Asymptotic Methods and Nonlinear Evolution Equations

Abstract

Many physical systems involving nonlinear wave propagation include the effects of dispersion, dissipation, and/or the inhomogeneous property of the medium. The governing equations are usually derived from conservation laws. In simple cases, these equations are hyperbolic. However, in general, the physical processes involved are so complex that the governing equations are very complicated and, hence, are not integrable by analytic methods. So, special attention is given to seeking mathematical methods which lead to a less complicated problem yet retain all of the important physical features. In recent years, several asymptotic methods have been developed for the derivation of the evolution equations, which describe how some dynamical variables evolve in time and space. So, we begin this chapter with one simple method of construction of the linear evolution equation from a given frequency-wavenumber dispersion relation of the form

$$ \omega = f\left( k \right). $$

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This relation is multiplied by −iU (k) exp [i (kx − ωt)] and integrated with respect to the wavenumber k from −∞ to ∞ to obtain the equation

$$ \frac{{\partial u}} {{\partial t}} = L\left( u \right) = i\smallint _{ - \infty }^\infty f\left( k \right)U\left( k \right)\exp \left[ {i\left( {kx - \omega t} \right)} \right]dk, $$

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where L(u) represents an operator and U (k) is an arbitrary function which is related to the function u (x, t) by the inverse Fourier transform

$$ u\left( {x,t} \right) = \smallint _{ - \infty }^\infty U\left( k \right)\exp \left[ {i\left( {kx - \omega t} \right)} \right]dk. $$

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