Generalized Camassa Holm Equation Research Articles

Classification of traveling waves to the generalized Camassa–Holm equation with dual-power nonlinearities

In this paper, we mainly devote to investigate the classification of traveling waves to the generalized Camassa–Holm equation with dual-power nonlinearities. Utilizing the celebrated approach of classification of traveling wave solution which was proposed by Jonatan Lenells [Traveling wave solutions of the Camassa-Holm equation. J Diff Equ. 2005;217(2):393–430]. We show a result concerns the regularity of traveling waves and then classify all traveling wave solutions.

Wave-breaking phenomena for the generalized Camassa–Holm equation with dual-power nonlinearities

In this paper, we mainly devote to investigate the generalized Camassa–Holm equation with dual-power nonlinearities. We first establish the local well-posedness by applying the Kato’s semigroup theory. Then, the precise blow-up result is obtained by using the transport equation theory and Moser-type estimates. Moreover, according to the different real-valued intervals in which the dispersive parameter s is located, the sufficient conditions which guarantee the occurrence of wave-breaking of solutions are studied. It is worth noting that we need to overcome the difficulty caused by complicated mixed dual-power nonlinear structure and balance the relationship between the various dispersive parameters to get corresponding convolution estimates.

Wave-breaking phenomena and Gevrey regularity for the weakly dissipative generalized Camassa–Holm equation

Wave-breaking phenomena and Gevrey regularity for the weakly dissipative generalized Camassa–Holm equation

Lyapunov-type stability criterion for periodic generalized Camassa–Holm equations

We study the Lyapunov stability of the periodic generalized Camassa–Holm equation in terms of the periodic/anti-periodic eigenvalues and the associated spectral intervals. Moreover, we establish a Lyapunov-type stability criterion based on the Floquet theory and a Lyapunov-type inequality.

Local well-posedness and decay for some generalized shallow water equations

Consideration herein is a big class of nonlocal partial differential equations including some classical shallow water models such as b-family equation, Novikov equation, rotation-Camassa-Holm equation, generalized Camassa-Holm equation and Camassa-Holm-Novikov equation over a whole line. Based on the local structure of the dynamics along the characteristics, a local well-posedness result is established for some initial data class containing certain non-smooth solitary wave solutions. Subsequently in this set certain decay persistence properties are also constructed. These partially extend some obtained results and bring the effect of nonlinearities to more extent.

Ill-posedness for a generalized Camassa–Holm equation with higher-order nonlinearity in the critical Besov space

In this paper, we prove that the Cauchy problem for a generalized Camassa–Holm equation with higher-order nonlinearity is ill-posed in the critical Besov space $$B^1_{\infty ,1}(\mathbb {R})$$ . It is shown in (J. Differ. Equ., 327:127-144,2022) that the Camassa–Holm equation is ill-posed in $$B^1_{\infty ,1}(\mathbb {R})$$ , here we turn our attention to a higher-order nonlinear generalization of Camassa–Holm equation proposed by Hakkaev and Kirchev (Commun Partial Differ Equ 30:761-781,2005). With newly constructed initial data, we get the norm inflation in the critical space $$B^1_{\infty ,1}(\mathbb {R})$$ which leads to ill-posedness.

Orbital stability of the sum of N peakons for the mCH-Novikov equation

ABSTRACT This paper investigates a generalized Camassa–Holm equation with cubic nonlinearities (alias the mCH-Novikov equation), which is a generalization of some special equations. The mCH-Novikov equation possesses well-known peaked solitary waves that are called peakons. The peakons were proved orbital stable by Chen et al. in [Stability of peaked solitary waves for a class of cubic quasilinear shallow-water equations. Int Math Res Not. 2022;1–33]. We mainly prove the orbital stability of the multi-peakons in the mCH-Novikov equation. In this paper, it is proved that the sum of N fully decoupled peaks is orbitally stable in the energy space by using energy argument, combining the orbital stability of single peakons and local monotonicity of the method.

Persistence properties and blow-up phenomena for a generalized Camassa–Holm equation

In this paper, we investigate a generalized Camassa–Holm equation. Firstly, we establish the persistence properties of strong solutions for the equation in weighted spaces L^{p}_{phi}=L^{p}(mathbb{R},phi ^{p},dx). Then we present some sufficient conditions of blow-up solutions assuming that the initial data satisfy certain conditions, which are more precise than those in the previous work.

Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities

This paper aims at studying a generalized Camassa–Holm equation under random perturbation. We establish a local well-posedness result in the sense of Hadamard, i.e., existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^s$$\\end{document} with s>3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s>3/2$$\\end{document} for x∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x\\in \\mathbb R$$\\end{document}. The analysis on continuous dependence on initial data for nonlinear stochastic partial differential equations has gained less attention in the literature so far. In this work, we first show that the solution map is continuous. Then we introduce a notion of stability of exiting time. We provide an example showing that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data. Finally, we analyze the regularization effect of nonlinear noise in preventing blow-up. Precisely, we demonstrate that global existence holds true almost surely provided that the noise is strong enough.

Blow-Up Criterion and Persistence Property to a Generalized Camassa–Holm Equation

In this paper, a generalized Camassa–Holm equation, which may be used to describe wave motion in the shallow water, is considered. Some dynamic properties are studied for the model. Firstly, a new blow-up criterion for the equation is established. Then, analytical solutions are presented for the first time by using a new method. Finally, we investigate the persistence property for strong solutions. The results we obtain complement earlier results in this direction.

An energy-preserving finite difference scheme with fourth-order accuracy for the generalized Camassa–Holm equation

In this article, an energy-preserving finite difference scheme for solving the generalized Camassa–Holm (gCH) equation with the dual-power law nonlinearities is proposed. We first show that the solution of the initial–boundary-value gCH equation is unique and continuously dependent on the initial condition, then we construct a linear energy-preserving difference scheme for the gCH equation. The proposed difference scheme is three-level implicit, and the numerical convergence order is O(τ2+h4). The energy conservation, unique solvability, convergence and stability of the finite difference scheme are rigorously proved by using the discrete energy method. Finally, some numerical examples show that the proposed numerical scheme is efficient and reliable.

Blow-up solution and analyticity to a generalized Camassa-Holm equation

<abstract><p>A generalized Camassa-Holm equation with higher order nonlinear term was studied. First, we give a new blow-up criterion and a new blow-up phenomenon to the Cauchy problem for the equation under some conditions. Then, we focus on the analytical solutions for the equation. Finally, we prove the analyticity of solution for the equation.</p></abstract>

Blow-up phenomenon to a generalized Camassa-Holm equation

Blow-up phenomenon to a generalized Camassa-Holm equation

Nonexistence of the CH-type peakon to a generalized Camassa-Holm equation

Nonexistence of the CH-type peakon to a generalized Camassa-Holm equation

Non-uniform Continuity of the Generalized Camassa–Holm Equation in Besov Spaces

In the paper, we gave a strengthening of our previous work in \cite{Li1} (J. Differ. Equ. 269 (2020)) and proved that the data-to-solution map for the Camassa-Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\R)$ with $s>\max\{1+1/{p},3/2\}$ and $(p,r)\in [1,\infty]\times[1,\infty)$. The method applies also to the b-family of equations which contain the Camassa-Holm and Degasperis-Procesi equations.

Stability of periodic peakons for a generalized-μ Camassa–Holm equation with quartic nonlinearities

In this paper, we study the stability of periodic peakons for a generalized-μ-Camassa–Holm (μQCH) equation with quartic nonlinearity. The equation is a μ-version of a generalized Camassa–Holm equation with quartic nonlinearities. First, we derive such equation via variational principle throw a modified Lagrangian, then we show the μQCH equation admits periodic peakons, and finally we prove that the periodic peakons of μQCH equation are orbitally stable under small perturbations in the energy space.

Several Dynamic Properties for the gkCH Equation

In this paper, we focus on a generalized Camassa–Holm equation (also known as a gkCH equation), which includes both the Camassa–Holm equation and Novikov equation as two special cases. Because of the potential applications in physics, we will further investigate the properties of the equation from a mathematical point of view. More precisely, firstly, we give a new wave-breaking phenomenon. Then, we present the theorem of existence and uniqueness of global weak solutions for the equation, provided that the initial data satisfy certain sign conditions. Finally, we prove the Hölder continuity of a solution map for the equation.

Stability of Peakons and Periodic Peakons for the mCH–Novikov–CH Equation

Peakons and periodic peakons are two kinds of special symmetric traveling wave solutions, which have important applications in physics, optical fiber communication, and other fields. In this paper, we study the orbital stability of peakons and periodic peakons for a generalized Camassa–Holm equation with quadratic and cubic nonlinearities (mCH–Novikov–CH equation). It is a generalization of some classical equations, such as the Camassa–Holm (CH) equation, the modified Camassa–Holm (mCH) equation, and the Novikov equation. By constructing an inequality related to the maximum and minimum of solutions with the conservation laws, we prove that the peakons and periodic peakons are orbitally stable under small perturbations in the energy space.

Blow-up phenomena and the local well-posedness and ill-posedness of the generalized Camassa–Holm equation in critical Besov spaces

Blow-up phenomena and the local well-posedness and ill-posedness of the generalized Camassa–Holm equation in critical Besov spaces

Exact kink wave solutions and peakons of a generalized Camassa–Holm equation

Utilizing the qualitative theory and bifurcation method to discuss a generalized Camassa–Holm equation ut−uxxt+(2ku+p+22up+1)x=(pup−1ux22+upuxx)x. In the case of p=2 and p=3, the bifurcation phase portraits and various new explicit kink wave solutions, peakons solutions and some implicit solutions of the equation are obtained.