly, in this situation there is no restriction besides the own algebraic construction (6).…”

“…vanishing boundary condition. A deformed vertex operator was introduced and the method was further applied to the whole mKdV hierarchy with nonvanishing boundary condition and to a hierarchy containing the Gardner equation [6]. Here in these notes, we shall provide further clarifications discussing nonvanishing boundary conditions for integrable hierarchies in general, constructed with the structure proposed in [7] (and references therein).…”

On the mKdV–Liouville hierarchy and its self-similarity reduction

“…The mKdV equation has been widely studied in the last decades and possess some well know solutions [1,2,3,4]. A generalization of the mKdV equation which combines the mKdV and the sinh-Gordon equations was proposed in [5] as a mixed integrable model.…”