Amplitude Equations at the Critical Points of Unstable Dispersive Physical Systems

• Published in: Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences Mathematical and physical sciences, 1981-06, Vol.377 (1769), p.185-219
• Main Authors: Gibbon, J. D., McGuinness, M. J.
• Format: Article
• Language: English
• Subjects: Amplitude; Baroclinic instability; Complex roots; Critical points; Critical values; Group velocity; Nonlinearity; Pendulums; Physics; Solitons
• ISSN: 1364-5021; 0080-4630; 1471-2946; 2053-9169
• DOI: 10.1098/rspa.1981.0121

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 Tm ═ εmt; Xm ═ εmx (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 kc is non-zero, a general result for this general class of unstable systems is that the typical amplitude equations are either of the form ( ∂/∂T1 + c1∂/∂X1) (∂/∂T1 + c2∂/∂X1) A ═ ±αA ─ βAB, ( ∂/∂T1 + c2∂/∂X1) B ═ (∂/∂T 1 + c1∂/∂X1) |A|2, or of the form ( ∂/∂T1 + c1∂/∂X1) (∂/∂T1 + c2∂/∂X1) A ═ ±αA - βA |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 ϕξƬ ═ ± 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 T1-time scale, this type operates on the T2-scale.