## Abstract

The amplitude equations that govern the motion of wavetrains near the critical point of unstable dispersive, weakly nonlinear physical systems are considered on slow time and space scales $T_m$ = $\epsilon^mt$; $X_m$ = $\epsilon^mx$ ($m$ = 1,2,...). Such systems arise when the dispersion relation for the harmonic wavetrain is purely real and complex conjugate roots appear when a control parameter ($\mu$) is varied. At the critical point, when the critical wavevector $k_c$ is non-zero, a general result for this general class of unstable systems is that the typical amplitude equations are either of the form $\Big(\frac{\partial}{\partial T_1}+c_1\frac{\partial}{\partial X_1}\Big)\Big(\frac{\partial} {\partial T_1}+c_2\frac{\partial}{\partial X_1}\Big) A = \pm \alpha A - \beta AB,$ $\Big(\frac{\partial}{\partial T_1} +c_2\frac{\partial}{\partial X_1}\Big) B =\Big(\frac{\partial}{\partial T_1}+c_1 \frac{\partial}{\partial X_1}\Big) |A|^2,$ or of the form $\Big(\frac{\partial}{\partial T _1}+c_1\frac{\partial}{\partial X_1}\Big)\Big(\frac{\partial}{\partial T_1}+c_2\frac{\partial}{\partial X_1}\Big) A = \pm \alpha A - \beta A |A|^2.$ The equations with the AB-nonlinearity govern for example the two-layer model for baroclinic instability and self-induced transparency (s.i.t.) in ultra-short optical pulse propagation in laser physics. The second equation occurs for the two-layer Kelvin-Helmholtz instability and a problem in the buckling of elastic shells. This second type of equation has been considered in detail by Weissman. The AB-equations are particularly important in that they are integrable by the inverse scattering transform and have a variety of multi-soliton solutions. They are also reducible to the sine-Gordon equation $\Phi_{\xi\tau} = \pm sin \Phi$ when A is real. We prove some general results for this type of instability and discuss briefly their applications to various other examples such as the two-stream instability. Examples in which dissipation is the dominant mechanism of the instability are also briefly considered. In contrast to the dispersive type which operates on the T$_1$-time scale, this type operates on the T$_2$-scale.

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