A method is proposed for obtaining traveling-wave solutions of nonlinear wave equations that are essentially of a localized nature. It is based on the fact that most solutions are functions of a hyperbolic tangent. This technique is straightforward to use and only minimal algebra is needed to find these solutions. The method is applied to selected cases.

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 h i t e 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 in6nity. Hence, exact closed form solutions can be obtained easily in various cases.

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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.Colorado School of Mines. Their hospitality is greatly appreciated. Thanks are due to Professor Frank Verheest for fruitful discussions.

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The basic set of equations describing nonlinear ion-acoustic waves in a cold collisionless plasma, in the limit of long wavelengths, is reconsidered. First, a travelling-wave solution is found up to third order by means of a straightforward perturbation approach based on the smallness of the wavenumber. As a result, a positive dressed solitary wave shows up, which is larger, taller and faster than the KdV soliton, the first-order result. Furthermore, the accuracy of this approach is tested and compared with previous result. Secondly, the reductive perturbation techique to study higher-order corrections is revised and adapted to the present problem.

In this paper a weakly nonlinear theory of wave propagation in superposed fluids in the presence of magnetic fields is presented. The equations governing the evolution of the amplitude of the progressive as well as the standing waves are reported. The nonlinear evolution of Rayleigh–Taylor instability (RTI) is examined in 2+1 dimensions in the context of Magnetohydrodynamics (MHD). This can be incorporated in studying the envelope properties of the 2+1 dimensional wave packet. We converted the resulting nonlinear equation (nonlinear Schrödinger (NLS) equation) for the evolution of the wave packets in 2+1 dimensions using the function transformation method into a sinh-Gordon equation and other nonlinear evolution equations. The latter depend only on one function ζ and we obtained several classes of general soliton solutions of these equations, leading to classes of soliton solutions of the 2+1 dimensional NLS equation. It contains some interesting specific solutions such as the N multiple solitons, the propagational breathers and the quadratic solitons, which contains the circular, elliptic and hyperbolic shape solitons. A stability analysis of these solutions is performed.

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