We show that the weakly dissipative Camassa–Holm, Degasperis–Procesi, Hunter–Saxton, and Novikov equations can be reduced to their non-dissipative versions by means of an exponentially time-dependent ...scaling. Hence, up to a simple change of variables, the non-dissipative and dissipative versions of these equations are equivalent. Similar results hold also for the equations in the so-called b-family of equations as well as for the two-component and μ-versions of the above equations.
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 initial value problem for a novel 4-parameter family of evolution equations, which are nonlinear and nonlocal and possess peakon traveling wave solutions, is studied on both the line and the ...circle. It is proved that this family of equations is well-posed in the sense of Hadamard when the initial data belong to the Sobolev spaces Hs with s>5/2. Also, it is shown that the data-to-solution map is not uniformly continuous. However, if Hs, s>5/2, is equipped with a weaker Hr norm, 0⩽r<s, then the solution map becomes Hölder continuous.
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.
To realize and comprehend the physical phenomena of nonlinear system, exploration of traveling wave solutions plays an important role. Among the class of dispersive PDEs of traveling wave solutions ...the Degasperis-Procesi (DP) equation comprises high order nonlinear derivatives and is considered as a well known model for shallow water dynamics having similar asymptotic accuracy as for the Camassa-Holm (CH) equation. In this study we investigate solutions of some high order nonlinear dispersive PDEs namely generalized Degasperis-Procesi (DP), Camassa-Holm (CH) and Korteweg-de Vries (KdV) equations by the use of Radial Basis Function (RBF) combined with Finite Differences (RBF-FD) and Pseudo-Spectral (RBF-PS) methods. For the time derivative approximation, the fourth-order Runge-Kutta (RK) technique is accomplished. The efficiency and accuracy of our suggested approaches are demonstrated using examples and results.
Under investigation in this work is the Degasperis–Procesi equation with a strong dispersive term, which can be investigated as a model for shallow water dynamics. With the help of distribution ...theory, a direct and effective way is used to succinctly study the multi-peakon solutions of the equation. These multi-peakon solutions are obtained in weak sense. In particular, several types of double-peakon solutions are discussed in detail. Moreover, the dynamic behaviors the double-peakon solutions are illustrated through some figures.
Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations are developed. Unlike the standard Degasperis–Procesi and Camassa–Holm equations, where multi-peakon ...solutions arise, the modification caused a change in the characteristic of these peakon solutions and changed it to bell-shaped solitons. The tanh method and the sine–cosine method are used to achieve this goal.
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.
PDF
