Wenguang Cheng, Ji Lin. Symmetry of traveling wave solutions for a Camassa-Holm type equation with higher-order nonlinearity[J]. Communications in Theoretical Physics, 2025, 77(7): 075002. DOI: 10.1088/1572-9494/ada913
1. Introduction
In this paper, we address the relation between symmetric waves and traveling waves for a Camassa-Holm (CH) type equation with higher-order nonlinearity
which was proposed by Grayshan and Himonas in [1]. Here $k\in {{\mathbb{N}}}^{+}$, $b\in {\mathbb{R}}$. The physical background of this relation is that of irrotational two-dimensional gravity water waves for a perfect fluid over a flat bottom [2]. This equation possesses peakon solutions [1] of the form $u(t,x)={c}^{\frac{1}{k}}{{\rm{e}}}^{-| x-ct| }$, c > 0. The local well-posedness, ill-posedness, blow-up criterions, global existence of strong and weak solutions, global analyticity, persistence properties, and asymptotic behavior of solutions to this equation have been investigated in [3-11]. Equation (1.1) has the nonlinearity of degree k + 1 and includes three integrable members, namely the CH equation (k = 1, b = 2)
The CH equation (1.2) was originally obtained by Fuchssteiner and Fokas [12] as an abstract bi-Hamiltonian equation with an infinite sequence of conservation laws, and subsequently re-obtained as an approximation of the incompressible Euler equations by Camassa and Holm [13] from physical principles (see [14] for a rigorous derivation). Moreover, the CH equation is a re-expression of the geodesic flow on the diffeomorphism group with the kinetic energy metric induced by the H1-norm [15], which indicates that it possesses the H1-norm conservation law. The CH equation has received much attention, and there have been numerous results on the well-posedness, blow-up via wave breaking, orbital stability and asymptotic stability of the peakon solutions, persistence properties, symmetry of traveling wave solutions, analyticity, Gevrey regularity, etc. [16-29].
The DP equation (1.3) was discovered by Degasperis and Procesi [30] via the method of asymptotic integrability. The DP equation has a similar structure and shares a lot of properties with the famous CH equation. It is also an approximation to the incompressible Euler equations, and its asymptotic accuracy is the same as for the CH equation [14]. It is completely integrable and admits the same explicit peakon solutions [31]. Both CH and DP equations can exhibit the wave-breaking phenomenon [19, 20, 32]. However, in contrast to the CH equation, the DP equation has shock peaked waves [33] of the form
Another important difference between these two equations is that the CH equation conserves the H1-norm of solutions while the DP equation does not [32].
The Novikov equation (1.4) is an integrable peakon model with cubic nonlinearity and can be viewed as a generalization of the CH equation or the DP equation. This equation was first derived by Novikov [34] from the classification of generalized CH-type equations having infinite hierarchies of higher symmetries. Recently it was re-derived in the context of shallow water waves [35]. After its derivation, the qualitative properties of solutions to the Novikov equation have been investigated intensively, such as the local well-posedness [36-39], global existence of strong and weak solutions [38, 40-42], blow-up [43, 44], orbital stability and asymptotic stability of the peakon solutions [45, 46].
Inspired by the work in [26], the purpose of this paper is to prove that horizontally symmetric weak solutions of equation (1.1) are traveling wave solutions. A key step of the proof is to give the form of a weak solution of (1.1), cf. Lemma 2.1. The detailed proof is given in the following section.
2. Symmetry of traveling wave solutions
In this section, we shall consider a unique x-symmetric weak solution of equation (1.1). Following [26], we will prove that such a solution must be a traveling wave. First, we recall the definitions of an x-symmetric solution and a weak solution of (1.1).
[26] A solution u is x-symmetric if there exists a function $a(t)\in {C}^{1}({{\mathbb{R}}}_{+})$ such that
where we have used (2.3) with φ(x) = Φc(t, x), which belongs to ${C}_{0}^{\infty }({\mathbb{R}})$, for any given t≥0. This finishes the proof of the lemma.
Next, we give our main result.
Let u(t, x) be x-symmetric. If u(t, x) is a unique weak solution of equation (1.1), then u(t, x) is a traveling wave.
Since ${C}_{0}^{\infty }({{\mathbb{R}}}_{+}\times {\mathbb{R}})$ is dense in ${C}_{0}^{1}({{\mathbb{R}}}_{+},{C}_{0}^{3}({\mathbb{R}}))$, we only consider the test function Φ in Definition 2.2 belonging to ${C}_{0}^{1}({{\mathbb{R}}}_{+},{C}_{0}^{3}({\mathbb{R}}))$. We denote
We introduce a mollifier ${\rho }_{\epsilon }\in {C}_{0}^{\infty }({{\mathbb{R}}}_{+})$ with ρε → δ(t - t0), the Dirac mass at a fixed t0, as ε → 0. Then we define the following test function
It is clear that u(t0, x) satisfies (2.3) with $c=\dot{a}({t}_{0})$. Lemma 2.1 indicates that $\tilde{u}(t,x)=u({t}_{0},x-\dot{a}({t}_{0})(t-{t}_{0}))$ is a traveling wave solution of equation (1.1). Since $\tilde{u}({t}_{0},x)=u({t}_{0},x)$, from the uniqueness assumption of the solution, we know that $\tilde{u}(t,x)=u(t,x)$ for all t > 0. The proof of Theorem 2.1 is hence completed.
Theorem 2.1 covers and generalizes the related results of the CH equation and the Novikov equation in [26, 47].
The work of WC is partially supported by the National Natural Science Foundation of China (Grant No. 12201417), and the Project funded by the China Postdoctoral Science Foundation (Grant No. 2023M733173). The work of JL is partially supported by the National Natural Science Foundation of China (Grant No. 12375006).
WeiL, QiaoZ J, WangY, ZhouS M2017 Conserved quantities, global existence and blow-up for a generalized Camassa-Holm equation Discrete Contin. Dyn. Syst.37 1733 1748
BarostichiR F, HimonasA A, PetronilhoG2017 Global analyticity for a generalized Camassa-Holm equation and decay of the radius of spatial analyticity J. Differ. Equ.263 732 764
HimonasA A, MisiołekG, PonceG, ZhouY2007 Persistence properties and unique continuation of solutions of the Camassa-Holm equation Comm. Math. Phys.271 511 522
ChenG, ChenR M, LiuY2018 Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation Indiana Univ. Math. J.67 2393 2433
ChenR M, LianW, WangD H, XuR Z2021 A rigidity property for the Novikov equation and the asymptotic stability of peakons Arch. Ration. Mech. Anal.241 497 533