This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.

Some asymptotic methods for strongly nonlinear equations

In this paper, a new method, called Exp-function method, is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations. The modified KdV equation and Dodd–Bullough–Mikhailov equation are chosen to illustrate the effectiveness and convenience of the suggested method.

Exp-function method for nonlinear wave equations

The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics

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.

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Linear and nonlinear waves

Variational iteration method-Some recent results and new interpretations

Variational iteration method has been favourably applied to various kinds of nonlinear problems. The main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper recent trends and developments in the use of the method are reviewed. Major applications to nonlinear wave equation, nonlinear fractional differential equations, nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed. The confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods. Variational iteration method is uniquely qualified to address this challenge. The flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions. This paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems. Both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations. A new iteration formulation is suggested to overcome the shortcoming. A very useful formulation for determining approximately the period of a nonlinear oscillator is suggested. Examples are given to illustrate the solution procedure.

Variational iteration method: New development and applications

Variational iteration method is used to construct solitary solutions and compacton-like solutions for nonlinear dispersive equations. The chosen initial solution (trial function) can be in compacton form or in soliton form with some unknown parameters which can be determined in the solution procedure. The compacton-like solution can be converted to solitary solution by suitable choice of a parameter, and vice versa.

Construction of solitary solution and compacton-like solution by variational iteration method

In this paper, the Exp-function method is used for seeking solitary solutions, periodic solutions, and compacton-like solutions of nonlinear differential equations. The combined KdV-MKdV equation and the Liouville equation are chosen to illustrate the effectiveness and convenience of the proposed method.

Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method

Exp-function method is used to find a unified solution of a nonlinear wave equation. Variant Boussinesq equations are selected to illustrate the effectiveness and simplicity of the method. A generalized solitary solution with free parameters is obtained.

Xu-Hong Wu

Papers

Exp-function method for nonlinear wave equations

Variational iteration method: New development and applications

Construction of solitary solution and compacton-like solution by variational iteration method

Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method

EXP-function method and its application to nonlinear equations