Amplitude equations at the critical points of unstable dispersive physical systems

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 ═ ε m t ; X m ═ ε m x ( 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 ( μ ) 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 ( ∂/∂ T 1 + c 1 ∂/∂ X 1 ) (∂/∂ T 1 + c 2 ∂/∂ X 1 ) A ═ ±α A ─ β AB , ( ∂/∂ T 1 + c 2 ∂/∂ X 1 ) B ═ (∂/∂ T 1 + c 1 ∂/∂ X 1 ) | A | 2 , or of the form ( ∂/∂ T 1 + c 1 ∂/∂ X 1 ) (∂/∂ T 1 + c 2 ∂/∂ X 1 ) A ═ ±α A - β 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 ϕ ξƬ ═ ± sin ϕ 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.