ON THE ASYMPTOTICS FOR CUBIC NONLINEAR SCHRÖDINGER EQUATIONS

林 仲夫

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation

\{ \begin{matrix}Lu = N (u) , x ∈ R, t > 1 \\ u (1, x) = u_{1} (x) , x ∈ R, \end{matrix} (0.1)

where L = i∂_{t} + \frac{1}{2}∂_{x}^{2},

N (u) = λ_{1}u^{3} + λ_{2}\overline{u}^{2}u + λ_{3}\overline{u}^{3},

λ_{j} ∈ C, j = 1, 2, 3. For the coefficients λ_{j} we assume that there exists θ_{0} > 0 such that

ℜ (\frac{λ_{1}}{\sqrt{3}}e^{2ir} − iλ_{2}e^{−2ir} + \frac{λ_{3}}{\sqrt{3}}e^{−4ir}) ≥ C > 0, (0.2)

ℑ (\frac{λ_{1}}{\sqrt{3}}e^{2ir} − iλ_{2}e^{−2ir} + \frac{λ_{3}}{\sqrt{3}}e^{−4ir}) r ≥ Cr^{2}, (0.3)

for all |r| < θ_{0}, and also we suppose that the initial data u_{1} (x) are such that

|arg e^{− \frac{i}{2} ξ^{2}} û_{1} (ξ)| < θ_{0}, inf_{|ξ|≤1} |û_{1} (ξ)| ≥ Cε, (0.4)

ε > 0 is a small constant depending on the size of the initial function in a suitable norm defined later. Note that conditions (0.2) - (0.3) can be fulfilled, for example, if we choose λ_{1} = −1, λ_{2} = λ_{3} = 0 and θ_{0} = \frac{π}{4} .

We let

(0.5) χ (ξ) = ℜ (\frac{λ_{1}}{\sqrt{3}} exp (2ir_{+} (ξ)) − iλ_{2} exp (−2ir_{+} (ξ)) + \frac{λ_{3}}{\sqrt{3}} exp (−4ir_{+} (ξ)))

and

|r_{+} (\frac{x}{t})| ≤ |θ| + ε^{\frac{1}{4}} , |W_{+} (\frac{x}{t})| ≤ 2ε.

To state our result precisely we now give some notations. Denote the usual Lebesgue space L^{p} = {ϕ ∈ S′; ∥ϕ∥_{p} < ∞}, where the norm ∥ϕ∥_{p} = (∫_{R} |ϕ (x)|^{p} dx)^{1/p} if 1 ≤ p < ∞ and ∥ϕ∥_{∞} = ess.sup_{x∈R} |ϕ (x)| if p = ∞. For simplicity we write ∥·∥ = ∥·∥_{2} . Weighted Sobolev space is

H^{m,k} = \{ϕ ∈ S′ : ∥ϕ∥_{m,k} ≡ ∥⟨x⟩^{k}⟨i∂⟩^{m}ϕ∥ < ∞\} , m, k ∈ R, ⟨x⟩ = \sqrt{1 + x^{2}}.

1This is a joint work with P.I.Naumkin

Date: 6 月 29 日, 2002.

We let K = FU (−t), where U (t) is the free Schrödinger group. The analytic function space A is defined

A = \{ϕ ∈ L^{2} : ∥ϕ∥_{A} ≡ ∑_{n=0}^{∞} \frac{ε^{n}}{n!} (∥I^{n}ϕ∥ + ∥KI^{n}ϕ∥_{∞}) < ∞\} .

Our main result is

Theorem 0.1. We suppose that conditions (0.2) - (0.4) are valid with sufficiently small ε > 0. We assume that the initial data u_{1} ∈ A_{1}. Then there exists an ε_{0} > 0 such that if ∥u_{1}∥_{A_{1}} ≤ ε, 0 < ε < ε_{0} , then there exists a unique solution u ∈ C ([1, ∞) , L^{∞}) of the Cauchy problem (0.1). Moreover there exist unique final state W_{+}, r_{+} ∈ L^{∞} and γ > 0 such that the following asymptotics for t → ∞

u (t, x) = \frac{(it)^{− \frac{1}{2}} W_{+} (\frac{x}{t}) e^{\frac{ix^{2}}{2t}}}{\sqrt{1 + χ (\frac{x}{t}) |W_{+} (\frac{x}{t})|^{2} log \frac{t^{2}}{t+x^{2}}}} + O (t^{− \frac{1}{2}} (1 + log \frac{t^{2}}{t + x^{2}} )^{− \frac{1}{2} −γ}) (0.6)

is valid uniformly with respect to x ∈ R,γ > 0, where χ (ξ) is given by (0.5).

大阪大学

E-mail address: nhayashi@math.wani.osaka-u.ac.jp