Welcome to visit Communications in Theoretical Physics,
Mathematical Physics

Symmetry of traveling wave solutions for a Camassa-Holm type equation with higher-order nonlinearity

  • Wenguang Cheng , 1, 2 ,
  • Ji Lin , 1, *
Expand
  • 1Department of Physics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
  • 2Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, China

*Author to whom any correspondence should be addressed.

Received date: 2024-09-11

  Revised date: 2024-12-16

  Accepted date: 2025-01-13

  Online published: 2025-03-27

Copyright

© 2025 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing. All rights, including for text and data mining, AI training, and similar technologies, are reserved.
This article is available under the terms of the IOP-Standard License.

Cite this article

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
$\begin{eqnarray}{u}_{t}-{u}_{txx}+(b+1){u}^{k}{u}_{x}-b{u}^{k-1}{u}_{x}{u}_{xx}-{u}^{k}{u}_{xxx}=0,\end{eqnarray}$
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)
$\begin{eqnarray}{u}_{t}-{u}_{txx}+3u{u}_{x}-2{u}_{x}{u}_{xx}-u{u}_{xxx}=0,\end{eqnarray}$
the Degasperis-Procesi (DP) equation (k = 1, b = 3)
$\begin{eqnarray}{u}_{t}-{u}_{txx}+4u{u}_{x}-3{u}_{x}{u}_{xx}-u{u}_{xxx}=0,\end{eqnarray}$
and the Novikov equation (k = 2, b = 3)
$\begin{eqnarray}{u}_{t}-{u}_{txx}+4{u}^{2}{u}_{x}-3u{u}_{x}{u}_{xx}-{u}^{2}{u}_{xxx}=0.\end{eqnarray}$
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
$\begin{eqnarray*}u(t,x)=-\frac{1}{t+k}\mathrm{sgn}(x){{\rm{e}}}^{-| x| },\quad k\gt 0.\end{eqnarray*}$
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

$\begin{eqnarray*}u(t,x)=u(t,2a(t)-x),\quad \forall \,t\in (0,\infty ),\end{eqnarray*}$
for almost every $x\in {\mathbb{R}}$. We call a(t) the symmetric axis of u.

We say u is a weak solution of equation (1.1) if $u\in C({{\mathbb{R}}}_{+},{H}^{1}({\mathbb{R}}))$ and it satisfies

$\begin{eqnarray}\begin{array}{l}\displaystyle \int {\displaystyle \int }_{{{\mathbb{R}}}_{+}\times {\mathbb{R}}}\left[u(1-{\partial }_{x}^{2}){\phi }_{t}-\frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3}\phi \right.\\ \quad +\left(\frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2}\right){\phi }_{x}\\ \quad \,\left.-\frac{1}{k+1}{u}^{k+1}{\phi }_{xxx}\right]{\rm{d}}t{\rm{d}}x=0,\end{array}\end{eqnarray}$
for all $\phi \in {C}_{0}^{\infty }({{\mathbb{R}}}_{+}\times {\mathbb{R}})$.

Further introducing the notation $\langle$ ·,· $\rangle$ for distributions, (2.1) becomes
$\begin{eqnarray}\begin{array}{l}\left\langle u,(1-{\partial }_{x}^{2}){\phi }_{t}\right\rangle -\left\langle \frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3},\phi \right\rangle \\ +\left\langle \frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2},{\phi }_{x}\right\rangle \\ -\left\langle \frac{1}{k+1}{u}^{k+1},{\phi }_{xxx}\right\rangle =0.\end{array}\end{eqnarray}$
Now we prove the following lemma.

Assume that $U(x)\in {H}^{1}({\mathbb{R}})$ and it satisfies

$\begin{eqnarray}\begin{array}{l}{\displaystyle \int }_{{\mathbb{R}}}\left[-cU(1-{\partial }_{x}^{2}){\varphi }_{x}-\frac{(b-k)(k-1)}{2}{U}^{k-2}{U}_{x}^{3}\varphi \right.\\ +\left(\frac{b+1}{k+1}{U}^{k+1}+\frac{3k-b}{2}{U}^{k-1}{U}_{x}^{2}\right){\varphi }_{x}\\ \quad \,\left.-\frac{1}{k+1}{U}^{k+1}{\varphi }_{xxx}\right]{\rm{d}}x=0,\end{array}\end{eqnarray}$
for all $\varphi \in {C}_{0}^{\infty }({\mathbb{R}})$. Then we obtain a weak solution of (1.1) in the following form
$\begin{eqnarray}u(t,x)=U(x-c(t-{t}_{0})),\end{eqnarray}$
for a fixed ${t}_{0}\in {\mathbb{R}}$.

We set without loss of generality that t0 = 0. We follow the arguments in [26] to get $u\in C({\mathbb{R}},{H}^{1}({\mathbb{R}}))$. Next we define

$\begin{eqnarray*}{\phi }_{c}(t,x)=\phi (t,x+ct),\end{eqnarray*}$
where $\phi \in {C}_{0}^{\infty }({{\mathbb{R}}}_{+}\times {\mathbb{R}})$. Then one has
$\begin{eqnarray}{\partial }_{x}({\phi }_{c})={({\phi }_{x})}_{c},\quad {\partial }_{t}({\phi }_{c})={({\phi }_{t})}_{c}+c{({\phi }_{x})}_{c}.\end{eqnarray}$
Suppose u(t, x) = U(x - ct), we have
$\begin{eqnarray}\begin{array}{rcl}\langle u,\phi \rangle & = & \langle U,{\phi }_{c}\rangle ,\quad \left\langle {u}^{k+1},\phi \right\rangle =\left\langle {U}^{k+1},{\phi }_{c}\right\rangle ,\\ \left\langle {u}^{k-2}{u}_{x}^{3},\phi \right\rangle & = & \left\langle {U}^{k-2}{U}_{x}^{3},{\phi }_{c}\right\rangle ,\\ \left\langle {u}^{k-1}{u}_{x}^{2},\phi \right\rangle & = & \left\langle {U}^{k-1}{U}_{x}^{2},{\phi }_{c}\right\rangle ,\end{array}\end{eqnarray}$
where U = U(x). From (2.5) and (2.6), we get
$\begin{eqnarray}\begin{array}{l}\left\langle u,(1-{\partial }_{x}^{2}){\phi }_{t}\right\rangle =\left\langle U,{\left((1-{\partial }_{x}^{2}){\phi }_{t}\right)}_{c}\right\rangle \\ =\left\langle U,(1-{\partial }_{x}^{2})({\partial }_{t}{\phi }_{c}-c{\partial }_{x}{\phi }_{c})\right\rangle ,\\ \left\langle \frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3},\phi \right\rangle \\ =\left\langle \frac{(b-k)(k-1)}{2}{U}^{k-2}{U}_{x}^{3},{\phi }_{c}\right\rangle ,\\ \left\langle \frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2},{\phi }_{x}\right\rangle \\ =\left\langle \frac{b+1}{k+1}{U}^{k+1}+\frac{3k-b}{2}{U}^{k-1}{U}_{x}^{2},{\partial }_{x}{\phi }_{c}\right\rangle ,\\ \left\langle \frac{1}{k+1}{u}^{k+1},{\phi }_{xxx}\right\rangle =\left\langle \frac{1}{k+1}{U}^{k+1},{({\phi }_{xxx})}_{c}\right\rangle \\ =\left\langle \frac{1}{k+1}{U}^{k+1},{\partial }_{x}^{3}{\phi }_{c}\right\rangle .\end{array}\end{eqnarray}$
Since U is independent of time, so it does not belong to the support of Φc for sufficiently large T. This implies
$\begin{eqnarray}\begin{array}{rcl}\left\langle U,(1-{\partial }_{x}^{2}){\partial }_{t}{\phi }_{c}\right\rangle & = & {\displaystyle \int }_{{\mathbb{R}}}U(x){\displaystyle \int }_{{{\mathbb{R}}}_{+}}{\partial }_{t}(1-{\partial }_{x}^{2}){\phi }_{c}\,{\rm{d}}t{\rm{d}}x\\ & & ={\displaystyle \int }_{{\mathbb{R}}}U(x)\left[(1-{\partial }_{x}^{2}){\phi }_{c}(T,x)\right.\\ & & \left.-(1-{\partial }_{x}^{2}){\phi }_{c}(0,x)\right]\,{\rm{d}}x=0,\\ \langle U,{\partial }_{t}{\phi }_{c}\rangle & = & {\displaystyle \int }_{{\mathbb{R}}}U(x){\displaystyle \int }_{{{\mathbb{R}}}_{+}}{\partial }_{t}{\phi }_{c}\,{\rm{d}}t{\rm{d}}x\\ & = & {\displaystyle \int }_{{\mathbb{R}}}U(x)\left[{\phi }_{c}(T,x)-{\phi }_{c}(0,x)\right]\,{\rm{d}}x=0.\end{array}\end{eqnarray}$
Then, (2.7) and (2.8) yield
$\begin{eqnarray*}\begin{array}{l}\,\,\left\langle u,(1-{\partial }_{x}^{2}){\phi }_{t}\right\rangle -\left\langle \frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3},\phi \right\rangle \\ +\left\langle \frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2},{\phi }_{x}\right\rangle \\ -\left\langle \frac{1}{k+1}{u}^{k+1},{\phi }_{xxx}\right\rangle \\ =\left\langle U,-c(1-{\partial }_{x}^{2}){\partial }_{x}{\phi }_{c}\right\rangle -\left\langle \frac{(b-k)(k-1)}{2}{U}^{k-2}{U}_{x}^{3},{\phi }_{c}\right\rangle \\ +\left\langle \frac{b+1}{k+1}{U}^{k+1}+\frac{3k-b}{2}{U}^{k-1}{U}_{x}^{2},{\partial }_{x}{\phi }_{c}\right\rangle \\ -\left\langle \frac{1}{k+1}{U}^{k+1},{\partial }_{x}^{3}{\phi }_{c}\right\rangle ={\displaystyle \int }_{{{\mathbb{R}}}_{+}}{\displaystyle \int }_{{\mathbb{R}}}\left[-cU(1-{\partial }_{x}^{2}){\partial }_{x}{\phi }_{c}\right.\\ -\frac{(b-k)(k-1)}{2}{U}^{k-2}{U}_{x}^{3}{\phi }_{c}\\ +\left(\frac{b+1}{k+1}{U}^{k+1}+\frac{3k-b}{2}{U}^{k-1}{U}_{x}^{2}\right){\partial }_{x}{\phi }_{c}\\ \left.-\frac{1}{k+1}{U}^{k+1}{\partial }_{x}^{3}{\phi }_{c}\right]{\rm{d}}x{\rm{d}}t=0,\end{array}\end{eqnarray*}$
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

$\begin{eqnarray*}{\phi }_{a}(t,x)=\phi (t,2a(t)-x),\quad a(t)\in {C}^{1}({\mathbb{R}}).\end{eqnarray*}$
It is easy to see that ${({\phi }_{a})}_{a}=\phi $ and
$\begin{eqnarray}\begin{array}{l}{\partial }_{x}{u}_{a}=-{({\partial }_{x}u)}_{a},\quad {\partial }_{x}{\phi }_{a}=-{({\partial }_{x}\phi )}_{a},\\ {\partial }_{t}{\phi }_{a}={({\partial }_{t}\phi )}_{a}+2\dot{a}{({\partial }_{x}\phi )}_{a},\end{array}\end{eqnarray}$
where $\dot{a}$ represents the time derivative of a(t). Moreover, we deduce that
$\begin{eqnarray}\begin{array}{l}\langle {u}_{a},\phi \rangle =\langle u,{\phi }_{a}\rangle ,\quad \left\langle {u}_{a}^{k-2}{({\partial }_{x}{u}_{a})}^{3},\phi \right\rangle =-\left\langle {u}^{k-2}{u}_{x}^{3},{\phi }_{a}\right\rangle ,\\ \left\langle {u}_{a}^{k+1},\phi \right\rangle =\left\langle {u}^{k+1},{\phi }_{a}\right\rangle ,\quad \left\langle {u}_{a}^{k-1}{({\partial }_{x}{u}_{a})}^{2},\phi \right\rangle =\left\langle {u}^{k-1}{u}_{x}^{2},{\phi }_{a}\right\rangle .\end{array}\end{eqnarray}$
Since u is x-symmetric, it follows from (2.9) and (2.10) that
$\begin{eqnarray}\begin{array}{l}\left\langle u,(1-{\partial }_{x}^{2}){\phi }_{t}\right\rangle =\left\langle u,{\left((1-{\partial }_{x}^{2}){\partial }_{t}\phi \right)}_{a}\right\rangle \\ =\left\langle u,(1-{\partial }_{x}^{2})({\partial }_{t}{\phi }_{a}+2\dot{a}{\partial }_{x}{\phi }_{a})\right\rangle ,\\ \left\langle \frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3},\phi \right\rangle \\ =-\left\langle \frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3},{\phi }_{a}\right\rangle ,\\ \left\langle \frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2},{\phi }_{x}\right\rangle \\ =-\left\langle \frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2},{\partial }_{x}{\phi }_{a}\right\rangle ,\\ \left\langle \frac{1}{k+1}{u}^{k+1},{\phi }_{xxx}\right\rangle =-\left\langle \frac{1}{k+1}{u}^{k+1},{\partial }_{x}^{3}{\phi }_{a}\right\rangle .\end{array}\end{eqnarray}$
From (2.2) we have
$\begin{eqnarray}\begin{array}{l}\left\langle u,(1-{\partial }_{x}^{2}){\phi }_{t}\right\rangle -\left\langle \frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3},\phi \right\rangle \\ +\left\langle \frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2},{\phi }_{x}\right\rangle -\left\langle \frac{1}{k+1}{u}^{k+1},{\phi }_{xxx}\right\rangle \\ =\left\langle u,(1-{\partial }_{x}^{2})({\partial }_{t}{\phi }_{a}+2\dot{a}{\partial }_{x}{\phi }_{a})\right\rangle +\left\langle \frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3},{\phi }_{a}\right\rangle \\ -\left\langle \frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2},{\partial }_{x}{\phi }_{a}\right\rangle \\ +\left\langle \frac{1}{k+1}{u}^{k+1},{\partial }_{x}^{3}{\phi }_{a}\right\rangle =0.\end{array}\end{eqnarray}$
Owing to ${({\phi }_{a})}_{a}=\phi $, we take Φ = Φa in (2.12) to obtain
$\begin{eqnarray*}\begin{array}{l}\left\langle u,(1-{\partial }_{x}^{2})({\partial }_{t}\phi +2\dot{a}{\partial }_{x}\phi )\right\rangle +\left\langle \frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3},\phi \right\rangle \\ \quad -\left\langle \frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2},{\partial }_{x}\phi \right\rangle \\ \quad +\left\langle \frac{1}{k+1}{u}^{k+1},{\partial }_{x}^{3}\phi \right\rangle =0,\end{array}\end{eqnarray*}$
which together with (2.2) gives
$\begin{eqnarray}\begin{array}{l}\left\langle u,\dot{a}(1-{\partial }_{x}^{2}){\partial }_{x}\phi \right\rangle +\left\langle \frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3},\phi \right\rangle \\ \quad \,-\left\langle \frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2},{\partial }_{x}\phi \right\rangle \\ \quad \,+\left\langle \frac{1}{k+1}{u}^{k+1},{\partial }_{x}^{3}\phi \right\rangle =0.\end{array}\end{eqnarray}$
We introduce a mollifier ${\rho }_{\epsilon }\in {C}_{0}^{\infty }({{\mathbb{R}}}_{+})$ with ρεδ(t - t0), the Dirac mass at a fixed t0, as ε → 0. Then we define the following test function
$\begin{eqnarray*}{\phi }_{\epsilon }(t,x)=\varphi (x){\rho }_{\epsilon }(t),\quad {\rm{forany}}\,\varphi \in {C}_{0}^{\infty }({\mathbb{R}}).\end{eqnarray*}$
Substituting the above test function into (2.13) gives rise to
$\begin{eqnarray}\begin{array}{l}{\displaystyle \int }_{{\mathbb{R}}}\left[(1-{\partial }_{x}^{2}){\partial }_{x}\varphi {\displaystyle \int }_{{{\mathbb{R}}}_{+}}\dot{a}u{\rho }_{\epsilon }(t)\,{\rm{d}}t\right]{\rm{d}}x\\ +{\displaystyle \int }_{{\mathbb{R}}}\left[\varphi {\displaystyle \int }_{{{\mathbb{R}}}_{+}}\frac{(b-k)(k-1)}{2}{u}^{k-2}{u}_{x}^{3}{\rho }_{\epsilon }(t)\,{\rm{d}}t\right]{\rm{d}}x\\ \quad -{\displaystyle \int }_{{\mathbb{R}}}\left[{\partial }_{x}\varphi {\displaystyle \int }_{{{\mathbb{R}}}_{+}}\left(\frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2}\right){\rho }_{\epsilon }(t)\,{\rm{d}}t\right]{\rm{d}}x\\ +\frac{1}{k+1}{\displaystyle \int }_{{\mathbb{R}}}\left({\partial }_{x}^{3}\varphi {\displaystyle \int }_{{{\mathbb{R}}}_{+}}{u}^{k+1}{\rho }_{\epsilon }(t)\,{\rm{d}}t\right){\rm{d}}x=0.\end{array}\end{eqnarray}$
Note that
$\begin{eqnarray*}\mathop{\mathrm{lim}}\limits_{\epsilon \to 0}{\int }_{{{\mathbb{R}}}_{+}}\dot{a}u{\rho }_{\epsilon }(t)\,{\rm{d}}t=\dot{a}({t}_{0})u({t}_{0},x),\quad {\rm{in}}\,{L}^{2}({\mathbb{R}}),\end{eqnarray*}$
and
$\begin{eqnarray*}\begin{array}{l}\mathop{\mathrm{lim}}\limits_{\epsilon \to 0}{\displaystyle \int }_{{{\mathbb{R}}}_{+}}{u}^{k-2}{u}_{x}^{3}{\rho }_{\epsilon }(t)\,{\rm{d}}t={u}^{k-2}({t}_{0},x){u}_{x}^{3}({t}_{0},x),\\ \mathop{\mathrm{lim}}\limits_{\epsilon \to 0}{\displaystyle \int }_{{{\mathbb{R}}}_{+}}\left(\frac{b+1}{k+1}{u}^{k+1}+\frac{3k-b}{2}{u}^{k-1}{u}_{x}^{2}\right){\rho }_{\epsilon }(t)\,{\rm{d}}t\\ \quad =\frac{b+1}{k+1}{u}^{k+1}({t}_{0},x)+\frac{3k-b}{2}{u}^{k-1}({t}_{0},x){u}_{x}^{2}({t}_{0},x),\\ \mathop{\mathrm{lim}}\limits_{\epsilon \to 0}{\displaystyle \int }_{{{\mathbb{R}}}_{+}}{u}^{k+1}{\rho }_{\epsilon }(t)\,{\rm{d}}t={u}^{k+1}({t}_{0},x),\end{array}\end{eqnarray*}$
in ${L}^{1}({\mathbb{R}})$. So, letting ε → 0 in (2.14), one gets
$\begin{eqnarray*}\begin{array}{l}{\displaystyle \int }_{{\mathbb{R}}}\dot{a}({t}_{0})u({t}_{0},x)(1-{\partial }_{x}^{2}){\partial }_{x}\varphi \,{\rm{d}}x+\frac{(b-k)(k-1)}{2}\\ \quad \,{\displaystyle \int }_{{\mathbb{R}}}{u}^{k-2}({t}_{0},x){u}_{x}^{3}({t}_{0},x)\varphi \,{\rm{d}}x\\ \quad -{\displaystyle \int }_{{\mathbb{R}}}\left(\frac{b+1}{k+1}{u}^{k+1}({t}_{0},x)+\frac{3k-b}{2}{u}^{k-1}({t}_{0},x){u}_{x}^{2}({t}_{0},x)\right){\partial }_{x}\varphi \,{\rm{d}}x\\ \quad \,+\frac{1}{k+1}{\displaystyle \int }_{{\mathbb{R}}}{u}^{k+1}({t}_{0},x){\partial }_{x}^{3}\varphi \,{\rm{d}}x=0.\end{array}\end{eqnarray*}$
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).

1
Grayshan K, Himonas A A 2013 Equations with peakon traveling wave solutions Adv. Dyn. Syst. Appl. 8 217 232

2
Johnson R S 1997 A Modern Introduction to the Mathematical Theory of Water Waves Cambridge Cambridge University Press

3
Himonas A A, Holliman C 2014 The Cauchy problem for a generalized Camassa-Holm equation Adv. Differ. Equ. 19 161 200

DOI

4
Himonas A A, Thompson R C 2014 Persistence properties and unique continuation for a generalized Camassa-Holm equation, J. Math. Phys. 55 091503

DOI

5
Li J L, Yu Y H, Zhu W P 2022 Sharp ill-posedness for the generalized Camassa-Holm equation in Besov spaces, J. Evol. Equ. 22 29

DOI

6
Chen L N, Guan C X 2021 Global solutions for the generalized Camassa-Holm equation Nonlinear Anal. Real World Appl. 58 103227

DOI

7
Yan K 2019 Wave breaking and global existence for a family of peakon equations with high order nonlinearity Nonlinear Anal. Real World Appl. 45 721 735

DOI

8
Wei L, Qiao Z J, Wang Y, Zhou S M 2017 Conserved quantities, global existence and blow-up for a generalized Camassa-Holm equation Discrete Contin. Dyn. Syst. 37 1733 1748

DOI

9
Guo Z G, Li X G, Yu C 2018 Some properties of solutions to the Camassa-Holm-type equation with higher-order nonlinearities J. Nonlinear Sci. 28 1901 1914

DOI

10
Guo Z G, Li K Q, Xu C B 2018 On a generalized Camassa-Holm type equation with (k + 1)-degree nonlinearities Z. Angew. Math. Mech. 98 1567 1573

DOI

11
Barostichi R F, Himonas A A, Petronilho G 2017 Global analyticity for a generalized Camassa-Holm equation and decay of the radius of spatial analyticity J. Differ. Equ. 263 732 764

DOI

12
Fuchssteiner B, Fokas A S 1981 Symplectic structures, their Bäcklund transformations and hereditary symmetries Phys. D 4 47 66

DOI

13
Camassa R, Holm D D 1993 An integrable shallow water equation with peaked solitons Phys. Rev. Lett. 71 1661 1664

DOI

14
Constantin A, Lannes D 2009 The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations Arch. Ration. Mech. Anal. 192 165 186

DOI

15
Kouranbaeva S 1999 The Camassa-Holm equation as a geodesic flow on the diffeomorphism group J. Math. Phys. 40 857 868

DOI

16
Brandolese L 2012 Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces Int. Math. Res. Not. 2012 5161 5181

DOI

17
Brandolese L 2014 Local-in-space criteria for blowup in shallow water and dispersive rod equations Comm. Math. Phys. 330 401 414

DOI

18
Constantin A 2000 Existence of permanent and breaking waves for a shallow water equation: a geometric approach Ann. Inst. Fourier (Grenoble) 50 321 362

DOI

19
Constantin A, Escher J 1998 Global existence and blow-up for a shallow water equation Ann. Sc. Norm. Super. Pisa Cl. Sci. 26 303 328

20
Constantin A, Escher J 1998 Wave breaking for nonlinear nonlocal shallow water equations Acta Math. 181 229 243

DOI

21
Danchin R 2001 A few remarks on the Camassa-Holm equation Differ. Integral Equ. 14 953 988

DOI

22
Li Y A, Olver P J 2000 Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation J. Differ. Equ. 162 27 63

DOI

23
Constantin A, Strauss W A 2000 Stability of peakons Comm. Pure Appl. Math. 53 603 610

DOI

24
Molinet L 2018 A Liouville property with application to asymptotic stability for the Camassa-Holm equation Arch. Ration. Mech. Anal. 230 185 230

DOI

25
Himonas A A, Misiołek G, Ponce G, Zhou Y 2007 Persistence properties and unique continuation of solutions of the Camassa-Holm equation Comm. Math. Phys. 271 511 522

DOI

26
Ehrnström M, Holden H, Raynaud X 2009 Symmetric waves are traveling waves Int. Math. Res. Not. 2009 4578 4596

DOI

27
Himonas A A, Misiołek G 2003 Analyticity of the Cauchy problem for an integrable evolution equation Math. Ann. 327 575 584

DOI

28
Luo W, Yin Z Y 2018 Gevrey regularity and analyticity for Camassa-Holm type systems Ann. Sc. Norm. Super. Pisa Cl. Sci. 18 1061 1079

DOI

29
Xin Z P, Zhang P 2000 On the weak solutions to a shallow water equation Comm. Pure Appl. Math. 53 1411 1433

DOI

30
Degasperis A, Procesi M 1999 Asymptotic integrability Symmetry and Perturbation Theory Degasperis A, Gaeta G Singapore World Scientific

31
Degasperis A, Holm D D, Hone A N W 2002 A new integrable equation with peakon solutions Theoret. Math. Phys. 133 1463 1474

DOI

32
Liu Y, Yin Z Y 2006 Global existence and blow-up phenomena for the Degasperis-Procesi equation Comm. Math. Phys. 267 801 820

DOI

33
Lundmark H 2007 Formation and dynamics of shock waves in the Degasperis-Procesi equation J. Nonlinear Sci. 17 169 198

DOI

34
Novikov V 2009 Generalizations of the Camassa-Holm equation J. Phys. A: Math. Theor. 42 342002

DOI

35
Chen R M, Hu T Q, Liu Y 2022 The shallow-water models with cubic nonlinearity J. Math. Fluid Mech. 24 49

DOI

36
Ni L D, Zhou Y 2011 Well-posedness and persistence properties for the Novikov equation J. Differ. Equ. 250 3002 3021

DOI

37
Himonas A A, Holliman C 2012 The Cauchy problem for the Novikov equation Nonlinearity 25 449 479

DOI

38
Wu X L, Yin Z Y 2012 Well-posedness and global existence for the Novikov equation Ann. Sc. Norm. Super. Pisa Cl. Sci. 11 707 727

DOI

39
Yan W, Li Y S, Zhang Y M 2012 The Cauchy problem for the integrable Novikov equation J. Differ. Equ. 253 298 318

DOI

40
Wu X L, Yin Z Y 2011 Global weak solutions for the Novikov equation J. Phys. A: Math. Theor. 44 055202

DOI

41
Lai S Y 2013 Global weak solutions to the Novikov equation J. Funct. Anal. 265 520 544

DOI

42
Chen G, Chen R M, Liu Y 2018 Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation Indiana Univ. Math. J. 67 2393 2433

DOI

43
Jiang Z H, Ni L D 2012 Blow-up phenomenon for the integrable Novikov equation J. Math. Anal. Appl. 385 551 558

DOI

44
Yan W, Li Y S, Zhang Y M 2013 The Cauchy problem for the Novikov equation Nonlinear Differ. Equ. Appl. 20 1157 1169

DOI

45
Liu X C, Liu Y, Qu C Z 2014 Stability of peakons for the Novikov equation J. Math. Pures Appl. 101 172 187

DOI

46
Chen R M, Lian W, Wang D H, Xu R Z 2021 A rigidity property for the Novikov equation and the asymptotic stability of peakons Arch. Ration. Mech. Anal. 241 497 533

DOI

47
Wu X L Global analytic solutions and traveling wave solutions of the Cauchy problem for the Novikov equation Proc. Amer. Math. Soc. 146 1537 1550

DOI

Outlines

/