W. Malfliet

The tanh method: I. Exact solutions of nonlinear evolution and wave equations

A systemized version of the tanh method is used to solve particular evolution and wave equations. If one deals with conservative systems, one seeks travelling wave solutions in the form of a finite series in tanh. If present, boundary conditions are implemented in this expansion. The associated velocity can then be determined a priori, provided the solution vanishes at infinity. Hence, exact closed form solutions can be obtained easily in various cases.

The tanh method: II. Perturbation technique for conservative systems

With the aid of the tanh method, nonlinear wave equations are solved in a perturbative way. First, the KdVBurgers equation is investigated in the limit of weak dispersion. As a result, a general shock wave profile, with a perturbative solitary-wave contribution superposed, emerges. For a particular choice of the parameters, a comparison with the exact solution is made. Further, the MKdVBurgers is investigated in the same limit and similar results are obtained.

The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations

The tanh (or hyperbolic tangent) method is a powerful technique to search for travelling waves coming out from one-dimensional nonlinear wave and evolution equations. In particular, in those problems where dispersive effects, reaction, diffusion and/or convection play an important role. To show the strength of the method, an overview is given to find out which kind of problems are solved with this technique and how in some nontrivial cases this method, adapted to the problem at hand, still can be applied. Single as well as coupled equations, arising from wave phenomena which appear in different scientific domains such as physics, chemical kinetics, geochemistry and mathematical biology, will be treated. Next, attention is focussed towards approximate solutions. As a result, solitary- and shock-wave profiles are derived together with the associated width and velocity. The same method can be easily extended so that difference-differential equations can be similarly solved. Finally, some extensions of the method are discussed.

The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction-diffusion equations

Abstract Tanh method is used to find travelling wave solutions for a single nonlinear reaction–diffusion equation. Moreover the extension of the tanh method (a simple transformation) is used to find travelling wave solutions for coupled nonlinear reaction–diffusion equations.

Travelling wave solutions of some classes of nonlinear evolution equations in (1 + 1) and (2 + 1) dimensions 1

The tanh method is proposed to find travelling wave solutions in (1+1) and (2+1) dimensional wave equations. It can be extended to solve a whole family of modified Korteweg-de Vries type of equations, higher dimensional wave equations and nonlinear evolution equations.

Travelling wave solutions of some classes of nonlinear evolution equations in (1 + 1) and higher dimensions

The tanh method is used to find travelling wave solutions to various wave equations. In particular, it is extended to solve a coupled set of (1 + 1) dimensional Korteweg-de Vries type of equations, (3 + 1) dimensional Korteweg-de Vries like equation and Liouvilles equation. Also the stability of those solutions is investigated.

Comment on “A new mathematical approach for finding the solitary waves in dusty plasma” [Phys. Plasmas 5, 3918 (1998)]

It is argued that in their recent paper Das and Sarma use a method derived by Malfliet without mentioning his work. Quite to the contrary, they claim to have developed the method themselves (some 5 years later) with their co-workers. Moreover, Das and Sarma use expansions without checking their validity and applicability, show little insight into the theory of integrable or nonintegrable evolution equations and claim to be among the pioneers in dusty plasma research whereas they came late to the field.

Bäcklund transformations and exact solutions for some nonlinear evolution equations in solar magnetostatic models

The Backlund transformations for some nonlinear evolution equations (the Liouville, the sine and sinh-Poisson equations) are constructed through the AKNS system in unified manner. The obtained Backlund transformations are used to generate new classes of solutions. The latter is employed to obtain an infinite sequence of solutions. Moreover, we generate an infinite sequence of additional solutions by employing the permutability theorem and give a general expression for the nth order solution. It turns out that this general expression can be employed to obtain solutions for the sine and sinh-Poisson equations. The final results are used to investigate some models in solar plasma physics. Conclusions and comments are given.

Travelling-wave solutions of coupled nonlinear evolution equations

The tanh technique is used to solve exactly a set of nonlinear coupled equations small describing a problem arising in geochemistry. Next a coupled problem originating from the field of (deterministic) random walk theory with reaction kinetics is investigated but now solved approximately. The latter solution corresponds quite well with numerical results.

Dressed shock waves in a nonlinear LC circuit

Nonlinear shock waves which appear in a nonlinear Toda lattice with internal dissipation have been studied. The tanh method is used as a perturbation approach together with the assumption of long wavelengths. Moreover, asymptotical behaviour is used to determine the velocity. Burgers-like wave solutions up to third order are derived. If the amplitude of the shock wave is given, the profiles are readjusted to match this boundary condition. The associated steepness and velocity of the solution can be written as a function of this boundary condition.