Generalized Degasperis Procesi Equation Research Articles

Stability analysis and novel solutions to the generalized Degasperis Procesi equation: An application to plasma physics.

In this work two kinds of smooth (compactons or cnoidal waves and solitons) and nonsmooth (peakons) solutions to the general Degasperis-Procesi (gDP) equation and its family (Degasperis-Procesi (DP) equation, modified DP equation, Camassa-Holm (CH) equation, modified CH equation, Benjamin-Bona-Mahony (BBM) equation, etc.) are reported in detail using different techniques. The single and periodic peakons are investigated by studying the stability analysis of the gDP equation. The novel compacton solutions to the equations under consideration are derived in the form of Weierstrass elliptic function. Also, the periodicity of these solutions is obtained. The cnoidal wave solutions are obtained in the form of Jacobi elliptic functions. Moreover, both soliton and trigonometric solutions are covered as a special case for the cnoidal wave solutions. Finally, a new form for the peakon solution is derived in details. As an application to this study, the fluid basic equations of a collisionless unmagnetized non-Maxwellian plasma is reduced to the equation under consideration for studying several nonlinear structures in the plasma model.

Smooth travelling wave solutions in a generalized Degasperis–Procesi equation

In this paper, by using a dynamical system-based approach, namely, the singular travelling wave method, we classify all the travelling wave solutions including periodic wave solutions, periodic peakon solutions, solitary wave solutions and compacton solutions etc. in the generalized Degasperis–Procesi (gDP) equation. All these travelling waves are smooth and classical solutions. The parameter conditions for the existence of these travelling waves are also obtained. Then we combine geometric singular perturbation theory with an “explicit”Melnikov method to analyze the persistence of solitary wave solutions under singular perturbation, i.e., the existence of solitary wave solutions in a perturbed gDP equation. By calculating the representations of the (unperturbed) homoclinic orbits and the associated Melnikov integral explicitly, the persistence of solitary wave solutions is shown and the speed of the wave is determined (to leading order).

Blow-up criteria for the generalized Degasperis–Procesi equation

ABSTRACT In this paper, we investigate the initial value problem of the generalized Degasperis–Procesi equation. We establish some new local-in-space blow-up criterion. Our results extend the corresponding results in the previous literature.

Global solutions and blow-up phenomena for a generalized Degasperis–Procesi equation

In this paper, we mainly investigate global solutions and blow-up phenomena for the Cauchy problem of a generalized Degasperis–Procesi equation. We develop a new approach to consider the equation for (1−∂x)u instead of (1−∂x2)u. We first prove a new conserved quantity ‖(1−∂x)u‖L1 and give the local characterization of this conserved quantity. These properties then allow us to improve considerably a previous global existence result in [38], and establish two new blow-up results for strong solutions to the equation. In the end, based on vanishing viscosity method and using the conserved quantity we deduce the existence of global weak solutions to the equation for any initial data belonging to L1∩BV.

Non-uniform dependence for a generalized Degasperis–Procesi equation

In the paper, we consider the Cauchy problem for a generalized Degasperis–Procesi equation. We prove that the data-to-solution map is not uniformly continuous.

On the Cauchy problem for a generalized Degasperis-Procesi equation

ABSTRACTWe first establish the local well-posedness for a generalized Degasperis-Procesi equation in nonhomogeneous Besov spaces. Then we present a global existence result for the equation. Moreover, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time.

The explicit periodic wave solutions and their limit forms for a generalized b-equation

In this paper, for b ∈ (−∞,∞) and b ≠ −1,−2, we investigate the explicit periodic wave solutions for the generalized b-equation ut + 2kux − uxxt + (1+b)u2ux = buxuxx + uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable α. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for α. When α tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = −1 or b = −2, we guess that the equation does not have any one of above solutions.

Non-uniform dependence on initial data for the generalized Degasperis–Procesi equation on the line

In this paper, we show that the solution map of the generalized Degasperis–Procesi (gDP) equation is not uniformly continuous in Sobolev spaces [Formula: see text] for [Formula: see text]. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part. It also exploits the fact that the gDP equation conserves a quantity which is equivalent to the [Formula: see text] norm.

Well-posedness and global existence for a generalized Degasperis–Procesi equation

We first establish the local existence and uniqueness of strong solutions for the Cauchy problem of a generalized Degasperis–Procesi equation in nonhomogeneous Besov spaces by using the Littlewood–Paley theory. Then, we prove the solution depends continuously on the initial data. Finally, we derive a blow-up criterion and present a global existence result for the equation.

The local well-posedness and stability to a nonlinear generalized Degasperis-Procesi equation

A nonlinear generalized Degasperis-Procesi equation is investigated. The local well-posedness of a strong solution for the equation in the Sobolev space $H^{s}(R)$ with $s>\frac{3}{2}$ is established. The $L^{1}(R)$ stability is obtained under certain assumptions on the initial data.

The Stability of Solutions for the Generalized Degasperis-Procesi Equation with Variable Coefficients

A generalized Degasperis-Procesi equation with variable coefficients is investigated. TheL1(R)stability of the strong solution for the equation is established under certain assumptions.

On theL1Stability to a Generalized Degasperis-Procesi Equation

A nonlinear generalized Degasperis-Procesi equation is investigated. Assuming that the strong solution of the equation is bounded in the sense ofL∞R-norm and the initial data belong to the spaceL1(R)∩L2(R), we prove that the solutions are stable in the spaceL1(R).

Variational Iteration Method for Solving the Generalized Degasperis-Procesi Equation

We introduce the variational iteration method for solving the generalized Degasperis-Procesi equation. Firstly, according to the variational iteration, the Lagrange multiplier is found after making the correction functional. Furthermore, several approximations ofun+1(x,t)which is converged tou(x,t)are obtained, and the exact solutions of Degasperis-Procesi equation will be obtained by using the traditional variational iteration method with a suitable initial approximationu0(x,t). Finally, after giving the perturbation item, the approximate solution for original equation will be expressed specifically.

The Cauchy problem for the generalized Degasperis-Procesi equation

In this paper, we investigate the Cauchy problem for the generalized Degasperis-Procesi equation in a Besov space. Firstly, we prove that the generalized Degasperis-Procesi equation is locally well posed in B sr with s >1+ 1 (or s ≥ 1+ 1 if r =1w ithp ∈ [1, +∞)). Secondly, we prove that the generalized Degasperis-Procesi equation possesses the peaked solitary wave which is the weak solution to the generalized Degasperis-Procesi equation. Thirdly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in B 3/2∞ . Fourthly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in H s (R )w iths < 3/2. Finally,

The entropy weak solution to a generalized Degasperis-Procesi equation

A nonlinear generalization of the Degasperis-Procesi equation is investigated. The well-posedness of entropy weak solutions for the Cauchy problem of the equation is established in the space L 1 (R)∩ L ∞ (R).MSC:35G25, 35L05.

Towered waves and anti-waves in the generalized Degasperis–Procesi equation

New traveling wave solutions of the generalized Degasperis–Procesi equation are investigated. The solutions are characterized by three parameters. Using an improved qualitative method, abundant traveling wave solutions, such as smooth waves, peaked waves, cusped waves, compacted waves, looped waves and fractal-like waves, are obtained. Especially, some strange composite wave solutions such as towered waves and their anti-waves are first given. We also study the limiting behavior of all periodic solutions as the parameters trend to some special values.

On the Cauchy problem for the periodic generalized Degasperis–Procesi equation

We mainly study the Cauchy problem of the periodic generalized Degasperis–Procesi equation. First, we establish the local well-posedness for the equation. Second, we give the precise blow-up scenario, a conservation law and prove that the equation has smooth solutions which blow up in finite time. Finally, we investigate the blow-up rate for the blow-up solutions.

On the Cauchy problem for a generalized Degasperis–Procesi equation

We mainly study the Cauchy problem of the generalized Degasperis–Procesi equation. First, we establish the local well posedness for the equation. Second, we give the precise blow-up scenario, a conservation law, and prove the equation has smooth solutions which blow up in finite time. Finally, we investigate the blow-up rate for the blow-up solutions.

Well-posedness and blow-up phenomena for the generalized Degasperis–Procesi equation

In this paper, we mainly study the Cauchy problem of the generalized Degasperis–Procesi equation. We establish the local well-posedness and give the precise blow-up scenario for the equation. Then we show that the equation has smooth solutions which blow up in finite time.

New solutions for the modified generalized Degasperis–Procesi equation

In this paper, using three distinct computational methods we obtain some new exact solutions for the generalized modified Degasperis–Procesi equation (mDP equation) u t - u xxt + ( b + 1 ) u 2 u x = bu x u xx + uu xxx . We show the graph of some of the new solutions obtained here with the aim to illustrate their physical relevance. Mathematica is used. Finally some conclusions are given.