Perturbational self-similar solutions for the 2-component Degasperis--Procesi system via a characteristic method

An, Hongli; Cheung, Ka Luen; Yuen, Manwai

doi:10.3906/mat-1509-83

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The Second Type Reduction: a A A Is A Decomposable T T T-dependent Matrix1

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2021

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“…where α i is governed by the generalized Emden equation with C i is the integration constant. We find that it is just a kind of elliptically symmetric solutions generalized what are obtained by An and Yuen [41]. In order to show the behaviors the solution may posses, we present the numerical simulations when N = 2.…”

Section: The Second Type Reduction: a A A Is A Decomposable T T T-dependent Matrixmentioning

confidence: 54%

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

An¹,

Hou²,

Yuen

2021

JNMP

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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 i 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.

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Section: The Second Type Reduction: a A A Is A Decomposable T T T-dependent Matrixmentioning

confidence: 54%

“…We notice that most existing papers deal with the case n = 1 or 2. Investigations on the multicomponent CH system mentioned above are rare, expect for the work done in [11,34,40,41]. In particular, little work has been done on seeking exact solutions.…”

Section: Introductionmentioning

confidence: 99%