Uniform adiabatic limit of Benney type systems
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Uniform adiabatic limit of Benney type systems Ad´ an J. Corcho and Juan C. Cordero
Abstract. In this paper, we show that solutions of the cubic nonlinear Schr¨ odinger equation are asymptotic limit of solutions to the Benney system. Due to the special characteristic of the one-dimensional transport equation, same result is obtained for solutions of the one-dimensional Zakharov and 1d-Zakharov–Rubenchik systems. Convergence is reached in the topology L2 (R) × L2 (R) and with an approximation in the energy space H 1 (R) × L2 (R). In the case of the Zakharov system, this is achieved without the condition ∂t n(x, 0) ∈ H˙ −1 (R) for the wave component, improving previous results. Mathematics Subject Classification. Primary 35Q55, 35Q60, Secondary 35B65. Keywords. Perturbed nonlinear Schr¨ odinger equation, Cauchy problem, Asymptotic behavior.
1. Introduction We consider a family of one-dimensional nonlinear dispersive systems, given by the following coupling equations: ⎧ 2 2 + ⎪ ⎨i∂t u + ∂x u = (τ |u| + αv + α z)u, (x, t) ∈ R × R , (1.1) ε∂t v + λ∂x v = β∂x |u|2 , ⎪ ⎩ ε∂t z + λ ∂x z = β ∂x |u|2 , where u is a complex-valued function, v and z are real-valued functions, the physical parameters τ, α, α , λ, λ , β, β are real numbers, and 0 < ε < 1. This model governs, on certain parameter regimes, the dynamics of many physical phenomena, and it is in the “neighborhood” of some other important models of the mathematical-physics, for example, the Zakharov system, the Davey–Stewartson system and the nonlinear Schr¨ odinger equation. We give further information about well-posedness concerning System (1.1) in Sect. 6. The family (1.1) contains, for instance, some cases of the non-resonant dynamics of small amplitude Alfven waves propagating in a plasma [9,20], modeled by the coupled equations: ⎧ 1 2 2 ⎪ ⎨i∂t u + ∂x u = k(c|u| − 2 aρ + ϕ)u, (1.2) ε∂t ρ + ∂x (ϕ − aρ) = −k∂x |u|2 , ⎪ ⎩ 1 2 ε∂t ϕ + ∂x (bρ − aϕ) = 2 k∂x |u| , where we have taken the frequency ω equal to 1, on the expanded flat wave front to generate u. This model is known as 1d-Zakharov–Rubenchik type system, which in the case b > 0 and b − a2 = 0, using A. J. Corcho was partially supported by CAPES and CNPq (307761/2016-9), Brazil. J. C. Cordero was partially supported by the Departamento de Matem´ aticas y Estad´ıstica, Universidad Nacional de Colombia, Sede Manizales. 0123456789().: V,-vol
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A. J. Corcho and J. C. Cordero
the transformation (see [17]) ρ = ψ1 + ψ2 , can be rewritten as
ϕ=
√
b(ψ1 − ψ2 ),
√ √ ⎧ 2 = (c|u|2 − ( b + a2 )ψ2 + ( b − a2 )ψ1 )u, ⎪ ⎨i∂t u + ∂x u √ a )∂ |u|2 , ε∂t ψ1 + ( b − a)∂x ψ1 = 12 (−1 + 2√ b x √ ⎪ ⎩ a ε∂t ψ2 − (a + b)∂x ψ2 = 12 (−1 − 2√ )∂ |u|2 . b x
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(1.3)
(1.4)
Another system included in the family (1.1) is the 1d-Zakharov system describing Langmuir turbulence [31], given by i∂t u + ∂x2 u = nu, (1.5) ε2 ∂t2 n − ∂x2 n = ∂x2 |u|2 , where 0 < ε = k/cs < 1, k is a positive parameter and cs the ionic sound speed. It can be set in the fo
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