Analytical Cartesian solutions of the 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 = ( αi α i ) 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.

Locations

• Journal of Nonlinear Mathematical Physics - View - PDF