Dispersive Wave Equations Research Articles

On the Well-Posedness Problem and the Scattering Problem for the Dullin-Gottwald-Holm Equation

In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, and the convergence of solutions to the DGH equation as α2→0 is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained in the DGH equation. After giving the condition of existing peakon in the DGH equation, it turns out to be nonlinearly stable for the peakon in the DGH equation.

An extended Jacobi elliptic function rational expansion method and its application to ([formula omitted])-dimensional dispersive long wave equation

With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ansatz, in which periodic solutions of nonlinear partial differential equations that can be expressed as a finite Laurent series of some of 12 Jacobi elliptic functions, is more powerful than exiting Jacobi elliptic function methods and is very powerful to uniformly construct more new exact periodic solutions in terms of rational formal Jacobi elliptic function solution of nonlinear partial differential equations. As an application of the method, we choose a (2+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.

Exact travelling wave solutions to some fifth order dispersive nonlinear wave equations

Exact travelling wave solutions to some nonlinear equations of fifth order derivatives are derived by using some accurate ansatz methods.

An algebro-geometric solution for a Hamiltonian system with application to dispersive long wave equation

By using an iterative algebraic method, we derive from a spectral problem a hierarchy of nonlinear evolution equations associated with dispersive long wave equation. It is shown that the hierarchy is integrable in Liouville sense and possesses bi-Hamiltonian structure. Two commutators, with zero curvature and Lax representations, for the hierarchy are constructed, respectively, by using two different systematic methods. Under a Bargmann constraint the spectral is nonlinearized to a completely integrable finite dimensional Hamiltonian system. By introducing the Abel–Jacobi coordinates, an algebro-geometric solution for the dispersive long wave equation is derived by resorting to the Riemann theta function.

A weighted least action principle for dispersive waves

We extend the least action principle to continuum systems. The data for the new principle consist of the intensity of the wave (or rather the wave action) at two instances of time. We define an appropriate Lagrangian, and formulate a variational problem in terms of it. The critical points of the functional are used to determine the wave’s phase. The theory is applicable to the semiclassical limit of a large class of dispersive wave equations. Associating the wave equation with a Liouville equation for the Wigner distribution function, we are able to extend the theory to include singular solutions such as caustics.

A parallel multi-subdomain strategy for solving Boussinesq water wave equations

This paper describes a general parallel multi-subdomain strategy for solving the weakly dispersive and nonlinear Boussinesq water wave equations. The parallelization strategy is derived from the additive Schwarz method based on overlapping subdomains. Besides allowing the subdomains to independently solve their local problems, the strategy is also flexible in the sense that different discretization schemes, or even different mathematical models, are allowed in different subdomains. The parallelization strategy is particularly attractive from an implementational point of view, because it promotes the reuse of existing serial software and opens for the possibility of using different software in different subdomains. We study the strategy’s performance with respect to accuracy, convergence properties of the Schwarz iterations, and scalability through numerical experiments concerning waves in a basin, solitary waves, and waves generated by a moving vessel. We find that the proposed technique is promising for large-scale parallel wave simulations. In particular, we demonstrate that satisfactory accuracy and convergence speed of the Schwarz iterations are obtainable independent of the number of subdomains, provided there is sufficient overlap. Moreover, existing serial wave solvers are readily reusable when implementing the parallelization strategy.

New interaction property of (2+1)-dimensional localized excitations from Darboux transformation

Using the binary Darboux transformation for the (2 + 1)-dimensional dispersive long wave equation, the “universal” variable separable formula is extended in a different way. From the extended formula, much more abundant localized excitations with arbitrary boundary conditions for the dispersive long wave equation can be obtained. The results obtained via the multi-linear variable separation approach are only a special case of the first step binary Darboux transformation. Two special interacting solutions are explicitly given. Especially, one of the examples exhibits a new interacting phenomenon: a localized solitary wave (dromion) can force an extended wave (solitoff) go back.

Exact Periodic and New Solitary Wave Solutions to the Generalization of Integrable (2+1)-dimensional Dispersive Long Wave Equations

A general solution involving three arbitrary functions is first obtained for the generalization of integrable (2+1)-dimensional dispersive long wave equations by means of WTC truncation method. Exact periodic wave solutions are then expressed as rational functions of the Jacobi elliptic functions. For the first time the interaction of Jacobi elliptic waves is studied and found to be nonelastic! Limit cases are studied and some interesting, new solitary structures are revealed. The interactions of between two dromions, between dromion and solitoff and between y -periodic solitons are all nonelastic, and x -periodic solitons can propagate steadily. It is shown that the Jacobi elliptic wave solutions can be viewed as the generalization of dromion, dromion-solitoff and periodic solitons.

New soliton-like solutions to the (2+1)-dimensional dispersive long wave equations

In this paper, we solve the (2+1)dimensional dispersive long wave equation by using a new modified algebraic method, and obtain abundant new exact solutions. These solutions contain solitonlike solutions, periodiclike solutions,hyperboliclike function solutions, Jacobilike elliptic function solutions and so on.

Non-isospectral scattering problems and truncation for hierarchies: Burgers and dispersive water waves

Non-isospectral scattering problems have been proven useful for several reasons, amongst them the information that they provide about Painlevé truncation for entire hierarchies of integrable partial differential equations (PDEs). We show in this paper how our approach provides in a very straightforward manner truncation results for two hierarchies in 1+1 dimensions, namely Burgers' hierarchy and the dispersive water wave hierarchy. Burgers' equation is well-known as a model for turbulence, and the dispersive water wave equations as a system governing shallow water waves. We also see how these results are easily extended to related hierarchies in 2+1 dimensions. Our results for the 2+1-dimensional Burgers' hierarchy, and for the 1+1 and 2+1-dimensional dispersive water wave hierarchies, are all new.

New Exact Solutions to (2+1)-Dimensional Dispersive Long Wave Equations**The project supported by the State Key Basic Research Program of China under Grant No. 1998030600 and National Natural Science Foundation of China under Grant Nos. 10072013 and 10072189

Based upon a further extended tanh method [Phys. Lett. A307 (2003) 269; Chaos, Solitons and Fractals 17 (2003) 669] and the symbolic computation system, Maple, we consider the (2+1)-dimensional dispersive long wave equations. We obtain many new solutions of the equation. These solutions contain soliton-like solutions, periodic form solutions, and some rational solutions.

Exact Solutions of Extended Boussinesq Equations

The problem addressed in this paper is the verification of numerical solutions of nonlinear dispersive wave equations such as Boussinesq-like system of equations. A practical verification tool for numerical results is to compare the numerical solution to an exact solution if available. In this work, we derive some exact solitary wave solutions and several invariants of motion for a wide range of Boussinesq-like equations using Maple software. The exact solitary wave solutions can be used to specify initial data for the incident waves in the Boussinesq numerical model and for the verification of the associated computed solution. The invariants of motions can be used as verification tools for the conservation properties of the numerical model.

A new general algebraic method with symbolic computation to construct new travelling wave solution for the (1 + 1)-dimensional dispersive long wave equation

With the aid of symbolic computation, a new algebraic method, named Riccati equation rational expansion (RERE) method, is devised for constructing multiple travelling wave solutions for nonlinear evolution equations (NEEs). Compared with most existing tanh methods for finding travelling wave solutions, the proposed method not only recover the results by most known algebraic methods, but also provides new and more general solutions. With the aid of symbolic computation, we choose (1 + 1)-dimensional dispersive long wave equation (DLWE) to illustrate our method. As a result, we obtain many types of solutions including rational form solitary wave solutions, triangular periodic wave solutions and rational wave solutions. The properties of the new solitary wave solutions are shown by some figures.

Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1 + 1)-dimensional dispersive long wave equation

Abstract Our Jacobi elliptic function rational expansion method is extended to be a more powerful method, called the extended Jacobi elliptic function rational expansion method, by using more general ansatz. The (1 + 1)-dimensional dispersive long wave equation is chosen to illustrate the approach. As a consequence, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. When the modulus m → 1, these doubly periodic solutions degenerate as soliton solutions. The method can be also applied to other nonlinear differential equations.

A new general algebraic method with symbolic computation to construct new doubly-periodic solutions of the (2 + 1)-dimensional dispersive long wave equation

For constructing more new exact doubly-periodic solutions in terms of rational form Jacobi elliptic function of nonlinear evolution equations, a new direct and unified algebraic method, named Jacobi elliptic function rational expansion method, is presented and implemented in a computer algebraic system. Compared with most of the existing Jacobi elliptic function expansion methods, the proposed method can be expected to obtain new and more general formal solutions. We choose a (2 + 1)-dimensional dispersive long wave equation to illustrate the method.

Using symbolic computation to construct travelling wave solutions to nonlinear partial differential equations

Based upon the symbolic computation and the coupled projective Riccati equation, the tanh function method is further improved. As its applications, Wu–Zhang equation (which describes a (2+1)-dimensional dispersive long wave) and the (1+1)-dimensional dispersive long wave equation obtained from Wu–Zhang equation by scaling transformation and symmetry reduction are chosen to illustrate the validity of the proposed approach.

A Generalized Algebraic Method for Constructing a Series of Explicit Exact Solutions of a (1+1)-Dimensional Dispersive Long Wave Equation**The project supported by the National Outstanding Youth Foundation of China under Grant No. 19925522 and National Natural Science Foundation of China under Grant No. 90203001

Making use of a new and more general ansatz, we present the generalized algebraic method to uniformly construct a series of new and general travelling wave solution for nonlinear partial differential equations. As an application of the method, we choose a (1+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by the method proposed by Fan [E. Fan, Comput. Phys. Commun. 153 (2003) 17] and find other new and more general solutions at the same time, which include polynomial solutions, exponential solutions, rational solutions, triangular periodic wave solutions, hyperbolic and soliton solutions, Jacobi and Weierstrass doubly periodic wave solutions.

Jacobi Elliptic Function Rational Expansion Method with Symbolic Computation to Construct New Doubly-periodic Solutions of Nonlinear Evolution Equations

A new Jacobi elliptic function rational expansion method is presented by means of a new general ansatz and is very powerful, with aid of symbolic computation, to uniformly construct more new exact doubly-periodic solutions in terms of rational form Jacobi elliptic function of nonlinear evolution equations (NLEEs). We choose a (2+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we obtain the solutions found by most existing Jacobi elliptic function expansion methods and find other new and more general solutions at the same time. When the modulus of the Jacobi elliptic functions m→1 or 0, the corresponding solitary wave solutions and trigonometric function (singly periodic) solutions are also found.

Well-posedness, global solutions and blowup phenomena for a nonlinearly dispersive wave equation

We establish the local well-posedness for a new nonlinearly dispersive wave equation which has solutions that exist for indefinite times as well as solutions that blowup infinite time. We also derive an explosion criterion for the equation, and we give a sharp estimate of the existence time for solutions with smooth initial data.

Existence of periodic travelling-wave solutions of nonlinear, dispersive wave equations

For general classes of nonlinear, dispersive wave equations, we discuss the existence of the analogue of the cnoidal-wave solutions of the Korteweg–de Vries equation. The solutions in question have travelling and periodic features. The topological-degree theory of positive operators on Banach spaces is applied.