Degasperis Procesi Equation Research Articles

Ghostpeakons and Characteristic Curves for the Camassa-Holm, Degasperis-Procesi and Novikov Equations

We derive explicit formulas for the characteristic curves associated with the multipeakon solutions of the Camassa-Holm, Degasperis-Procesi and Novikov equations. Such a curve traces the path of a fluid particle whose instantaneous velocity equals the elevation of the wave at that point (or the square of the elevation, in the Novikov case). The peakons themselves follow characteristic curves, and the remaining characteristic curves can be viewed as paths of 'ghostpeakons' with zero amplitude; hence, they can be obtained as solutions of the ODEs governing the dynamics of multipeakon solutions. The previously known solution formulas for multipeakons only cover the case when all amplitudes are nonzero, since they are based upon inverse spectral methods unable to detect the ghostpeakons. We show how to overcome this problem by taking a suitable limit in terms of spectral data, in order to force a selected peakon amplitude to become zero. Moreover, we use direct integration to compute the characteristic curves for the solution of the Degasperis-Procesi equation where a shockpeakon forms at a peakon-antipeakon collision. In addition to the theoretical interest in knowing the characteristic curves, they are also useful for plotting multipeakon solutions, as we illustrate in several examples.

Adaptive moving knots meshless method for Degasperis-Procesi equation with conservation laws

The nonlinear Degasperis-Procesi (DP) equation allows shock wave solutions and possesses bi-Hamiltonian properties. Considering both of the two properties, we introduce a numerical scheme which involves three key features. First, a stable moving knots strategy is introduced to simulate the shock waves with less knots. Second, the moving knots equation together with the DP equation are simulated using the meshless multi-quadric (MQ) quasi-interpolation method, which allows scattered data approximation. Third, the proposed moving knots meshless scheme preserves the conservation laws. Both theoretical analysis and numerical examples demonstrate that the algorithm saves the computational time, improves the accuracy, and possesses a long-time tracking capability.

A general family of multi-peakon equations and their properties

A general family of peakon equations is introduced, involving two arbitrary functions of the wave amplitude and the wave gradient. This family contains all of the known breaking wave equations, including the integrable ones: Camassa–Holm equation, Degasperis–Procesi equation, Novikov equation, and FORQ/modified Camassa–Holm equation. One main result is to show that all of the equations in the general family possess weak solutions given by multi-peakons which are a linear superposition of peakons with time-dependent amplitudes and positions. In particular, neither an integrability structure nor a Hamiltonian structure is needed to derive N-peakon weak solutions for arbitrary N > 1. As a further result, single peakon travelling-wave solutions are shown to exist under a simple condition on one of the two arbitrary functions in the general family of equations, and when this condition fails, generalized single peakon solutions that have a time-dependent amplitude and a time-dependent speed are shown to exist. An interesting generalization of the Camassa–Holm and FORQ/modified Camassa–Holm equations is obtained by deriving the most general subfamily of peakon equations that possess the Hamiltonian structure shared by the Camassa–Holm and FORQ/modified Camassa–Holm equations. Peakon travelling-wave solutions and their features, including a variational formulation (minimizer problem), are derived for these generalized equations. A final main result is that two-peakon weak solutions are investigated and shown to exhibit several novel kinds of behaviour, including the formation of a bound pair consisting of a peakon and an anti-peakon that have a maximum finite separation.

Long-time asymptotics for the Degasperis–Procesi equation on the half-line

We analyze the long-time asymptotics for the Degasperis--Procesi equation on the half-line. By applying nonlinear steepest descent techniques to an associated $3 \times 3$-matrix valued Riemann--Hilbert problem, we find an explicit formula for the leading order asymptotics of the solution in the similarity region in terms of the initial and boundary values.

Local Discontinuous Galerkin Methods for the $$\mu $$ μ -Camassa–Holm and $$\mu $$ μ -Degasperis–Procesi Equations

In this paper, we develop and analyze a series of conservative and dissipative local discontinuous Galerkin (LDG) methods for the $$\mu $$ -Camassa–Holm ( $$\mu $$ CH) and $$\mu $$ -Degasperis–Procesi ( $$\mu $$ DP) equations. The conservative schemes for both two equations can preserve discrete versions of their own first two Hamiltonian invariants, while the dissipative ones guarantee the corresponding stability. The error estimates of both LDG schemes for the $$\mu $$ CH equation are given. Comparing with the error estimates for the Camassa–Holm equation, some important tools are used to handle the unexpected terms caused by its particular Hamiltonian invariants. Moreover, a priori error estimates of two LDG schemes for the $$\mu $$ DP equation are also proven in detail. Numerical experiments for both equations in different circumstances are provided to illustrate the accuracy and capability of these schemes and give some comparisons about their performance on simulations.

Dynamics of conservative peakons in a system of Popowicz

We consider a two-component Hamiltonian system of partial differential equations with quadratic nonlinearities introduced by Popowicz, which has the form of a coupling between the Camassa–Holm and Degasperis–Procesi equations. Despite having reductions to these two integrable partial differential equations, the Popowicz system itself is not integrable. Nevertheless, as one of the authors showed with Irle, it admits distributional solutions of peaked soliton (peakon) type, with the dynamics of N peakons being determined by a Hamiltonian system on a phase space of dimension 3N. As well as the trivial case of a single peakon (N=1), the case N=2 is Liouville integrable. We present the explicit solution for the two-peakon dynamics, and describe some of the novel features of the interaction of peakons in the Popowicz system.

General Degasperis-Procesi equation and its solitary wave solutions

We consider the general Degasperis-Procesi model of shallow water out-flows. This six parametric family of conservation laws contains, in particular, KdV, Benjamin-Bona-Mahony, Camassa-Holm, and Degasperis-Procesi equations. The main result consists of criterions which guarantee the existence of solitary wave solutions: solitons and peakons (“peaked solitons”).

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.

Numerical Solution of Modified Forms of Camassa-Holm and Degasperis-Procesi Equations via Quartic B-Spline Collocation Method

In this paper, a collocation finite difference scheme based on Quartic B-spline function is developed for solving non-linear modified Camassa-Holm and Degasperis-Procesi equations. A finite difference scheme and Quartic B-spline function are used to discretize the time and spatial derivatives, respectively. The obtained numerical results are compared with the exact analytical solutions and some methods existing in literature. The numerical solutions of proposed non-linear equations are acquired without any linearization technique. The convergence of the method is proved of order \((\Delta t + h^2)\). The efficiency of the proposed scheme is demonstrated through illustrative examples. The presented scheme is realized to be a very reliable alternate method to some existing schemes for such physical problems.

On the integrability of Degasperis–Procesi equation: Control of the Sobolev norms and Birkhoff resonances

We consider the dispersive Degasperis–Procesi equation ut−uxxt−cuxxx+4cux−uuxxx−3uxuxx+4uux=0 with c∈R∖{0}. In [15] the authors proved that this equation possesses infinitely many conserved quantities. We prove that there are infinitely many of such constants of motion which control the Sobolev norms and which are analytic in a neighborhood of the origin of the Sobolev space Hs with s≥2, both on R and T. By the analysis of these conserved quantities we deduce a result of global well-posedness for solutions with small initial data and we show that, on the circle, the formal Birkhoff normal form of the Degasperis–Procesi at any order is action-preserving.

Degasperis–Procesi peakon dynamical system and finite Toda lattice of CKP type

In this paper, we propose a finite Toda lattice of CKP type (C-Toda) together with a Lax pair. Our motivation is based on the fact that the Camassa–Holm (CH) peakon dynamical system and the finite Toda lattice may be regarded as opposite flows in some sense. As an intriguing analogue to the CH equation, the Degasperis–Procesi (DP) equation also supports the presence of peakon solutions. Noticing that the peakon solution to the DP equation is expressed in terms of bimoment determinants related to the Cauchy kernel, we impose opposite time evolution on the moments and derive the corresponding bilinear equation. The corresponding quartic representation is shown to be a continuum limit of a reduced discrete CKP equation, due to which we call the obtained equation a finite Toda lattice of CKP type. Then, a nonlinear version of the C-Toda lattice together with a Lax pair is derived. As a result, it is shown that the DP peakon lattice and the finite C-Toda lattice form opposite flows under a certain transformation.

Unfamiliar Aspects of Bäcklund Transformations and an Associated Degasperis–Procesi Equation

We summarize the results of our recent work on B\"acklund transformations (BTs), particularly focusing on the relationship of BTs and infinitesimal symmetries. We present a BT for an associated Degasperis-Procesi (aDP) equation and its superposition principle, and investigate the solutions generated by application of this BT. Following our general methodology, we use the superposition principle of the BT to generate the infinitesimal symmetries of the aDP equation.

Global weak solutions for a generalized Camassa–Holm equation

AbstractIn this paper we mainly investigate the Cauchy problem of a generalized Camassa–Holm equation. We first derive two global existence results and two blow‐up results from the relationship between the Degasperis–Procesi equation and the generalized Camassa–Holm equation. We then prove the existence and uniqueness of global weak solutions under some certain sign conditions on the initial data.

Ill-posedness of the Camassa–Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space H3/2 and even in the Besov space Bp,r1+1/p for p∈[1,∞],r∈(1,∞]. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5,14,16]).

On the finite time singularities for a class of Degasperis–Procesi equations

As we all know, the conservation laws take an important part in deriving wave breaking for Degasperis–Procesi (DP) type equation. In the article, we give a new method to study singularities in finite time for a class of DP equations with high nonlinear terms. The paper tells us that the result of wave breaking can be established without the conservation law.

Degenerate form of a three-component Degasperis–Procesi equation

We discuss the short-wave limit of a new three-component Degasperis–Procesi equation, and construct infinitely many conserved quantities for the degenerate system. We study the reductions for a spectral problem of the degenerate system, which are shown to contain spectral problems of some famous or new systems found recently.

On a new two-component $b$-family peakon system with cubic nonlinearity

In this paper, we propose a two-component \begin{document}$b$\end{document} -family system with cubic nonlinearity and peaked solitons (peakons) solutions, which includes the celebrated Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and its two-component extension as special cases. Firstly, we study single peakon and multi-peakon solutions to the system. Then the local well-posedness for the Cauchy problem of the system is discussed. Furthermore, we derive the precise blow-up scenario and global existence for strong solutions to the two-component \begin{document}$b$\end{document} -family system with cubic nonlinearity. Finally, we investigate the asymptotic behaviors of strong solutions at infinity within its lifespan provided the initial data decay exponentially and algebraically.

Liouville theorems for periodic two-component shallow water systems

We establish Liouville-type theorems for periodic two-component shallow water systems, including a two-component Camassa-Holm equation (2CH) and a two-component Degasperis-Procesi (2DP) equation. More presicely, we prove that the only global, strong, spatially periodic solutions to the equations, vanishing at some point $(t_0, x_0)$, are the identically zero solutions. Also, we derive new local-in-space blow-up criteria for the dispersive 2CH and 2DP.

Numerical solution of the Degasperis–Procesi equation by the cubic B-spline quasi-interpolation method

In this paper, a numerical scheme is presented to solve the non-dissipative Degasperis–Procesi equation based on the u-p formulation. The cubic B-spline quasi-interpolation coupled with the finite difference method is applied to approximate the spatial derivatives and an optimal third order TVD Runge–Kutta method to estimate the time derivative of the dependent variable. The accuracy and effectiveness of the proposed method are validated by six classical problems. Numerical results indicate that the proposed scheme is simple, easy to implement with high accuracy.