ASYMPTOTICAL ANALYSIS OF SOME COUPLED NONLINEAR WAVE EQUATIONS

We consider coupled nonlinear equations modelling a family of travelling wave solutions. The goal of our work is to show that the method of internal averaging along characteristics can be used for wide classes of coupled non-linear wave equations such as Korteweg-de Vries, Klein – Gordon, Hirota – Satsuma, etc. The asymptotical analysis reduces a system of coupled non-linear equations to a system of integro – differential averaged equations. The averaged system with the periodical initial conditions disintegrates into independent equations in non-resonance case. These equations describe simple weakly non-linear travelling waves in the non-resonance case. In the resonance case the integro – differential averaged systems describe interaction of waves and give a good asymptotical approximation for exact solutions.

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Evolution of near-soliton initial conditions in non-linear wave equations through their Bäcklund transforms

Chaos Solitons & Fractals ◽

10.1016/s0960-0779(04)00449-7 ◽

2005 ◽

Vol 23 (5) ◽

pp. 1841-1854 ◽

Cited By ~ 5

Author(s):

G TSIGARIDAS ◽

A FRAGOS ◽

I POLYZOS ◽

M FAKIS ◽

A IOANNOU ◽

...

Keyword(s):

Wave Equations ◽

Initial Conditions ◽

Linear Wave ◽

Non Linear ◽

Linear Wave Equations

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A note on improved F -expansion method combined with Riccati equation applied to nonlinear evolution equations

Royal Society Open Science ◽

10.1098/rsos.140038 ◽

2014 ◽

Vol 1 (2) ◽

pp. 140038 ◽

Cited By ~ 24

Author(s):

Md. Shafiqul Islam ◽

Kamruzzaman Khan ◽

M. Ali Akbar ◽

Antonio Mastroberardino

Keyword(s):

Riccati Equation ◽

Wave Equations ◽

Evolution Equations ◽

Nonlinear Evolution ◽

Vries Equation ◽

Expansion Method ◽

Travelling Wave ◽

Nonlinear Evolution Equations ◽

Nonlinear Wave Equations ◽

Travelling Wave Solutions

The purpose of this article is to present an analytical method, namely the improved F -expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.

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POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891606000732 ◽

2006 ◽

Vol 03 (01) ◽

pp. 81-141 ◽

Cited By ~ 5

Author(s):

PIOTR T. CHRUŚCIEL ◽

SZYMON ŁȨSKI

Keyword(s):

Initial Data ◽

Wave Equations ◽

Space Time ◽

Einstein Equations ◽

Nonlinear Wave Equations ◽

Linear Wave ◽

Cauchy Problems ◽

Time Dimension ◽

Wave Maps ◽

Corner Conditions

The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

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