Analytical Cartesian solutions of the multi-component Camassa-Holm equations

DOI

10.1080/14029251.2019.1591725How to use a DOI?

Keywords

Solution; Analytical Cartesian solution; Camassa-Holm equation; Curve integration theory; Multi-component Camassa-Holm equations

Abstract

Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is given by u = b(t) + A(t)x and no extra constraint on the dimension N is required. The advantage of our method is that we turn the process of analytically solving MCCH equations into algebraically constructing the suitable matrix A(t). As the applications, we obtain some interesting results: 1) If u is a linear transformation on x ∈ ℝ^N, then p takes a quadratic form of x. 2) If A = f(t)I + D with D^T = −D, we obtain the spiral solutions. When N = 2, the solution can be used to describe “breather-type” oscillating motions of upper free surfaces. 3) If A=(α˙iαi)N×N, we obtain the generalized elliptically symmetric solutions. When N = 2, the solution can be used to describe the drifting phenomena of the shallow water flow.

Copyright

© 2019 The Authors. Published by Atlantis and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Hongli An
AU  - Liying Hou
AU  - Manwai Yuen
PY  - 2021
DA  - 2021/01/06
TI  - Analytical Cartesian solutions of the multi-component Camassa-Holm equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 255
EP  - 272
VL  - 26
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1591725
DO  - 10.1080/14029251.2019.1591725
ID  - An2021
ER  -