Perturbational self-similar solutions for multi-dimensional camassa-holm-type equations

In this article, we sutdy a multi-component Camassa-Holm-type system. By employing the characteristic method, we obtain a class of perturbational self-similar solutions with elliptical symmetry, whose velocity components are governed by the generalized Emden equations. In particular, when n = 1, these solutions constitute a generalization of that obtained by Yuen in [38]. Interestingly, numerical simulations show that the analytical solutions obtained can be used to describe the drifting phenomena of shallow water flows. In addition, the method proposed can be extended to other mathematical physics models such as higher-dimensional Hunter-Saxton equations and Degasperis-Procesi equations.

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