Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity

Abstract

We present a compact parametric representation of the smooth bright multisolution solutions for the modified Camassa-Holm (mCH) equation with cubic nonlinearity. We first transform the mCH equation to an associated mCH equation through a reciprocal transformation and then find a novel Bäcklund transformation between solutions of the associated mCH equation and a model equation for shallow-water waves (SWW) introduced by Ablowitz {\it at al}. We combine this result with the expressions of the multisoliton solutions for the SWW and modified Korteweg-de Vries equations to obtain the multisoliton solutions of the mCH equation. Subsequently, we investigate the properties of the one- and two-soliton solutions as well as the general multisoliton solutions. We show that the smoothness of the solutions is assured only if the amplitude parameters of solitons satisfy certain conditions. We also find that at a critical value of the parameter beyond which the solution becomes singular, the soliton solution exhibits a different feature from that of the peakon solution of the CH equation. Then, by performing an asymptotic analysis for large time, we obtain the formula for the phase shift and confirm the solitonic nature of the multisoliton solutions. Last, we use the Bäcklund transformation to derive an infinite number of conservation laws of the mCH equation.

22 pages, 3 figures, Ver. 1 Minor corrections

References in corpus (2)

• Dark solitons of the Qiao's hierarchy
• The Peakon Limit of the N-Soliton Solution of the Camassa-Holm Equation

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• A Riemann-Hilbert approach to the modified Camassa-Holm equation with nonzero boundary conditions
• Smooth multisoliton solutions and their peakon limit of Novikov's Camassa-Holm type equation with cubic nonlinearity
• A view of the peakon world through the lens of approximation theory
• Multisoliton solutions of the two-component Camassa-Holm system and their reductions
• Liouville integrability of conservative peakons for a modified CH equation
• Liouville correspondences between multi-component integrable hierarchies
• A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions