Abstract
The dispersionless Whitham modulation equations in 2+1 (two space dimensions and time) are reviewed and the instabilities identified. The modulation theory is then reformulated, near the Lighthill instability threshold, with a slow phase, moving frame and different scalings. The resulting nonlinear phase modulation equation near the Lighthill surfaces is a geometric form of the 2+1 two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multi-periodic, quasi-periodic and multi-pulse localized solutions. For illustration the theory is applied to a complex nonlinear 2+1 Klein-Gordon equation which has two Lighthill surfaces in the manifold of periodic travelling waves.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.
Highlights
Modulation of periodic travelling waves, in classicalWhitham modulation theory in 2 + 1, results in a closed system of firstorder nonlinear PDEs ⎫qT − ΩX = 0, pT − ΩY = 0, ⎪⎪⎪⎪⎪⎬and pX − qY = 0 AT + BX + CY = ⎪⎪⎪⎪⎪⎭ 0. (1.1)The fourth equation is conservation of wave action with A, B and C functions of (Ω, q, p), the modulation frequency and wavenumbers, with T = εt, X = εx and Y = εy slow time and space variables
The basic state is considered stable when this phase modulation equation is hyperbolic in time ( L < 0 and Aω M > 0) and unstable otherwise
Unfolding and scaling the coefficients leads to the following canonical form for the emergent 2 + 1 two-way Boussinesq equation uτ τ
Summary
Whitham modulation theory in 2 + 1 (two space dimensions and time), results in a closed system of firstorder nonlinear PDEs. The basic state is considered stable when this phase modulation equation is hyperbolic in time ( L < 0 and Aω M > 0) and unstable otherwise. The three key diagnostics in the second-order phase equation (2.4) are computed to be ⎫ In these expressions, has been set to zero, after the derivatives are taken, as this is the case needed in the nonlinear modulation. The phase equation (2.4) is hyperbolic in the time direction when L < 0 and Aω M > 0 or g11g22(1 + 3g11ω2 + 3g22k2) < 0 and g11g33(1 + 3g11ω2 + g22k2) < 0 This set is non-empty in the canonical case (3.2). A general theory will be developed for abstract Lagrangian PDEs, and it is applied to this example
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