Asymptotic Expansion of Small Analytic Solutions to the Quadratic Nonlinear Schrödinger Equations in Two-dimensional Spaces

We study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spaces i∂ t u + (1/2)∆u = ᏺ(u), (t, x) ∈ R × R 2 ; u(0,x) = ϕ(x), x ∈ R 2 , where ᏺ(u) = 2 j,k=1 (λ jk (∂ x j u)(∂ x k u)+µ jk (∂ x j u)(∂ x k u)), where λ jk ,µ jk ∈ C. We prove that if the initial data ϕ satisfy some analyticity and smallness conditions in a suitable norm, then the solution of the above Cauchy problem has the asymptotic representation in the neighborhood of the scattering states. 1. Introduction. We consider the large time asymptotic behavior of small analytic solutions to the Cauchy problem for the derivative nonlinear Schrödinger equation in two-dimensional spaces


Introduction.
We consider the large time asymptotic behavior of small analytic solutions to the Cauchy problem for the derivative nonlinear Schrödinger equation in two-dimensional spaces with quadratic nonlinearity where λ jk ,µ jk ∈ C. In [8], we proved the global in time existence of small analytic solutions to the Cauchy problem (1.1) and showed that the usual scattering states exist.In [3], a global existence theorem of small solutions to (1.1) with λ jk = 0 was shown in the usual weighted Sobolev space by using the method of normal forms by Shatah [12].In the present paper, we continue to study the asymptotic behavior in time of solutions to the Cauchy problem (1.1) and obtain the asymptotic expansion of solutions in the neighborhood of the scattering states.We use the following classification of the scattering problem.If the usual scattering states exist in L 2 sense, then we call the scattering problem a super-critical problem.If the usual scattering states do not exist and the L 2 norm of the nonlinearity decays like Ct −δ , then we call the problem a critical one, when δ = 1 and a sub-critical one, when 0 < δ < 1.The problem under consideration is classified as super-critical since the usual scattering states were shown, in [8], to exist in L 2 .In [10], the asymptotic expansion was obtained in the neighborhood of scattering states for small solutions to the nonlinear nonlocal Schrödinger equations with nonlinearities of Hartree type ᏺ(u) = u(t, x) dy|x − y| −δ u(t, y) 2 (1.3) in the super-critical case 1 < δ < n.The critical case δ = 1 was treated in [13], where the asymptotic expansion of small solutions in the neighborhood of the modified scattering states was obtained.In the case of critical power nonlinearity ᏺ(u) = |u| 2 u in onedimensional spaces, the asymptotic expansion of solutions was constructed in [11].In [5,6], the sub-critical scattering problem in one-dimensional spaces was studied for the nonlinear Schrödinger equation with power nonlinearity ᏺ(u) = t 1−δ |u| 2 u and Hartree type nonlinearity (1.3) with 0 < δ < 1. Roughly speaking, they used the asymptotic expansion in the neighborhood of the final states to the transformed equations for the new dependent variable (in the case of the power type nonlinearity).
Thus the asymptotic expansions of solutions to the nonlinear Schrödinger equations were studied extensively in the case of the nonlinear terms without derivatives of unknown function and satisfying the gauge condition (i.e., having the self-conjugate property ᏺ(u) = e −iθ ᏺ(e iθ u) for any θ ∈ R).The present paper is concerned with the derivative nonlinear Schrödinger equations which do not satisfy the gauge condition.The presence of derivatives in the nonlinear term implies the so-called derivative loss and the absence of the gauge condition makes it difficult to estimate the norm involving the operator = x + it∇, which plays a crucial role in the large time asymptotic behavior of solutions to the nonlinear Schrödinger equations.To overcome these obstacles, we use the analytic function spaces A m,p b defined in (1.9) and the operators ᏼ = x •∇+2t∂ t and ᏽ = x •∇+it∆.
To state our result precisely, we now give notation and function spaces.We denote , where the operators Ω (j,k) = x j ∂ k − x k ∂ j act as the angular derivatives.These operators help us to obtain the time decay properties of the linear Schrödinger evolution group where Ᏺφ ≡ φ(ξ) = (1/2π) e −i(x•ξ) φ(x)dx denotes the Fourier transform of the function φ(x), and Ᏺ −1 is the inverse Fourier transformation defined by Ᏺ −1 φ ≡ φ(x) = (1/2π) e i(x•ξ) φ(ξ)dξ.Note that the free Schrödinger evolution group ᐁ(t) also can be represented as ᐁ(t) = ᏹ(t)Ᏸ(t)Ᏺᏹ(t), where ᏹ(t) = exp(ix 2 /2t), the dilation operator is (Ᏸ(t)φ)(x) = (i/t)φ(x/t), then the inverse free Schrödinger evolution group is written as is the inverse dilation operator.We define the extended vectors Γ = (ᏼ, Ω, ∇), Γ = (ᏼ + 2, Ω, ∇), and Θ = (ᏽ, Ω, ∇).We have the following relations: where where δ are used freely in the paper.We denote the usual Lebesgue space by L p (R 2 ) with the norm The weighted Sobolev space is defined by where We write for simplicity ) and the norm φ m,k = φ m,k,2 .Now we define the analytic function space where the vector Here the summation is over all admissible multi-indices α.We often use the summations convention if it does not cause confusion.By [s] we denote the largest integer less than or equal to s.Let C(I; B) be the space of continuous functions from a time interval I to a Banach space B. We denote different positive constants by the same letter C. We introduce the following functional spaces where where η > 0 is sufficiently small.We define the constants Let u 0 (t) = ᐁ(t)u + with some final state u + ∈ L 2 and u n (t), n = 1, 2,..., be the solution to the final problem for the linear Schrödinger equations From [8] we see that if the initial data ϕ ∈ A 3 a are such that x j ϕ ∈ A 2 a for j = 1, 2 and the norm ϕ Now we state the main result in this paper.
Theorem 1.1.We assume that the initial data ϕ ∈ A 3 a are such that x j ϕ ∈ A 2 a for j = 1, 2 and the norm ϕ of the Cauchy problem (1.1).Moreover, the estimates and the asymptotics are valid for all t ≥ 1, where w ∈ (0, 1/2) and where C is a positive constant independent of n and b j .
We assume in Theorem 1.1 that 0 < a < 1.This ensures that the function space A 3 a for the initial data is not empty, as in [1,2], we can see that our result is valid for the initial function φ, which has analytic continuation Φ to the domain where ϑ ∈ (0,π/2), sinϑ = C 2 , and C 1 ,C 2 ∈ (a, 1).For example, we can take 1/(1+x 4 ), e −x 2 as the initial data for the Cauchy problem (1.1). Denote µ jk ξ j ξ k t ∞ e 3itξ 2 /4 dτ iτ .
For the convenience of the reader we now give the outline of the proof of Theorem 1.1.As in [8] we apply the operator Ᏺᐁ(−t) to (1.1) to get where , and R is a remainder term since in [8] we proved the estimate R ≤ Then we show that the first term of the integral I(t, ξ)dt is also convergent in view of the oscillating factor E. Roughly speaking, in [8] the following estimate was shown: Similarly, we have However, the right-hand sides of (1.25) and (1.26) contain an additional operator Θ (derivative loss with respect to derivative Θ).This is the reason why we used the analytic function spaces involving generalized derivative Γ , enabling us to get an additional regularity with respect to operator Γ , hence we obtain the estimate for all t > s > 0. The last estimate implies existence of the usual scattering states u + .Method of analytic function spaces involving usual derivatives was used by many authors (e.g., see [4,9]) and analytic function spaces involving the generalized derivatives was used in [7].By the definition of u n (t) we have with ᏸ where R is the remainder term since R Y b n+1 ≤ Cε n+2 |t| −1−(n+1)w .In Section 3, we will prove that the other three terms are estimated by Cε n+1 |t| −1−nw in the norm Y b n+1 .Then via the inequality for any |α| we obtain the desired result.The rest of the paper is organized as follows.In Section 2, we state some preliminary estimates concerning the analytic functional spaces A m b .Section 3 is devoted to the proof of Theorem 1.1 and Corollary 1.2.

Preliminary estimates.
We summarize some lemmas proved in [7,8], which are necessary to prove the theorem. where ) Lemma 2.2.The following commutation relations are valid: where is true.

Proof of Theorem 1.1.
We consider the linear Schrödinger equation Since u 0 (t) is a solution of linear Schrödinger equation it is easy to see that u 0 (t) Y b 0 ≤ C 0 ε.Then by induction we assume that Multiplying both sides of (3.1) by Γ α+δ , where = Ᏺᏹᐁ(−t), we get where Applying the operator ᐂ(−t) = Ᏺ −1 ᏹ(t)Ᏺ to both sides of (3.3) we obtain by virtue of identity (2.7) Then we write the identity By Lemma 2.5, we have the estimate (3.6) Thus we can rewrite (3.4) in the form where the remainder term R 1 (t) can be estimated by virtue of (3.6) and Lemmas 2.1, 2.2, 2.3, and 2.4 as follows: we have (3.10) By virtue of the identity Hence by (3.7) we get If |δ − γ| < |γ| we exchange f and g in the right-hand side of (3.11).By virtue of the equality tE ν ) we obtain the identity (3.12) Therefore, we get from (3.11) where 3 j=1 I j , and (3.15)By Hölder's inequality, the identity j = ᐁ(t)x j ᐁ(−t), and Lemmas 2.1, 2.2, 2.3, and 2.4 we get the estimates for |δ| ≤ 3.In the same way we obtain (3.17) In view of (3.10), (3.16), and (3.17) we have the estimate Multiplying both sides of (3.13) by Ψ (t), integrating with respect to the space variables, and taking the imaginary part of the result, we obtain the inequality Then integration with respect to t in view of (3.18) yields Applying the operator x l to both sides of (3.1), we get Therefore, by virtue of (3.24) and (3.25) we get for any n ∈ N. We next prove the last part of the theorem by induction.By the definition of u n (t) we have with ᏸ where and R consists of quadratic nonlinearities involving u k u l and ūk ūl with k+l ≥ n.We have In the same way as in the proof of (3.7) we estimate We have by (3.10) and the fact that b

.33)
Hence by the assumption and (3.28) We also have by (3.28) Proof of Corollary 1.2.By (1.1) we have and the property of the solution of (1.1) we see that R ≤ Cε 2 |t| −1−wt .In the same way as in the proof of Theorem 1.1 we have By the definition of u we see that where (3.41) We iterate this procedure to get (3.42) with u + 0 (t, ξ) = u + (ξ).This completes the proof of Corollary 1.2.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
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