Singularities in finite time of a 3-component Camassa–Holm equations

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Abstract

Wave breaking (bounded solutions with unbounded derivative) of the hyperbolic equation is important and interesting to physicist and mathematician. In this article, thanks to the conservation law, we derive some new wave breakings of a class of strong solutions for a 3-component Camassa–Holm equation.

Introduction

In this article, we devote to studying the wave breaking for the Cauchy problem of a 3-component Camassa–Holm (CH) equation tζζxu+2ζux+(ζv+ζw)x+ηvx+ςwx=0,tηηxv+2ηvx+(ηu+ηw)x+ζux+ςwx=0,tςςxw+2ςwx+(ςu+ςv)x+ζux+ηvx=0,which was introduced by Qu and Fu in [1] to study multipeakons, where the potential ζ=uuxx, η=vvxx and ς=wwxx, (t,x)R+×R, and the subscripts denote the partial derivatives.

If we choose the functions v=w=0, then system (1.1) becomes the following classical CH equation tζ+uζx+2uxζ=0,ζ=uuxx,which comes from an asymptotic approximation to the Hamiltonian for the Green–Naghdi equations in shallow water theory. The CH equation models the unidirectional propagation of shallow water waves over a flat bottom [2], [3], and also is a model for the propagation of axially symmetric waves in hyperelastic rods [4]. It has a bi-Hamiltonian structure [5] and is completely integrable [6], and with a Lax pair based on a linear spectral problem of second order. Also, there exist smooth soliton solutions on a non-zero constant background [7]. Compared with KdV equation, the CH equation not only approximates unidirectional fluid flow in Euler’s equations [8] at the next order beyond the KdV equation but also there exist blow-up phenomena of the strong solution and global existence of strong solution [9], [10], [11], [12]. It is remarkable that the CH equation has peaked solitons of the form u(t,x)=ce|xct|,cR [7], which are orbital stable [13], and n-peakon solutions [14] u(t,x)=j=1npj(t)exp(|xqj(t)|), where the positions qj and amplitudes pj satisfy the system of ODEs qj̇=k=1npkexp(|qjqk|),pj̇=pjk=1npksgn(qjqk)exp(|qjqk|),where j=1,,n. The CH equation has attracted a lot of interest in the past twenty years for various reasons [6], [10], [13], [15], [16], [17], [18], [19].

In 2012, Hu etc. considered the Cauchy problem of equation (1.1) [20]. By the Kato’s semigroup theory, they established the local well-posedness in Hs(R), then they obtained precise blow-up scenario and the conservation law by energy estimates. In particular, assume the derivative of initial data is negative, they showed that the strong solution blows up in finite time and derive the blow-up rate of blow-up solution. In article [21], by the theory of transport equations, Wu studied the well-posedness of Eq. (1.1) in critical Besov space B2,132, he also showed the exponential decay of strong solution and derived a class of traveling wave solutions to Eq. (1.1). Moreover, he got a new blow-up phenomena of Eq. (1.1). In this paper, we will give some new wave breaking of a family of solution to system (1.1), if there exists a constant M>0 such that the initial data satisfy R(u0x+v0x+w0x)2n+1dx<M,for nN/{0}. Thus we obtain a class of blow-up solutions for different n of system (1.1), which covers the blow-up results in [21].

Section snippets

Wave breaking of system (1.1)

In this section, basing on the conservation law of strong solution and blow-up scenario of Eq. (1.1), our aim is to investigate the wave breaking of system (1.1). Comparing with the two results of blow-up phenomena which are obtained in [20], we give another new wave-breaking of a family of solutions. With the potential functions ζ=uuxx, η=vvxx and ς=wwxx, it is easy to transform the system (1.1) into its formally equivalent differential form tu+(u+v+w)ux+(1x2)1(uvx+uwx)+x(1x2)1F=0,t

Acknowledgments

This work was partially supported by NSFC, PR China (Grant No.: 11771442) and the Fundamental Research Funds for the Central University, PR China (WUT: 2021III056JC). The authors thank the professor Boling Guo for his helpful discussions and constructive suggestions.

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