Generalized Camassa Holm Equation Research Articles

Global conservative and dissipative solutions of the generalized Camassa-Holm equation

This paper is devoted to the continuation of solutions to the generalized Camassa-Holm equationbeyond wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear system. This formulation allows one to continue the solution after collision time, giving either a global conservative solution where the energy is conserved for almost all times or a dissipative solution where energy may vanish from the system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global conservative or dissipative solutions, which depend continuously on the initial data.

The orbital stability of the solitary wave solutions of the generalized Camassa–Holm equation

In this paper, we consider the orbital stability of smooth solitary wave solutions of the generalized Camassa–Holm equation. By constructing the functional extremum problem and using the orbital stability theory presented by Grillakis, Shatah, Strauss and Bona, and Souganidis, we show that the solitary wave solutions of the generalized Camassa–Holm equation are orbitally stable or unstable as determined by the sign of a discriminant. The conclusions presented by the previous authors, such as Hakkaev and Kirchev, Constantin and Strauss, can be considered as a special case of our results.

An optimal distributed control problem of the viscous generalized Camassa–Holm equation

In this work, an optimal distributed control problem of the viscous generalized Camassa–Holm equation is considered. By the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the investigational system. The necessary condition for optimality is established for the controlled object in the fixed final horizon case and, subsequently, a remark on how to apply the obtained results is made as an illustration.

The Extended Tanh Method for Compactons and Solitons Solutions for the CH(<i>n</i>,2<i>n</i> – 1,2<i>n</i>,–<i>n</i>) Equations

In this paper, by using the sine-cosine method, the extended tanh-method, and the rational hyperbolic functions method, we study a class of nonlinear equations which derived from a fourth order analogue of generalized Camassa-Holm equation. It is shown that this class gives compactons, solitary wave solutions, solitons, and periodic wave solutions. The change of the physical structure of the solutions is caused by variation of the exponents and the coefficients of the derivatives.

Self-adjointness of a generalized Camassa–Holm equation

It is well known that the Camassa–Holm equation possesses numerous remarkable properties characteristic for KdV type equations. In this paper we show that it shares one more property with the KdV equation. Namely, it is shown in [1,2] that the KdV and the modified KdV equations are self-adjoint. Starting from the generalization [3] of the Camassa–Holm equation [4], we prove that the Camassa–Holm equation is self-adjoint. This property is important, e.g. for constructing conservation laws associated with symmetries of the equation in question. Accordingly, we construct conservation laws for the generalized Camassa–Holm equation using its symmetries.

The existence of weak solutions for a generalized Camassa-Holm equation

A Camassa-Holm type equation containing nonlinear dissipative effect is investigated. A sufficient condition which guarantees the existence of weak solutions of the equation in lower order Sobolev space $H^s$ with $1 \leq s \leq \frac{3}{2}$ is established by using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself.

EXPLICIT PERIODIC WAVE SOLUTIONS AND THEIR BIFURCATIONS FOR GENERALIZED CAMASSA–HOLM EQUATION

In this paper, through qualitative analysis and integration, we study the explicit periodic wave solutions and their bifurcations for the generalized Camassa–Holm equation [Formula: see text] When the parameter k satisfies k < 3/8 and the constant wave speed c satisfies [Formula: see text], we obtain two types of explicit periodic wave solutions, elliptic smooth periodic wave solution and elliptic periodic blow-up solutions. These solutions include a bifurcation parameter α which has four bifurcation values αi(i = 1, 2, 3, 4). When α tends to the bifurcation values, the elliptic periodic wave solutions become three types of other solutions, the hyperbolic smooth solitary wave solution, the hyperbolic blow-up solution and the trigonometric periodic blow-up solution. Especially, a new bifurcation phenomenon is found, that is, the periodic blow-up solution can become a smooth solitary wave solution when α varies. When k > 3/8, we guess that there is no other explicit solution except the explicit periodic blow-up solution.

Explicit peaked wave solutions to the generalized Camassa-Holm equation

By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.

Existence of weak solutions in lower order Sobolev space for a Camassa–Holm-type equation

A generalized Camassa–Holm equation containing a nonlinear dissipative effect is investigated. The existence of the weak solution of the equation in lower order Sobolev space Hs with is established by using the techniques of the pseudoparabolic regularization and some a priori estimates derived from the equation itself.

The local well-posedness and existence of weak solutions for a generalized Camassa–Holm equation

A generalization of the Camassa–Holm equation, a model for shallow water waves, is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space H s ( R ) with s > 3 2 is established via a limiting procedure. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space H s with 1 < s ⩽ 3 2 is developed.

On Solutions to a Two-Component Generalized Camassa-Holm Equation

We consider a two-component Camassa–Holm system which arises in shallow water theory. We analyze a wave breaking mechanism and the global existence of solutions. First, we discuss the local well posedness and a blow up mechanism, then establish some new blow up criteria for this system formulated either on the line or with space-periodic initial conditions. Finally, the existence of global solutions is analyzed.

Bifurcations of travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa–Holm equation

In this paper, by using the bifurcation theory of dynamical systems for a class of nonlinear fourth order variant of a generalized Camassa–Holm equation, the existence of solitary wave solutions, breaking bounded wave solutions, compacton solutions and non-smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.

Optimal control of the viscous generalized Camassa–Holm equation

In this paper, we study the optimal control problem for the viscous generalized Camassa–Holm equation. We deduce the existence and uniqueness of weak solution to the viscous generalized Camassa–Holm equation in a short interval by using Galerkin method. Then, by using optimal control theories and distributed parameter system control theories, the optimal control of the viscous generalized Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous generalized Camassa–Holm equation is proved.

Variable Wave-form Periodic Solutions for the Dispersive Wave Equation of Generalized Camassa-Holm Equation Type

Variable Wave-form Periodic Solutions for the Dispersive Wave Equation of Generalized Camassa-Holm Equation Type

Travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa–Holm equation

In this paper, we study a class of nonlinear fourth order analogue of a generalized Camassa–Holm equation by using sine–cosine method. It is shown that this class gives compactons and solitary patterns solutions.

Explicit solutions of two nonlinear dispersive equations with variable coefficients

A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is developed to construct the exact solutions for a generalized Camassa–Holm equation and a nonlinear dispersive equation with variable coefficients. It is shown that the variable coefficients of the derivative terms in the equations cause the qualitative change in the physical structures of the solutions.

Exact travelling wave solutions of a generalized Camassa–Holm equation using the integral bifurcation method

In this paper, a generalized Camassa–Holm equation is studied by using the integral bifurcation method. Many travelling waves such as peaked compacton, compacton, peaked solitary wave, solitary wave and kink-like wave are found. In some parameter conditions, exact parametric representations of these travelling waves in explicit form and implicit form are obtained.

Peakons, solitary patterns and periodic solutions for generalized Camassa–Holm equations

This Letter deals with a generalized Camassa–Holm equation and a nonlinear dispersive equation by making use of a mathematical technique based on using integral factors for solving differential equations. The peakons, solitary patterns and periodic solutions are expressed analytically under various circumstances. The conditions that cause the qualitative change in the physical structures of the solutions are highlighted.

Blow-up of solution of an initial boundary value problem for a generalized Camassa–Holm equation

In this Letter, we study the following initial boundary value problem for a generalized Camassa–Holm equation { u t − u x x t + 3 u u x − 2 u x u x x − u u x x x + k ( u − u x x ) x = 0 , t ⩾ 0 , x ∈ [ 0 , 1 ] , u ( 0 , t ) = u ( 1 , t ) = u x ( 0 , t ) = u x ( 1 , t ) = 0 , t ⩾ 0 , u ( 0 , x ) = u 0 ( x ) , x ∈ [ 0 , 1 ] , where k is a real constant. We establish local well-posedness of this closed-loop system by using Kato's theorem for abstract quasilinear evolution equation of hyperbolic type. Then, by using multiplier technique, we obtain a conservation law which enable us to present a blow-up result.

Periodic blow-up solutions and their limit forms for the generalized Camassa–Holm equation

In this paper, we consider the generalized Camassa–Holm equation u t + 2 ku x - u xxt + au 2 u x = 2 u x u xx + uu xxx . Under substitution ξ = x - ct , some new explicit periodic wave solutions and their limit forms are presented through some special phase orbits. These periodic wave solutions tend to infinity on ξ - u plane periodically. Thus we call them periodic blow-up solutions. To our knowledge, such periodic blow-up solutions have not been found in any other equations.