Abstract
The KdV equation can be derived via multiple scaling analysis for the approximate description of long waves in dispersive systems with a conservation law. In this paper, we justify this approximation for a system with unstable resonances by proving estimates between the KdV approximation and true solutions of the original system. By working in spaces of analytic functions, the approach will allow us to handle more complicated systems without a detailed discussion of the resonances and without finding a suitableenergy.
Highlights
We consider the Boussinesq-Klein-Gordon (BKG) system∂t2u = α2∂x2u + ∂t2∂x2u + ∂x2(auuu2 + 2auvuv + avvv2), (1)∂t2v = ∂x2v − v + buuu2 + 2buvuv + bvvv[2], (2)where u = u(x, t), v = v(x, t), x, t ∈ R, and coefficients α > 0, auu, . . . , bvv ∈ R
The amplitude A(X, T ) ∈ R depends on the long temporal variable T = ε3t and on the long spatial variable X = ε(x − αt)
We have that ∂T A can be expressed via the right-hand side of the KdV equation in terms of A, . . . , ∂X3 A
Summary
In this situation the KdV approximation makes wrong predictions It can only make correct predictions if initially εnA1 and εnB1 are chosen exponentially small w.r.t. ε, cf Assumption (5) in Theorem 1.1. For the BKG system in [CS11] for α < 2, i.e., in case of no additional quadratic resonances, i.e., in case ωu(k) = ωv(k) for all k ∈ R, a KdV approximation result has been established. It has been explained in [BCS17] how to establish an approximation result for all α ≥ 2 in case of stable quadratic resonances, i.e., in case that the eigenvalues of M are purely imaginary or have negative real part It is the purpose of this paper to cover the case of unstable quadratic resonances, i.e, when M has at least one eigenvalue with positive real part. Different constants which can be chosen independently of the small perturbation parameter 0 < ε2 1 are denoted with the same symbol C
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