The attractor on viscosity Degasperis–Procesi equation

In this paper the dynamical behaviors of a dispersive shallow water equation with viscosity, viscosity Degasperis–Procesi equation, are investigated. The existence of global solution to viscosity Degasperis–Procesi equation in L² under the periodical boundary condition is studied and the existence of the global attractor of semi-group to solution on viscosity Degasperis–Procesi equation in H² is obtained.     

Global weak solutions and breaking waves to the Degasperis–Procesi equation with linear dispersion

We study here an initial-value problem for the Degasperis–Procesi equation with a strong dispersive term, which is an approximation to the incompressible Euler equations for shallow water waves. We first determine the blow-up set of breaking waves to the equation. We then prove the existence and uniqueness of global weak solutions to the equation with certain initial profiles.     

Global weak solutions and wave breaking phenomena to the periodic Degasperis–Procesi equation with strong dispersion

Considered herein is the initial-value problem for the periodic Degasperis–Procesi equation with a strong dispersive term that is an approximation to the incompressible Euler equation for shallow water waves. The existence and uniqueness of global weak solutions are established. Moreover, the periodic peakon and shockpeakon solutions of the equation are constructed.     

Stability of peakons for the Degasperis‐Procesi equation

The Degasperis‐Procesi equation can be derived as a member of a one‐parameter family of asymptotic shallow‐water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa‐Holm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis‐Procesi equation on the line. By constructing a Lyapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations. © 2007 Wiley Periodicals, Inc.     

Dressing Method for the Degasperis–Procesi Equation

The soliton solutions of the Degasperis–Procesi equations are constructed by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by Hirota's method.     

Initial–boundary value problems for conservation laws with source terms and the Degasperis–Procesi equation

We consider conservation laws with source terms in a bounded domain with Dirichlet boundary conditions. We first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition. Equipped with this formulation, we go on to establish the well-posedness of entropy solutions to the initial–boundary value problem. The proof utilizes the kinetic formulation and the averaging lemma. Finally, we make use of these results to demonstrate the well-posedness in a class of discontinuous solutions to the initial–boundary value problem for the Degasperis–Procesi shallow water equation, which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term.     

Maximum principle for optimal distributed control of viscous weakly dispersive Degasperis–Procesi equation

This paper is concerned with the optimal distributed control of the viscous weakly dispersive Degasperis–Procesi equation in nonlinear shallow water dynamics. It is well known that the Pontryagin maximum principle, which unifies calculus of variations and control theory of ordinary differential equations, sets up the theoretical basis of the modern optimal control theory along with the Bellman dynamic programming principle. In this paper, we commit ourselves to infinite dimensional generalizations of the maximum principle and aim at the optimal control theory of partial differential equations. In contrast to the finite dimensional setting, the maximum principle for the infinite dimensional system does not generally hold as a necessary condition for optimal control. By the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the controlled viscous weakly dispersive Degasperis–Procesi equation. The necessary optimality condition is established for the problem in fixed final horizon case. Finally, a remark on how to utilize the obtained results is also made. Copyright © 2015 John Wiley & Sons, Ltd.     

On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis–Procesi equation

We consider the problem of the existence of the global solutions and formation of singularities for a b-family of equations which includes the Camassa–Holm and Degasperis–Procesi equation. We also consider the problem of the uniformly continuity of Degasperis–Procesi equation.     

Periodic solutions to the Cahn–Hilliard equation with constraint

This paper is concerned with the multidimensional Cahn–Hilliard equation with a constraint. The existence of periodic solutions of the problem is mainly proved under consideration by the viscosity approach. More precisely, with the help of the subdifferential operator theory and Schauder fixed point theorem, the existence of solutions to the approximation of the original problem is shown, and then the solution is obtained by using a passage‐to‐limit procedure based on a prior estimate. Copyright © 2015 John Wiley & Sons, Ltd.     

New periodic solitary-wave solutions for the Benjiamin Ono equation

In this paper, the new periodic solitary wave and doubly periodic solutions for (1 + 1)-dimensional Benjiamin Ono equation are obtained, using the bilinear method and extended homoclinic test approach. These results demonstrate that the integrable system has richly dynamical behavior even if it is (1 + 1)-dimensional.     

Periodic and solitary traveling wave solutions for the generalized Kadomtsev–Petviashvili equation

This paper is concerned with traveling waves for the generalized Kadomtsev–Petviashvili equation \input amssym.tex $(w_{t}+w_{\xi\xi\xi}+f(w)_{\xi})_{\xi}=w_{yy},(\xi,y)\in{\Bbb R}^{2}, t\in{\Bbb R}$ , i.e. solutions of the form . We study both, solutions periodic in and solitary waves, which are decaying in x, and their interrelations. In particular, we prove the existence of a sequence of k‐periodic solutions, \input amssym.def $k\in{\Bbb N}$ , which is uniformly bounded in norm and converges to a solitary wave in a suitable topology. This result also holds for the corresponding ground states, i.e. solutions with minimal energy. Copyright © 1999 John Wiley & Sons, Ltd.     

Application of variational iteration method for modified Camassa‐Holm and Degasperis‐Procesi equations

In this article, the variational iteration method (VIM) is used to obtain approximate analytical solutions of the modified Camassa‐Holm and Degasperis‐Procesi equations. The method is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or Adomian polynomials. The results reveal that the VIM is very effective. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Periodic wave solutions and asymptotic analysis of the Hirota–Satsuma shallow water wave equation

In this paper, we devise a simple way to explicitly construct the Riemann theta function periodic wave solution of the nonlinear partial differential equation. The resulting theory is applied to the Hirota–Satsuma shallow water wave equation. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function. We obtain the one‐periodic and two‐periodic wave solutions of the equation. The relations between the periodic wave solutions and soliton solutions are rigorously established. Copyright © 2014 John Wiley & Sons, Ltd.     

Periodic solutions of the Hopf equation

We consider the long-time dynamics of approximate solutions of the boundary-value problem for the Hopf equation on a finite segment. Together with the initial conditions, for instance, we impose the zero Dirichlet conditions on both ends of the segment. In this case, all features of solutions associated with the intersections of characteristics are accumulated on a strip bounded by the vertical characteristics emitted from the boundary points.     

Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation

In this paper we consider a generalized Kadomtsev-Petviashvili equation in the form It is shown that the solutions blow up in finite time for the supercritical power of nonlinearity with the ratio of an even to an odd integer. Moreover, it is shown that the solitary waves are strongly unstable if ; that is, the solutions blow up in finite time provided they start near an unstable solitary wave.     

Periodic solutions for the Landau-Lifshitz-Gilbert equation

Ferromagnetic materials tend to develop very complex magnetization patterns whose time evolution is modeled by the so-called Landau-Lifshitz-Gilbert equation (LLG). In this paper, we construct time-periodic solutions for LLG in the regime of soft and small ferromagnetic particles which satisfy a certain shape condition. Roughly speaking, it is assumed that the length of the particle is greater than its hight and its width. The approach is based on a perturbation argument and the spectral analysis of the corresponding linearized problem as well as the theory of sectorial operators.     

Periodic solutions of the Lyness max equation

We investigate the periodic nature of solutions of a “max-type” difference equation sometimes referred to as the “Lyness max” equation. The equation we consider is x_n+1=max{x_n,A}/x_n−1, n=0,1,…, where A is a positive real parameter and the initial conditions are arbitrary positive numbers. We also present related results for a similar equation sometimes referred to as the “period 7 max” equation.