Abstract
The aim of this paper is to study singular dynamics of solutions of Camassa-Holm equation. Based on the semigroup theory of linear operators and Banach contraction mapping principle, we prove the asymptotic stability of the explicit singular solution of Camassa-Holm equation.
Highlights
Consider the well-known Camassa-Holm equation as follows: mt + c0ux + umx + 2mux = 0, (1.1)where (t, x) ∈ + ×, u = u (t, x) is the velocity of fluid, m is the momentum given by=m m= (t, x) u (t, x) −α 2uxx (t, x), c0 ∈ is the critical speed and α ∈ relates to the length scale
Based on the semigroup theory of linear operators and Banach contraction mapping principle, we prove the asymptotic stability of the explicit singular solution of Camassa-Holm equation
Since it is rare to see the explicit stable blowup solutions of Camassa-Holm equation, in this paper, we study the stability of the explicit solution of (1.2) as follows: u (t, x)
Summary
Consider the well-known Camassa-Holm equation as follows (see [1]): mt + c0ux + umx + 2mux = 0,. Where (t, x) ∈ + × , u = u (t, x) is the velocity of fluid, m is the momentum given by. Given the initial value as u (0, x) = u0 ( x) for x ∈. The Camassa-Holm equation describes unidirectional propagation of surface water waves in shallow water area. For the global well-posedness and stability of solutions, we recommend that the reader refers to [2]-[9], etc. Since it is rare to see the explicit stable blowup solutions of Camassa-Holm equation, in this paper, we study the stability of the explicit solution of (1.2) as follows (see [20]): u (t, x).
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