The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c.560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d’Alembert (1717-83) who found the first solution to the linear wave equation, see section (3.2). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet’s conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.

/pdf/linear-and-nonlinear-waves-4ds9thgmph.pdf

Linear and nonlinear waves

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.

https://iopscience.iop.org/article/10.1088/0031-8949/54/6/003/pdf

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

One method for finding exact solutions of nonlinear differential equations

A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F -expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 + 1 dimensional Jimbo–Miwa equation is treated, together with a Backlund transformation.

https://shell.cas.usf.edu/~wma3/MaL-CSF2009.pdf

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation

The tanh method for traveling wave solutions of nonlinear equations

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.

https://iopscience.iop.org/article/10.1088/0031-8949/54/6/004/pdf

The tanh method: II. Perturbation technique for conservative systems

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.

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

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 Schrodinger (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.

Willy Malfliet

Papers

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

The tanh method: II. Perturbation technique for conservative systems

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

Nonlinear Dispersive Rayleigh–Taylor Instabilities in Magnetohydrodynamic Flows

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