Semigroup estimates and stability/instability results for the linearized three waves interaction equations

We consider the three waves interaction system and its linearization (the “pump-wave approximation”). We give some estimates on the semigroup as well as stability or instability results for the linearized problem in suitable norms. We work in the whole space and with periodic boundary condition, and our analysis relies on energy estimates and not on the complete integrability of the system. Key-words: three waves interaction equations, pump-wave approximation, stability, eigenvalue, semigroup. MSC (2010): 35Q99, 35F05, 35F20. 1 The three waves interaction equations In this paper, we consider the three waves interaction system in dimension d  ∂tA1 + v1 · ∇A1 + iσ1Ā2Ā3 = 0 ∂tA2 + v2 · ∇A2 + iσ2Ā3Ā1 = 0 ∂tA3 + v3 · ∇A3 + iσ3Ā1Ā2 = 0. (TWI) The amplitudes A` are complex-valued, the speeds v` are fixed vectors in R, and the coupling constants σ` are real-valued. To take into account a domain with periodic condition or in the whole space, the system (TWI) will be posed in Ω = ΩΛ ≡ d ∏ j=1 R/(2πΛjZ), with Λ = (Λj)1≤j≤d ∈ (0,+∞] and the convention that R/(2πΛjZ) = R if Λj = +∞. The system (TWI) appears in various areas of physics: plasma physics, fluid mechanics, optics, acoustics, mechanical or electrical oscillations... We refer, for instance, to the book [6] and to the paper [8]. One classical way to derive the system (TWI) is as follows. Consider a scalar wave equation with quadratic nonlinearity ∂tu+Qu+R(u) = 0 (1) (for example the KdV equation, the KP equation... but this can be extended to systems). The operators Q and R are pseudo-differential operators of symbol iQ = iQ(k) ∈ iR and iR = iR(k) ∈ iR respectively, and the dispersion relation for the linearized equation is ω = Q(k). If some wave numbers (k1, k2, k3) form a resonant triad, i.e. k1 + k2 + k3 = 0, ω1 + ω2 + ω3 = 0, with ω` ≡ Q(k`), ` = 1, 2, 3, 1 ha l-0 08 09 13 4, v er si on 1 8 Ap r 2 01 3 Author manuscript, published in "Revista Matematica Complutense 25 (2012) 285-333" DOI : 10.1007/s13163-011-0070-y