Camassa Holm Equation Research Articles

Zero-filter limit issue for the Camassa–Holm equation in Besov spaces

Zero-filter limit issue for the Camassa–Holm equation in Besov spaces

Explorations of certain nonlinear waves of the Boussinesq and Camassa–Holm equations using physics-informed neural networks

The Boussinesq and Camassa–Holm equations are, respectively, used to describe the bidirectional and unidirectional motions of small-amplitude waves on shallow water surfaces, but their wave dynamics remain a challenge for almost all conventional numerical methods. In this paper, we use physics-informed neural networks (PINNs), a mesh-free deep learning method, to accurately predict the soliton (peakon) interaction or rogue wave behaviours of both equations with only a few initial and boundary data, providing a method for solving certain numerically unstable fluid systems or extreme wave solutions of regular fluid systems. It is revealed that by decomposing both of the equations into lower-order coupled systems, one can obtain higher-precision wave behaviours, especially for the Camassa–Holm equation. To retrieve certain unknown hydraulic parameters based on the observed wave data, e.g. the water depth, we use a multiple PINN method and stably identify the dynamic parameters in the Boussinesq equation through the association of localized soliton and rogue wave solutions. Furthermore, we compare the PINNs with conventional high-precision time-splitting Fourier spectral (TSFS) method and find that to achieve the same split feature of the Y-shapedsoliton of the Boussinesq equation, PINNs require only one-third of the initial and boundary data of the TSFS method.

Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations

By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.

Novel solitary wave solutions in dual-mode simplified modified Camassa-Holm equation in shallow water waves

Novel solitary wave solutions in dual-mode simplified modified Camassa-Holm equation in shallow water waves

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.

Non-uniform convergence of solution for the Camassa–Holm equation in the zero-filter limit

Non-uniform convergence of solution for the Camassa–Holm equation in the zero-filter limit

Existence and uniqueness of the global conservative weak solutions for a cubic Camassa-Holm type equation

The extended cubic Camassa-Holm equation can be derived as asymptotic model from shallow-water approximation to the 2D incompressible Euler equations. This model encompasses both quadratic and cubic nonlinearities and the solution of the extended cubic Camassa-Holm equation possible development of singularities in finite time. This paper is devoted to the continuation of solutions to the extended cubic Camassa-Holm equation beyond wave breaking, and the global existence and uniqueness of the Hölder continuous energy conservative solutions for the Cauchy problem of the extended cubic Camassa-Holm type equation are investigated. An equivalent semilinear system was first introduced by a new set of independent and dependent variables, which can resolve all singularities due to possible wave breaking. Returning to the original variables, depending continuously on the initial data, the existence of the global conservative weak solutions can be obtained. Moreover, by analyzing the evolution of the quantities u and v=2arctan⁡ux along each characteristic, the uniqueness of the global conservative solutions for the Cauchy problem with general initial data u0∈H1(R) was proved.

Reducibility of the dispersive Camassa-Holm equation with unbounded perturbations

Considered herein is the reducibility of the quasi-periodically time dependent linear dynamical system with a diophantine frequency vector ω∈O0⊂Rν. This system is derived from linearizing the dispersive Camassa-Holm equation with quasi-linear perturbations at a small amplitude quasi-periodic function. It is shown that there is a set O∞⊂O0 of asymptotically full Lebesgue measure such that for any ω∈O∞, the system can be reduced to the one with constant coefficients by a quasi-periodic linear transformation. The strategy adopted in this paper consists of two steps: (a) A reduction based on the orders of the pseudo differential operators in the system which conjugates the linearized operator to a one with constant coefficients up to a small remainder; (b) A perturbative reducibility scheme which completely diagonalizes the remainder of the previous step. The main difficulties in the reducibility we need to tackle come from the operator J=(1−∂xx)−1∂x, which induces the symplectic structure of the dispersive Camassa-Holm equation.

Global existence of dissipative solutions to the Camassa–Holm equation with transport noise

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term.

Some new decay properties of solutions for a modified Camassa–Holm equation with cubic nonlinearity

In this article, we derive some new asymptotic properties of solution for a modified Camassa–Holm equation with cubic nonlinearity in weighted Sobolev spaces. Specifically, we prove that the strong solutions of Eq. (1.2) will maintain the corresponding decay properties provided the initial data ω0(x), ω0,x(x) decay logarithmically, algebraically and exponentially, respectively, which improves and extends the persistence properties of strong solutions to Eq. (1.2) obtained by Wu and Guo in paper [J. Math. Phys. 55, 081504 (2014)]. This method can be used to study the asymptotic decay properties for a class of Camassa–Holm equations.

Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation

Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation

Finite-dimensional global attractor for the three-dimensional viscous Camassa-Holm equations with fractional diffusion on bounded domains

Finite-dimensional global attractor for the three-dimensional viscous Camassa-Holm equations with fractional diffusion on bounded domains

Orbital stability of the periodic peakons for the higher-order modified Camassa-Holm equation

Orbital stability of the periodic peakons for the higher-order modified Camassa-Holm equation

The new complex travelling wave solutions of the simplified modified camassa holm equation

The new complex travelling wave solutions of the simplified modified camassa holm equation

A coupled scheme based on uniform algebraic trigonometric tension B-spline and a hybrid block method for Camassa-Holm and Degasperis-Procesi equations

In this article, high temporal and spatial resolution schemes are combined to solve the Camassa-Holm and Degasperis-Procesi equations. The differential quadrature method is strengthened by using modified uniform algebraic trigonometric tension B-splines of order four to transform the partial differential equation (PDE) into a system of ordinary differential equations. Later, this system is solved considering an optimized hybrid block method. The good performance of the proposed strategy is shown through some numerical examples. The stability analysis of the presented method is discussed. This strategy produces a saving of CPU-time as it involves a reduced number of grid points.

On the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail.

Asymptotics for peakon-antipeakon collision in a general Camassa-Holm model

We analyze a peakon - antipeakon collision for essentially non-integrable versions of the Camassa-Holm equation. Using the matching and weak asymptotics methods, we construct an asymptotic solution that satisfies both a one-parameter family of equations (with a parameter r), and two energy laws. It is shown that the original waves with amplitudes A1>0 and A2<0 are reflected upon collision and form new waves with amplitudes B1<0 and B2>0 such that, depending on the parameter r, either B1=A2 and B2=A1, or Bi are arbitrary numbers provided B1+B2=A1+A2.

Numerical study of fractional Camassa–Holm equations

A numerical study of fractional Camassa–Holm equations is presented. Smooth solitary waves are constructed numerically. Their stability is studied as well as the long time behavior of solutions for general localized initial data from the Schwartz class of rapidly decreasing functions. The appearance of dispersive shock waves is explored.

The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlevé asymptotics

Based on the ∂‾-generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlevé asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space H4,2(R). With a new scale (y,t) and a Riemann-Hilbert problem associated with the initial value problem, we derive different long time asymptotic expansions for the solutions of the CH equation in different space-time solitonic regions. The half-plane {(y,t):−∞<y<∞,t>0} is divided into four asymptotic regions: 1. Fast decay region, y/t∈(−∞,−1/4) with an error O(t−1/2); 2. Modulation-soliton region, y/t∈(2,+∞), the result can be characterized with an modulation-solitons with residual error O(t−1/2); 3. Zakhrov-Manakov region, y/t∈(0,2) and y/t∈(−1/4,0). The asymptotic approximation is characterized by the dispersion term with residual error O(t−3/4); 4. Two transition regions, |y/t|≈2 and |y/t|≈−1/4, the asymptotic results are described by the solution of Painlevé II equation with error order O(t−1/2).

Singular solutions of the r-Camassa–Holm equation

This paper introduces the r-Camassa–Holm (r-CH) equation, which describes a geodesic flow on the manifold of diffeomorphisms acting on the real line induced by the metric. The conserved energy for the problem is given by the full norm. For r = 2, we recover the Camassa–Holm equation. We compute the Lie symmetries for r-CH and study various symmetry reductions. We introduce singular weak solutions of the r-CH equation for and demonstrates their robustness in numerical simulations of their nonlinear interactions in both overtaking and head-on collisions. Several open questions are formulated about the unexplored properties of the r-CH weak singular solutions, including the question of whether they would emerge from smooth initial conditions.