Related papers: Peakons
We consider a new partial differential equation, of a similar form to the Camassa-Holm shallow water wave equation, which was recently obtained by Degasperis and Procesi using the method of asymptotic integrability. We prove the exact…
In this paper, we study orbital stability of peakons for the generalized modified Camassa-Holm (gmCH) equation, which is a natural higher-order generalization of the modified Camassa-Holm (mCH) equation, and admits Hamiltonian form and…
We give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form $\ddot{u}\,u + \frac{1}{2}\dot{u}^2 + F'(u) =0$, where $F$ is an analytic function. Our motivation stems from the fact that in…
Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the…
It is well-known that the celebrated Camassa-Holm equation has the peaked solitary waves, which have been not reported for other mainstream models of shallow water waves. In this letter, the closed-form solutions of peaked solitary waves of…
A famous feature of the Camassa-Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation,…
We present an inverse scattering approach for computing n-peakon solutions of the Degasperis-Procesi equation (a modification of the Camassa-Holm (CH) shallow water equation). The associated non-self-adjoint spectral problem is shown to be…
We consider the CH-KP-I equation. For this equation we prove the existence of steady solutions, which are solitary in one horizontal direction and periodic in the other. We show that such waves bifurcate from the line solitary wave…
The main purpose of the paper is to provide a survey of our recent studies on soliton solutions of the Kadomtsev-Petviashvili (KP) equation. The classification is based on the far-field patterns of the solutions which consist of a finite…
We use a sticky particle method to show global existence of (energy) conservative sticky $N$-peakon solutions to the modified Camassa-Holm equation. A dispersion regularization is provided as a selection principle for the uniqueness of…
In this paper, we study an integrable Camassa-Holm (CH) type equation with quadratic nonlinearity. The CH type equation is shown integrable through a Lax pair, and particularly the equation is found to possess a new kind of peaked soliton…
We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz, which has the appearance of a two-field coupling between the Camassa-Holm and Degasperis-Procesi equations. The latter equations are both…
A wide class of nonlinear dispersive wave equations are shown to possess a novel type of peakon solution in which the amplitude and speed of the peakon are time-dependent. These novel dynamical peakons exhibit a wide variety of different…
The paper deals with the construction of the asymptotic soliton-like and the asymptotic peakon-like solutions to the modified Camassa-Holm equation with variable coefficicents and a singular perturbation. This equation is a generalization…
We study the generalization of the dispersionless Kadomtsev - Petviashvili (dKP) equation in n+1 dimensions and with nonlinearity of degree m+1, a model equation describing the propagation of weakly nonlinear, quasi one dimensional waves in…
We show that wave breaking occurs with positive probability for the Stochastic Camassa-Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads…
Peakons are singular, soliton-like solutions to nonlinear wave equations whose dynamics can be studied using ordinary differential equations (ODEs). The Degasperis-Procesi equation (DP) is an important example of an integrable PDE…
In this paper, we present a dispersive regularization for the modified Camassa-Holm equation (mCH) in one dimension, which is achieved through a double mollification for the system of ODEs describing trajectories of $N$-peakon solutions.…
The Euler-Poincar\'e differential (EPDiff) equations and the shallow water (SW) equations share similar wave characteristics. Using the Hamiltonian structure of the SW equations with flat bottom topography, we establish a connection between…
We present a study on the nonlinear dynamics of a disturbance to the laminar state in non-rotating axisymmetric Poiseuille pipe flows. The associated Navier-Stokes equations are reduced to a set of coupled generalized Camassa-Holm type…