Scattering of solitons on resonance

We investigate a propagation of solitons for nonlinear Schrodinger equation under small driving force. The driving force passes the resonance. The process of scattering on the resonance leads to changing of number of solitons. After the resonance the number of solitons depends on the amplitude of the driving force. Nonlinear Schrodinger equation (NLSE) is a mathematical model for wide class of wave phenomenons from signal propagation into optical fibre [1, 2] to surface wave propagation [3]. This equation is integrable by inverse scattering transform method [4] and can be considered as an ideal model equation. The perturbations of this ideal model lead to nonintegrable equations. Here we consider such nonintegrable example which is NLSE perturbed driving force. The most known class of the solutions of NLSE is solitons [4]. The structure of this kind of solutions is not changed in a case of nonperturbed NLSE. The perturbations usually lead to modulation of parameters of solitons [5, 6]. Number of solitons does not change. In this work we investigate a new effect called scattering of solitons on resonance. We consider the process of scattering in detail and obtain the connection formula between pre-resonance and post-resonance solutions. In general case the passage through resonance leads to changing of the number of solitons. This effect is based on the soliton generation due to passage through resonance by external driving force [7]. We found that the scattering of solitary waves on resonance is a general effect for nonlinear equations described the wave propagation. In this work we investigate this effect for the simplest model. It allows to show the essence of this effect without unnecessary details. This work was supported by RFBR 03-01-00716, grant for Sci. Schools 1446.2003.1 and INTAS 03-51-4286 1 STATEMENT OF THE PROBLEM AND RESULT 2 This paper has the following structure. The first section contains the statement of the problem and the main result. The second section contains the asymptotic construction in the pre-resonance domain. In the third section we construct the asymptotic solution in the neighborhood of the resonance curve. The fourth section of the paper is devoted to construction of the post-resonance asymptotics. Asymptotics are constructed by multiple scale method [8] and matched [9]. 1 Statement of the problem and result Let us consider the perturbed NLSE i∂tΨ+ ∂ 2 xΨ+ |Ψ|Ψ = εfe 2 , 0 < ε ≪ 1. (1) The phase of the driving force is S/ε = ωt. The amplitude f = f(εx) is a smooth and rapidly vanished function. In the simplest case the phase is linear function with respect to t S/ε = ωt, ω = const. In general situation the constant frequency of the driving force does not lead to scattering of solitons. Let us investigate the driving force with slowly varying frequency. The most simplest dependence on t for ω has a form ω = εt/2. The amplitude f of the driving force admit an additional dependency on εt but it leads to complicated formulas and no more. Let us formulate the result of this work. Below we use the following variables xj = ε x, tj = ε t, j = 1, 2. Let the asymptotic solution of (1) be Ψ(x, t, ε) = ε 1 u (x1, t2) +O(ε ) as t2 < 0, where 1 u (x1, t2) satisfies ∂t2 1 u +∂ x1 1 u +| 1 u | 1 u= 0 and initial condition 1 u |t2=t0 = h1(x1), t = const < 0. Then in the domain t2 > 0 the asymptotic solution of (1) has a form Ψ(x, t, ε) = ε 1 v (x1, t2) +O(ε ), (2)