On the Nonlocal Symmetries of the μ-Camassa–Holm Equation

Abstract

The µ-Camassa-Holm (µCH) equation is a nonlinear integrable partial differential equation closely related to the Camassa-Holm and the Hunter-Saxton equations. This equation admits quadratic pseudo-potentials which allow us to compute some first-order nonlocal symmetries. The found symmetries preserve the mean of solutions. Finally, we discuss also the associated µCH equation.

Nonlocal symmetries and conservation laws of nonlocal Camassa-Holm type equations

“…By the infinitesimal criterion for symmetries [6,8] , System (2.1) admits the Symmetry (2.4) provided φ, ψ, and η satisfies the following equations…”

“…Those models admit Lax-pairs and bi-Hamiltonian structures, thus they are completely integrable. It is well-known that nonlocal symmetries of integrable equations are closely related to their properties of integrability [4,6] , conservation laws [6,8] , Bäcklund transformation and Darboux transformation [5,9] and Hirota's bilinear method [11] , etc. So it is of interest to explore the existence of their nonlocal symmetries.…”

Generalized Nonlocal Symmetries of Two-Component Camassa–Holm and Hunter–Saxton Systems

“…The two-component Camassa-Holm and Hunter-Saxton systems also have drawn much attention and have multi-peakon solitons [5]. As an extension of the Camassa-Holm equation, µ Camassa-Holm type equations also have geometric integrability and a bi-Hamiltonian structure, drawing much attention [17][18][19][20][21][22]. The outline of this paper is as follows.…”

A new kind of nonlocal symmetry for the μ‐Camassa‐Holm equation with linear dispersion

“…This model admits Lax‐pairs and bi‐Hamiltonian structures and is therefore completely integrable. It is well known that the nonlocal symmetries of integrable equations are closely related to their properties of integrability,() conservation laws,() Bäcklund transformation, Darboux transformation,() and Hirota's bilinear method . Therefore, it is of interest to explore the existence of their nonlocal symmetries.…”