Integrable Singular Integral Evolution Equations

Abstract

Integro-differential evolution equations (IDEE’s) appear in several areas of applied Science. For instance, in a fluid dynamical context, equation

$$\begin{array}{*{20}{c}} {{{u}_{t}} + au{{u}_{x}} + \int\limits_{{ - \infty }}^{\infty } {dx\prime K(x - x\prime ){{u}_{{x\prime }}}(x\prime ,t) = 0,} } & {K(x): = \int\limits_{{ - \infty }}^{\infty } {\tfrac{{dk}}{{2\pi }}c(k){{e}^{{ikx}}}} } \\ \end{array}$$

(1.1)

is a simple mathematical model combining the breaking term uu_x of shallow water theory with a linear dispersion w(k) = kc(k) [1]. As observed in [2],[3], in a fluid of total depth D characterized by a thin termocline located at depth d and in a long wave regime, the dispersion takes the form [4]

$$c(k) = {{c}_{0}}(1 - \frac{{kd}}{2}(\coth (kD) - \frac{1}{{kD}}))$$

(1.2)

and gives rise to the IDEE

$${{u}_{t}} + ({{c}_{0}} + \frac{d}{{2D}}){{u}_{x}} + au{{u}_{x}} + \frac{d}{{4D}}P\int\limits_{{ - \infty }}^{\infty } {dx\prime \coth (\frac{\pi }{{2D}}(x\prime - x)){{u}_{{x\prime x\prime }}}(x\prime ,t) = 0,}$$

(1.3)

where the symbol P∫ denotes principal value integrals.