Using a generalized framework that consists of evolution of the solution to the Camassa- Holm equation and its energy measure, we establish the global-in-time orbital stability of peakons with respect to the perturbed (energy) conservative solutions to the Camassa-Holm equation. Especially, we extend the H1-stability result obtained by Constantin and Strauss (Comm. Pure Appl. Math., 53(5), 603-610, 2000) globally-in-time, even after the perturbed solutions experience wave breaking. In addition, our result also shows that the singular part of the energy measure of the perturbed solutions will remain stable for all times.
The $μ$-Camassa-Holm ($μ$CH) equation is a nonlinear integrable partial differential equation closely related to the Camassa-Holm equation. We prove that the periodic peaked traveling wave solutions (peakons) of the $μ$CH equation are orbitally stable.
We use a sticky particle method to show global existence of (energy) conservative sticky $N$-peakon solutions to the modified Camassa-Holm equation. A dispersion regularization is provided as a selection principle for the uniqueness of conservative $N$-peakon solutions. The dispersion limit avoids the collision between peakons, and numerical results show that the dispersion limit is exactly the sticky peakons. At last, when the splitting of peakons is allowed, we give an example to show the non-uniqueness of conservative solutions.
We compute explicitly the peakon-antipeakon solution of the Camassa-Holm equation $u_t-u_{txx}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ in the non-symmetric and $α$-dissipative case. The solution experiences wave breaking in finite time, and the explicit solution illuminates the interplay between the various variables.
In this paper, we study the following generalized Camassa-Holm equation with both cubic and quadratic nonlinearities: $$ m_{t}+k_{1}(3uu_{x}m+u^2m_{x})+k_{2}(2mu_{x}+m_{x}u)=0, \quad m=u-u_{xx}, $$ which is presented as a linear combination of the Novikov equation and the Camassa-Holm equation with constants $k_{1}$ and $k_{2}$. The model is a cubic generalization of the Camassa-Holm equation. It is shown that the equation admits single-peaked soliton and periodic peakons.
We show that wave breaking occurs for the modified Camassa-Holm (mCH) equation. Next we classify all traveling wave solutions of the modified Camassa-Holm equation in the weak sense via parametrization of their maxima, minima and wave velocity constants. This equation is shown to admit in addition to more popular solutions like smooth traveling waves and peakons, some not so well-known traveling waves as, for example, kinks, cuspons, composite waves and stumpons. Moreover, explicit peakons in terms of Jacobian elliptic functions are found.
This paper is devoted to an integrable two-component Camassa-Holm system with cubic nonlinearity, which includes the cubic Camassa-Holm equation (also called the Fokas-Olver-Rosenau-Qiao equation) as a special case. The one peaked solitons (peakons) and two peakon solutions are described in an explicit formula. Then, the local well-posedness for the Cauchy problem of the system is studied. Moreover, we target at the precise blow-up scenario for strong solutions to the system, and establish a new blow-up result with respect to the initial data.
In this paper we mainly investigate the Cauchy problem of a three-component Camassa-Holm system. By using the method of approximation of smooth solutions, a regularization technique and the special structure of the system, we prove the existence of global weak solutions to the system.
We give a proof for the uniqueness of dissipative solution for the Camassa-Holm equation with some peakon-antipeakon initial data following Dafermos' earlier resut in [5] on the Hunter-Saxton equation. Our result shows that two existing global existence frameworks, through the vanishing viscosity method by Xin-Zhang in [11] and the transformation of coordinate method for dissipative solutions by Bressan-Constantin in [3], give the same solution, for a special but typical initial data forming finite time gradient blowup.
In this paper, we investigate the orbital stability of peakons for a modified Camassa-Holm equation with cubic nonlinearity derived from the two-dimensional Euler equation. By overcoming the difficulties caused by one of the complicated conservation laws and introducing a new auxiliary function, we prove the orbital stability of peakons for the equation.
In this paper, we present a dispersive regularization for the modified Camassa-Holm equation (mCH) in one dimension, which is achieved through a double mollification for the system of ODEs describing trajectories of $N$-peakon solutions. From this regularized system of ODEs, we obtain approximated $N$-peakon solutions with no collision between peakons. Then, a global $N$-peakon solution for the mCH equation is obtained, whose trajectories are global Lipschitz functions and do not cross each other. When $N=2$, the limiting solution is a sticky peakon weak solution. By a limiting process, we als…
In this paper, we study orbital stability of peakons for the generalized modified Camassa-Holm (gmCH) equation, which is a natural higher-order generalization of the modified Camassa-Holm (mCH) equation, and admits Hamiltonian form and single peakons. We first show that the single peakon is the usual weak solution of the PDEs. Some sign invariant properties and conserved densities are presented. Next, by constructing the corresponding auxiliary function $h(t,\,x)$ and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate sol…
Consideration here is a generalized $μ$-type integrable equation, which can be regarded as a generalization to both the $μ$-Camassa-Holm and modified $μ$-Camassa-Holm equations. It is shown that the proposed equation is formally integrable with the Lax-pair and the bi-Hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary-gravity waves. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established by the viscosity method. Existence of peaked traveling-wave solutions and formation of singularities of soluti…
We study special properties of solutions to the IVP associated to the Camassa-Holm equation on the line related to the regularity and the decay of solutions. The first aim is to show how the regularity on the initial data is transferred to the corresponding solution in a class containing the "peakon solutions". In particular, we shall show that the local regularity is similar to that exhibited by the solution of the inviscid Burger's equation with the same initial datum. The second goal is to prove that the decay results obtained in a paper of Himonas, Misiołek, Ponce, and Zhou extend to the c…
We continue our investigation on the asymptotic stability of the peakon. In a first step we extend our asymptotic stability result [29] in the class of functions whose negative part of the momentum density is supported in ] -- $\infty$, x 0 ] and the positive part in [x 0 , +$\infty$[ for some x 0 $\in$ R. In a second step this enables us to prove the asymptotic stability of well-ordered train of antipeakons-peakons and, in particular, of the antipeakon-peakon profile. Finally, in the appendix we prove that in the case of a non negative momentum density the energy at the left of any given poin…
We prove a Liouville property for uniformly almost localized (up to translations) H 1-global solutions of the Camassa-Holm equation with a momentum density that is a non negative finite measure. More precisely, we show that such solution has to be a peakon. As a consequence, we prove that peakons are asymptotically stable in the class of H 1-functions with a momentum density that belongs to M + (R). Finally, we also get an asymptotic stability result for train of peakons.
The present work is mainly concerned with global existence for the two-component Camassa-Holm system and the modified two-component Camassa-Holm system. By discovering new conservative quantities of these systems, we prove several new global existence results for these two-component shallow water systems.
We prove that the peakons are asymptotically H 1-stable, under the flow of the Degasperis-Procesi equation, in the class of functions with a momentum density that belongs to M + (R). The key argument is a rigidity result for uniformly in time exponentially decaying global solutions that is shared by the Holm-Staley b-family of equations for b $\ge$ 1. This extends previous results obtained for the Camassa-Holm equation (b = 2).
In this article we study a new kind of unbounded solutions to the Novikov equation, found via a Lie symmetry analysis. These solutions exhibit peakon creation, i.e., these solutions are smooth up until a certain finite time, at which a peak is created. We show that the functions are still weak solutions for those times where the peak lives. We also find similar unbounded solutions with peakon creation in the related Camassa-Holm equation, by making an ansatz inspired by the Novikov solutions. Finally, we see that the same ansatz for the Degasperis-Procesi equation yields unbounded solutions wh…
In this paper we mainly investigate the Cauchy problem of a generalized Camassa-Holm equation. First by this relationship between the Degasperis-Procesi equation and the generalized Camassa-Holm equation, we then obtain two global existences result and two blow-up result. Then, we prove the existence and uniqueness of global weak solutions.
Lie symmetry group method is applied to study for the higher order Camassa-Holm equation. The symmetry group and its optimal system are given. Furthermore, preliminary classification of its group invariant solutions, symmetry reduction and nonclassical symmetries are investigated. Finally conservation laws for the higher order Camassa-Holm equation are presented.
In this paper, we investigate the orbital stability problem of peakons for a modified Camassa-Holm equation with both quadratic and cubic nonlinearity. This equation was derived from integrable theory and admits peaked soliton (peakon) and multipeakon solutions. By constructing two suitable piecewise functions, we establish the polynomial inequality relating to two conserved quantities and the maximum of the solution to this equation. The error estimate between the maximum of the solution and the peakon then follows from the structure of the polynomial inequality. Finally, we prove that a wave…
We study global conservative solutions of the Cauchy problem for the Camassa-Holm equation $u_t-u_{txx}+κu_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with nonvanishing and distinct spatial asymptotics.
The aim of this paper is to answer the question left in \cite{GL} (Math. Z. (2015) 281). We prove that the zero-filter limit of the Camassa-Holm equation is the Burgers equation in the same topology of Sobolev spaces as the initial data.
The Camassa-Holm equation and its two-component Camassa-Holm system generalization both experience wave breaking in finite time. To analyze this, and to obtain solutions past wave breaking, it is common to reformulate the original equation given in Eulerian coordinates, into a system of ordinary differential equations in Lagrangian coordinates. It is of considerable interest to study the stability of solutions and how this is manifested in Eulerian and Lagrangian variables. We identify criteria of convergence, such that convergence in Eulerian coordinates is equivalent to convergence in Lagran…
In this paper, a generalized nonlinear Camassa-Holm equation with time- and space-dependent coefficients is considered. We show that the control of the higher order dispersive term is possible by using an adequate weight function to define the energy. The existence and uniqueness of solutions are obtained via a Picard iterative method.
A novel $π$-Camassa--Holm system is studied as a geodesic flow on a semidirect product obtained from the diffeomorphism group of the circle. We present the corresponding details of the geometric formalism for metric Euler equations on infinite-dimensional Lie groups and compare our results to what has already been obtained for the usual two-component Camassa--Holm equation. Our approach results in well-posedness theorems and explicit computations of the sectional curvature.
We investigate the measures of dissipation and accretion related to the weak solutions of the Camassa-Holm equation. Demonstrating certain properties of nonunique characteristics, we prove a new representation formula for these measures and conclude about their structural features, such us singularity with respect to the Lebesgue measure. We apply these results to gain new insights into the structure of weak solutions, proving in particular that measures of accretion vanish for dissipative solutions of the Camassa-Holm equation.
In this paper, we prove the existence of a global entropy weak solution $u\in H^1(\mathbb{R})$ and $\partial_{x}u\in L^1(\mathbb{R})\cap BV(\mathbb{R})$ for the Cauchy problem of a generalized Camassa-Holm equation by the viscous approximation method.
In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation that includes the Camassa-Holm as well as the Novikov equation on the line. We present a new and unified method to prove the sharp ill-posedness for the generalized Camassa-Holm equation in $B^s_{p,\infty}$ with $s>\max\{1+1/p, 3/2\}$ and $1\leq p\leq\infty$ in the sense that the solution map to this equation starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^s_{p,\infty}$. Our results cover and improve the previous work given in [J. Li, Y. Yu, W. Zhu, Ill-posedness for the Camassa-Holm and…
In this paper, we consider the solution map of the initial value problem to the two-component Camassa-Holm equation on the line. We prove that the solution map of this problem is not uniformly continuous in Sobolev spaces $H^s(\R)\times H^{s-1}(\R)$ for $s>3/2$.
In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation proposed by Hakkaev and Kirchev (2005) \cite{Hakkaev 2005}. We prove that the solution map of the generalized Camassa-Holm equation is not uniformly continuous on the initial data in Besov spaces. Our result include the present work (2020) \cite{Li 2020,Li 2020-1} on Camassa-Holm equation with $Q=1$ and extends the previous non-uniform continuity in Sobolev spaces (2015) \cite{Mi 2015} to Besov spaces. In addition, the non-uniform continuity in critical space $B_{2, 1}^{\frac{3}{2}}(\mathbb{R})$ is the firs…
We study the propagation of energy density in finite-energy weak solutions of the Camassa-Holm and related equations. Developing the methods based on generalized nonunique characteristics, we show that the parts of energy related to positive and negative slopes are one-sided weakly continuous and of bounded variation, which allows us to define certain measures of dissipation of both parts of energy. The result is a step towards the open problem of uniqueness of dissipative solutions of the Camassa-Holm equation.
In the paper, by constructing a initial data $u_{0}\in B^σ_{p,\infty}$ with $σ-2>\max\{1+\frac 1 p, \frac 3 2\}$, we prove that the corresponding solution to the higher dimensional Camassa-Holm equations starting from $u_{0}$ is discontinuous at $t=0$ in the norm of $B^σ_{p,\infty}$, which implies that the ill-posedness for the higher dimensional Camassa-Holm equations in $B^σ_{p,\infty}$.
Considered herein is the initial-value problem for the generalized periodic Camassa-Holm equation which is related to the Camassa-Holm equation and the Hunter-Saxton equation. Sufficient conditions guaranteeing the development of breaking waves in finite time are demonstrated. On the other hand, the existence of strong permanent waves is established with certain initial profiles depending on the linear dispersive parameter in a range of the Sobolev spaces. Moreover, the admissible global weak solution in the energy space is obtained.
In this paper, we consider the Cauchy problem for the fractional Camassa-Holm equation which models the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. Using Kato's semigroup approach for quasilinear evolution equations, we prove that the Cauchy problem is locally well-posed for data in $H^{s}({\Bbb R})$, $s>{\frac{5}{2}}$.
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 $B^1_{\infty,1}(\R)$. It is shown in (J. Differ. Equ., 327:127-144,2022) that the Camassa-Holm equation is ill-posed in $B^1_{\infty,1}(\R)$, 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 $B^1_{\infty,1}(\R)$ which leads to ill-p…
In this paper, we present a construction of $u_{0}\in B^{1}_{\infty,1}$ and get the local ill-posedness for the rotation Camassa-Holm equation modelling the equatorial water waves with the weak Coriolis effect by proving the norm inflation.
The present paper is mainly concerned with the blow-up phenomena and exponential decay of solution for a three-component Camassa-Holm equation. Comparing with the result of Hu, ect. in the paper[1], a new wave-breaking solution is obtained. The results of exponential decay of solution in our paper cover and extent the corresponding results in [12, 19, 22].
We prove that dissipative weak solutions of the Camassa-Holm equation are unique. Thus we complete the global well-posedness theory of this celebrated model of shallow water, initiated by a general proof of existence in [Z. Xin, P. Zhang Comm. Pure Appl. Math. 53 (2000)]. As the dissipative weak solutions, being viscosity solutions, seem to constitute the most physically relevant class of solutions in the wave-breaking regime, the result provides mathematical rationale for the feasibility of the Camassa-Holm equation as a model of water waves encompassing both soliton interactions and wave-bre…
This paper is devoted to the Cauchy problem for the modified multi-component Camassa-Holm system in higher dimensions. On the one hand, we establish an almost complete local well-posedness results for the system in the framework of Besov spaces. On the other hand, several blow-up criteria of strong solutions to the system are derived by using the Littlewood-Paley decomposition and the energy method.
This work is concerned with the Gevrey regularity and analyticity of the solution to a weakly dissipative Camassa-Holm system. We first demonstrate the local Gevery regularity and analyticity of this equation. Then, we disscuss the continuity of the data-to-solution map. Finally, we obtain the global Gevery regularity of this system in Gevery class $G_σ$ with $σ\geq 1$ in time.
It is well-known that peakons in the Camassa-Holm equation are $H^1$-orbitally stable thanks to the presence of conserved quantities and properties of peakons as constrained energy minimizers. By using the method of characteristics, we prove that piecewise $C^1$ perturbations to peakons grow in time in spite of their stability in the $H^1$-norm. We also show that the linearized stability analysis near peakons contradicts the $H^1$-orbital stability result, hence passage from linear to nonlinear theory is false in $H^1$.
We first establish local well-posedness for a periodic 2-component Camassa-Holm equation with vorticity. We then present a global existence result for strong solutions to the equation. We finally obtain several blow-up results and the blow-up rate of strong solutions to the equation.
In the paper, we consider the initial value problem to the Camassa-Holm equation in the real-line case. Based on the local well-posedness result and the lifespan, we proved that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces in the sense of Hadamard. Our obtained result improves considerably the result in \cite{H-K}.
Following conservative solutions of the two-component Camassa-Holm system $u_t-u_{txx}+3uu_x-2u_xu_{xx}-uu_{xxx}+ρρ_x=0$, $ρ_t+(uρ)_x=0$ along characteristics, we determine if wave breaking occurs in the nearby future or not, for initial data $u_0\in H^1(\mathbb R)$ and $ρ_0\in L^2(\mathbb R)$.
For the real-line case, it is shown that both the Camassa--Holm \cite{Guo} and Novikov equations \cite{Li-arx} are ill-posed in $B_{\infty,1}^{1}$. In this paper, by presenting a new construction of initial data which leads to the norm inflation phenomena, we prove that both the periodic Camassa--Holm and Novikov equations are also ill-posed in $B_{\infty,1}^{1}$.
In this paper, we study blow-up phenomena for the rotation-two-component Camassa- Holm (R2CH) system, a model of equatorial water waves that includes the effect of the Coriolis force. We first derive the precise blow-up scenario for R2CH with ? > 0: Then, we present two new blow-up results for strong solutions to the system and then the precise blow-up rate for strong solutions to the system is given.
In this paper, we give a construction of $u_0\in B^σ_{p,\infty}$ such that corresponding solution to the Camassa-Holm equation starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^σ_{p,\infty}$, which implies the ill-posedness for this equation in $B^σ_{p,\infty}$. We also apply our method to the b-equation and Novikov equation.
We discuss the recent paper by A. Darós and L.K. Arruda (On the instability of elliptic traveling wave solutions of the modified Camassa-Holm equation, J. Diff. Equat., 266 (2019), 1946-1968). Our intention is to correct some imperfections left by the authors and present the orbital stability of periodic snoidal waves in the energy space.
Considered in this paper is the generalized Camassa-Holm-Novikov equation with high order nonlinearity, which unifies the Camassa-Holm and Novikov equations as special cases. We show that the solution map of generalized Camassa-Holm-Novikov equation is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^s(\mathbb{R})$ with $s>\max\{1+\frac{1}{p}, \frac{3}{2}\}$, $1\leq p, r< \infty$ as well as in critical space $B_{2, 1}^{\frac{3}{2}}(\mathbb{R}).$ Our result covers and improves the previous work given by Li et al. \cite{Li 2020, 1Li 2020, Li 2021}(J. Differ. Equ. 269 (2020)…
This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa-Holm system on the real line. We obtain the global weak solutions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well.
In this paper we prove the local well-posedness of the Camassa-Holm equation on the real line in the space of continuously differentiable diffeomorphisms with an appropriate decaying condition. This work was motivated by G. Misiolek who proved the same result for the Camassa-Holm equation on the periodic domain. We use the Lagrangian approach and rewrite the equation as an ODE on the Banach space. Then by using the standard ODE technique, we prove existence and uniqueness. Finally, we show the continuous depdendence of the solution on the initial data by using the topological group property of…
Compared with the two-component Camassa-Holm system, the modified two-component Camassa-Holm system introduces a regularized density which makes possible the existence of solutions of lower regularity, and in particular of multipeakon solutions. In this paper, we derive a new pointwise invariant for the modified two-component Camassa-Holm system. The derivation of the invariant uses directly the symmetry of the system, following the classical argument of Noether's theorem. The existence of the multipeakon solutions can be directly inferred from this pointwise invariant. This derivation shows t…
