Degasperis Procesi Research Articles

Asymptotic stability of the Degasperis–Procesi antipeakon–peakon profile

The aim of this paper is to prove that the Degasperis–Procesi antipeakon–peakon profile is asymptotically stable for all time. We start by proving the asymptotic stability of a single Degasperis–Procesi peakon and antipeakon with respect to perturbations having a momentum density that is first negative and then positive. Then this result is extended towards a well-ordered trains of antipeakons–peakons under such perturbations. In particular, the asymptotic stability of the antipeakon–peakon profile holds.

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.

A Degasperis–Procesi equation II with multi-peakon solutions

Based on a 3 × 3 matrix spectral problem, a new hierarchy of nonlinear evolution equations is obtained by means of the zero-curvature equation and the Lenard recursion equation. With the help of the trace identity, the Hamiltonian structures of this hierarchy are established. Then a new nonlinear wave equation, called the Degasperis–Procesi equation II, is derived from a negative flow, which admits exact solutions with N-peakons. Finite-dimensional dynamical systems related to N-peakon solutions and an infinite sequence of conserved quantities of the Degasperis–Procesi equation II are obtained.

New blow-up criterion for the Degasperis–Procesi equation with weak dissipation

New blow-up criterion for the Degasperis–Procesi equation with weak dissipation

A Rigidity Property for the Novikov Equation and the Asymptotic Stability of Peakons

We consider weak solutions of the Novikov equation that lie in the energy space $$H^1$$ with non-negative momentum densities. We prove that a special family of such weak solutions, namely the peakons, is $$H^1$$ -asymptotically stable. Such a result is based on a rigidity property of the Novikov solutions which are $$H^1$$ -localized and the corresponding momentum densities are localized to the right, which extends the earlier work of Molinet (Arch Ration Mech Anal 230:185–230, 2018; Nonlinear Anal Real World Appl 50:675–705, 2019) for the Camassa–Holm and Degasperis–Procesi peakons. The main new ingredients in our proof consist of exploring the uniform in time exponential decay property of the solutions from the localization of the $$H^1$$ energy and redesigning the localization of the total mass from the finite speed of propagation property of the momentum densities.

Jacobi wavelet collocation method for the modified Camassa–Holm and Degasperis–Procesi equations

Matrices representations of integrations of wavelets have a major role to obtain approximate solutions of integral, differential and integro-differential equations. In the present work, operational matrix representation of rth integration of Jacobi wavelets is introduced and to find these operational matrices, all details of the processes are demonstrated for the first time. Error analysis of offered method is also investigated in present study. In the planned method, approximate solutions are constructed with the truncated Jacobi wavelets series. Approximate solutions of the modified Camassa–Holm equation and Degasperis–Procesi equation linearized using quasilinearization technique are obtained by presented method. Applicability and accuracy of presented method is demonstrated by examples. The proposed method is also convergent even when a minor number of grid points. The numerical results obtained by offered technique are compatible with those in the literature.

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).

A super Degasperis-Procesi equation and related integrable systems.

Based on a 4 × 4 matrix spectral problem, a super Degasperis-Procesi (DP) equation is proposed. We show that under a reciprocal transformation, the super DP equation is related to the first negative flow of a super Kaup-Kupershmidt (KK) hierarchy, which turns out to be a particular reduction of a super Boussinesq hierarchy. The bi-Hamiltonian structure of the super Boussinesq hierarchy is established and subsequently produces a Hamiltonian structure, as well as a conjectured symplectic formulation of the super KK hierarchy via suitable reductions. With the help of the reciprocal transformation, the bi-Hamiltonian representation of the super DP equation is constructed from that of the super KK hierarchy. We also calculate a positive flow of the super DP hierarchy and explain its relations with the super KK equation. Infinitely many conservation laws are derived for the super DP equation, as well as its positive flow.

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.

Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model

Before going further with fractional derivative which is constructed by Rabotnov exponential kernel, there exist many questions that are not addressed. In this paper, we try to recapitulate all the fundamental calculus, which we can obtain with this new fractional operator. The problems in this paper are to determine the solutions of the fractional differential equations where the second members are constant functions, polynomial functions, exponential functions, trigonometric functions, or Mittag-Leffler functions. For all the fractional differential equations, the obtained solutions are represented graphically. The Laplace transform of the fractional derivative with Rabotnov exponential kernel is the primary tool in the investigations. Finally, we give the fundamental solution to the nonlinear time-fractional modified Degasperis–Procesi equation by considering the fractional operator with Rabotnov exponential kernel.

Novel approach for modified forms of Camassa–Holm and Degasperis–Procesi equations using fractional operator

In the present study, we consider the q-homotopy analysis transform method to find the solution for modified Camassa–Holm and Degasperis–Procesi equations using the Caputo fractional operator. Both the considered equations are nonlinear and exemplify shallow water behaviour. We present the solution procedure for the fractional operator and the projected solution procedure gives a rapidly convergent series solution. The solution behaviour is demonstrated as compared with the exact solution and the response is plotted in 2D plots for a diverse fractional-order achieved by the Caputo derivative to show the importance of incorporating the generalised concept. The accuracy of the considered method is illustrated with available results in the numerical simulation. The convergence providence of the achieved solution is established in -curves for a distinct arbitrary order. Moreover, some simulations and the important nature of the considered model, with the help of obtained results, shows the efficiency of the considered fractional operator and algorithm, while examining the nonlinear equations describing real-world problems.

Reducible KAM Tori for the Degasperis\u2013Procesi Equation

We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis–Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash–Moser nonlinear iteration, pseudo differential calculus and normal form techniques. In the present case the complicated symplectic structure, the weak dispersive effects of the linear flow and the presence of strong resonant interactions require a novel set of ideas. The main points are to exploit the integrability of the unperturbed equation, to look for special wave packet solutions and to perform a very careful algebraic analysis of the resonances. Our approach is quite general and can be applied also to other 1d integrable PDEs. We are confident for instance that the same strategy should work for the Camassa–Holm equation.

Analytical methods: Nonlinear longitudinal wave equation in a magnetoelectro-elastic circular rod, foam drainage and modified Degasperis–Procesi models arising in nonlinear water wave models

We form the analytical solitary wave solutions with the execution of generalized direct algebraic technique on three well-known nonlinear wave models, namely, called foam drainage, longitudinal magnetoelectro-elastic circular rod and modified Degasperis–Procesi equations. The derived solutions are hyperbolic functions in which some are plotted graphically on meticulous values to the parameters which provides the basic knowledge to understand physical significant of these three wave models. The obtained solutions show the efficiency and precision of our scheme. These derived new results prove that our novel technique is awfully effective and can be productive as a instrument solving for sundry other nonlinear evolution equations.

Bäcklund Transformations for the Degasperis-Procesi Equation

We consider the Backlund transformation for the Degasperis-Procesi (DP) equation. Using the reciprocal transformation and the associated DP equation, we construct the Backlund transformation for the DP equation involving both dependent and independent variables. We also obtain the corresponding nonlinear superposition, which we use together with the Backlund transformation to derive some soliton solutions of the DP equation.

An efficient computational technique for time-fractional modified Degasperis-Procesi equation arising in propagation of nonlinear dispersive waves

In this paper, an efficient hybrid numerical scheme which is based on a joint venture of the q-homotopy analysis method and Sumudu transform is applied to investigate the time-fractional modified Degasperis-Procesi (DP) equation. The present study considers the Caputo fractional derivative. The fractional order modified DP model is very important and plays a great role in study of ocean engineering and science. The proposed scheme provides a beautiful opportunity for proper selection of the auxiliary parameter ℏ and the asymptotic parameter ρ ( ≥ 1) to handle mainly the differential equations of nonlinear nature. The offered scheme produces the solution in the shape of a convergent series in a large admissible domain which is helpful to regulate the region of convergence of a series solution. The proposed work computes the approximate analytical solution of the fractional modified DP equation systematically and also presents graphically the variation of the obtained solution for diverse values of the fractional parameter β.

Multisoliton solutions of the Degasperis–Procesi equation and its shortwave limit: Darboux transformation approach

We propose a new approach for calculating multisoliton solutions of the Degasperis–Procesi equation and its shortwave limit by combining a reciprocal transformation with the Darboux transformation of the negative flow of the Kaup–Kupershmidt hierarchy. In particular, different specifications of the soliton parameters lead to two different types of soliton solutions of the Degasperis–Procesi equation.

An optimized compact reconstruction weighted essentially non-oscillatory scheme for Degasperis-Procesi equation

This article presents a two-step iterative method that uses u – P formulation to study a Degasperis-Procesi (DP) equation, with which the DP equation is decomposed into the nonlinear advection equation and the Helmholtz equation. The first-order derivative terms in the advection equation are approximated by an Optimized Compact Reconstruction Weighted Essentially Non-Oscillatory (OCRWENO) scheme that reduces dispersion and dissipation errors and suppresses oscillation near discontinuities. Besides, high-order symplectic Runge-Kutta scheme is used for time marching. A rigorous analysis of the dispersion and dissipation errors are provided for the OCRWENO scheme. Single smooth soliton solution is investigated to check the accuracy and the order of the proposed method. Peakon, peakon-peakon, peakon-antipeakon and shockpeakon solutions of the DP equation are then predicted. Finally, wave breaking phenomena of smoothed initial condition from the DP equation are addressed. In addition, two conservative quantities associated with the bi-Hamiltonian form of the DP equation are calculated to demonstrate capabilities of the present method.

The solitary wave solutions of the nonlinear perturbed shallow water wave model

In this paper, we study the nonlinear perturbed shallow water wave model, which satisfies the asymptotic integrability condition with this family: the KdV equation, Camassa–Holm equation and Degasperis–Procesi equation. We establish the existence of solitary wave solutions for the perturbed nonlinear dispersive equations by means of the geometric singular perturbation and invariant manifold theory.

A finite difference scheme for smooth solutions of the general Degasperis–Procesi equation

AbstractThe general Degasperis–Procesi equation (gDP) describes the evolution of the water surface in a unidirectional shallow water approximation. We propose a finite‐difference scheme for this equation that preserves some conservation and balance laws. In addition, the stability of the scheme and the convergence of numerical solutions to exact solutions for solitons are proved. Numerical experiments confirm the theoretical conclusions. For essentially nonintegrable versions of the gDP equation, it is shown that solitons and antisolitons collide almost elastically: they retain their shape after interaction, but a small “tail”, the so‐called “radiation”, appears.

The shock peakon wave solutions of the general Degasperis–Procesi equation

This research paper applies the modified Khater method and the generalized Kudryashov method to the general Degasperis–Procesi (DP) equation, which is used to describe the dynamical behavior of the shallow water outflows. Some shock peakon wave solutions are obtained by using these methods. Moreover, some figures are sketched for these solutions to explain more physical properties of the general DP equation and to figure out the coincidence between different types of obtained solutions. The stability property by using the features of the Hamiltonian system is tested to some obtained solutions to show their ability for applying in the model’s applications. The obtained solutions were verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows their power and effectiveness for applying to many different forms of the nonlinear evolution equations with an integer and fractional order.