Energy transfer model for the derivative nonlinear Schrodinger equations on the torus

We consider the nonlinear derivative Schrodinger equation with a quintic nonlinearity, on the one dimensional torus. We exhibit that the nonlinear dynamic properties consist of four frequency modes initially excited, whose frequencies include the resonant clusters and phase matched resonant interactions of nonlinearities. This phenomena arrests energy transfers between low and high modes, which are quantified by a growth in the Sobolev norm.


Introduction
In this paper we consider the derivative nonlinear Schrödinger equation with quintic nonlinearity: i∂u + ∂ 2 x u = −iλu 2 ∂ x u + µ|u| 4 u, (t, x) ∈ R × T, u(0, x) = u 0 (x), x ∈ T, (1.1) where u = u(t, x) : R × T → C, λ, µ ∈ R, and T = R/2πZ is the torus. Our aim is to prove that there exist solutions of (1.1) which initially oscillate, and illustrate the energy transfer from one coupled oscillator to another. The derivative nonlinear Schrödinger equation in the completely integrable form is written as As in [10], by using the gauge transform v(t, x) = u(t, x)e −iG [u] , * This work was supported by JSPS KAKENHI Grant Number 10322794.
we derive a equivalent equation of (1.2) for v as follows: The equations (1.2) and (1.3) are completely integrable Hamiltonian equations, having infinitely many conservation laws [2]. The well-posedness results for the equation (1.2), in particular considering solutions in the Sobolev space H s , were studied in many works (e.g., [10,16,17]). If u 0 ∈ H s with s ≥ 1/2, local well-posedness was established by Herr [10]. This combined with the conservation laws, can often yield the corresponding global well-posedness result for data in H s for s ≥ 1, which is small in L 2 . Extension of the global well-posedness with low regularity data was obtained in the case s > 1/2 by Su Win [17], by using the I-method developed in [3,4,5]. A further extension was given by [11,12,16]. For instance, in [11] Grünrock and Herr proved the local well-posedness result for initial data in Fourier-Lebesgue space H s r (1.1), in the range s ≥ 1/2, 4/3 < r < 2, where the space H s r is defined by the norm Later on in [12] Nahmod, Oh, Rey-Bellt and Staffilani established the almost sure global well-posedness of (1.2) with random initial data in H s r with s > 1/2 and certain 4/3 < r < 2. Below s = 1/2, in [16] Takaoka obtained the existence and time continuity of solution in H s with some s < 1/2. We recall that the initial value problem of the equation (1.2) is ill-posed in H s provided s < 1/2 (see [1]). In the case of the line case setting, analogous results to the well-posedness of the equation (1.2) were taken from the references [4,5,8,9,13,14,15].
The above technology can also be applied to the equation in (1.1) to obtain the local and global well-posedness in H s . Indeed, the initial value problem (1.1) is locally well-posed in H s for all s ≥ 1/2. In order to capture the global solution in the energy class s ≥ 1, we need a priori estimate. Formally, solutions of (1.1) satisfy the conservations laws, that is, the following three quantities are constant in time: where m[u] = |u| 2 (mass density), Note that in the case µ > −5λ 2 /16, the energy E[u](t) assigns positive sign. In practice if µ > −λ 2 /16, the absorption of the second term λ 4 |u| 2 Im(u∂ x u) into the first and third terms yields for some c > 0, and we conclude We refer to the cases µ > −5λ 2 /16 and µ < −5λ 2 /16 as that the nonlinearity are respectively defocusing and focusing. By (1.5) and L 2 -norm conservation (1.4), in the case µ > −5λ 2 /16 we have the global well-posedness for any data in H s for any s ≥ 1, whereas in the case µ ≤ −5λ 2 /16 we require small in L 2 .
In the present paper, we concentrate on defocusing case as prototype problem. The purpose of this paper is to investigation the dynamics on resonant clusters for (1.1) in the underlying nonlinear energy transfers. Notice that such energy exchange processes have been seen with the defocusing cubic nonlinear Schrödinger equation on two spacial dimension by Colliander, Keel, Staffilani, Takaoka and Tao [6], and with the quintic nonlinear Schrödinger equation on one spacial dimension by Grébert and Thomann [7]. We summarize these results in Remark 1.4 and Remark 1.5, respectively. Now it is similar in [7], we begin with a resonant set associated to (1.1). We are interested in whether there exist solutions that have frequencies of the energy exchange observed between the Fourier modes when only four are initially excited. Definition 1.1. For (M, N ) ∈ Z 2 with λ(M + N ) > 0, we will use the notation and M * = max{|M |, |N |}. With numbers α j , β j (j = 1, 2) satisfying Let P L and P H be frequency cut-off operators at low and high frequency modes, respectively, defined as follows: and P H f = (I − P L )f . The main result of this paper is the following theorem. There exist c * ∈ (0, 1/2) and T > 0 satisfying T → 0 as 1 |µ| → 0 such that the following holds: there exist 2T -periodic function K(t) : R → (0, 1) and a solution to (1.1) Moreover for δ ∈ [1/2, 1) and |t| ≪ min{ 1 |λ|M * , 1 Remark 1.2. One notices that the periodicity rate T obtained in Theorem 1.1 has the bound 1 c|µ| < T < c |µ| , where the constant c > 1 is independent of λ, µ, M and N . Notice that 1 |µ| ≫ min{ 1 |λ|M * , 1 λ 2 +|µ| }, provided |λ|M * ≫ |µ| or |λ| ≫ 1. So it is unclear what the time trajectory of e(t, x) in (1.7) describes beyond the time interval |t| ≪ min{ 1 |λ|M * , 1 λ 2 +|µ| } in that setting. Remark 1.3. The above theorem gives the approximation for |t| ≪ min{ 1 |λ|M * , 1 λ 2 +|µ| }, where masses a βj are excited initially compared these of a αj , then move decreasingly to that of a αj , in periodically. i∂ t u + ∆u = |u| 2 u, one can obtain the low-to-high energy cascade scenario, see [6]. In the proof, an instability phenomenon of Arnold diffusion phenomena is considered in perturbed Hamiltonian systems. Indeed, for any s > 1 and K ≫ 1 ≫ ε there exist a global solution u(t) of 2d-NLS on T 2 and time T satisfying that Remark 1.5. In [7], one striking example of a similar situation is considered for the quintic nonlinear Schrödinger equation on one spacial dimension: For (k, n) ∈ Z 2 with k = 0, a resonant set Λ 1 corresponding to (1.8) is the following: 2 ) = (n + 3k, n, n + k, n + 4k). Comparing to the result stated in Theorem 1.1, they obtained dynamical properties for (1.8) in stronger sense. More precisely, it is obtained that there exist T > 0, ν 0 > 0, 2T -periodic function K : R → (0, 1) with K(0) ≤ 1/4 and K(T ) ≥ 3/4 so that for all 0 < ν < ν 0 and s ∈ R, there exists a solution of (1.8) satisfying The proof is centered on the resonant normal form approach. The formula (1.7) looks very similar to that of (1.9), but this conveys different information because of obstructions by the derivative nonlinearity in (1.1).
We now turn to equation adjusting the amplitude scaling: which transforms the equation (1.1) in the case λ = 20(M + N ) to the following: where Apply this to Theorem 1.1, we obtain the immediate corollary, which is equivalent to the counterpart of Theorem 1.1.
There exist c * ∈ (0, 1/2) and T > 0 satisfying T → 0 as 1 |µ ′ | ( M+N λ ′ ) 2 → 0 such that the following holds: there exist 2T -periodic function K(t) : R → (0, 1) and a global solution to (1.10) Moreover for δ ∈ [1/2, 1) and |t| ≪ min{ Theorem 1.1 is obtained by analyzing an infinitely dimensional system of ordinary differential equations that include the resonant components on nonlinearity of (1.1). We first make the Fourier transform to the equation (1.1) and build a resonant approximation model (toy model), which is an ordinary differential system. This is accomplished by investigating the long time dynamics of toy model. The error estimate is proved to approximate well the solution of (1.1) by that of the toy model by using the bootstrap argument. Notation 1.1. We prefer to use the notation · = (1 + | · | 2 ) 1/2 . The over doṫ a(t) denotes the time derivative of a(t).
The Fourier transform with respect to the space variable (discrete Fourier transform) is defined by We abbreviate F x f as f .
with the obvious modification when q = ∞. We also use L p t ℓ q ξ in a similar manner.
We use c, C to denote various constants that does not depend on λ, µ, M and N . We use A B to denote A ≤ CB for some constant C > 0. Similarly, we write A ≪ B to mean A ≤ cB for some small constant c > 0.
The rest of this paper is organized as follows. In Section 2, we reduce the equation (1.1) to that for the Fourier coefficients, by taking the advantage of conservation laws (1.4). In Section 3, we construct an appropriate ordinary differential equation (ODE) system as theoretical toy model for (1.1). In Section 4, we solve the ODE system. In Section 5, the equation that we derived in the previous section should approximate the dynamics of the first differential equation (1.1).
. Taking the momentum conservation (1.4) via Fourier transformation, we obtain On substituting this into the second term on the right-hand side of (2.3), we have the following form , it oscillates in space-time at different frequencies, which illustrates hidden smoothing effect. Thus for the third term on the right-hand side of (2.4), we divide the sum into two cases: We also divide the sum in the last term on the right-hand side of (2.4) into two cases; The contribution of the subset (i) to the last term on the right-hand side of (2.4) is equal to The contribution of the subset (ii) to the last term on the right-hand side of (2.4) is equal to The contribution of the subset (iii) to the last term on the right-hand side of (2.4) is equal to Collecting the above results, we have that * a ξ1 a ξ2 a ξ3 a ξ4 a ξ5 e it(ξ 2 From this, the formula (2.4) is described as The cubic interactions through resonance in the first term on the right-hand side of (2.5) are self-interaction, which enhance the phase oscillations. In next section, we seek the contribution of quintic resonant interactions to the last term on the right-hand side of (2.5).
Having discussed in Section 2, we define the resonant truncation ODE model of (2.5) by with initial data where ξ ∈ Λ. One of the most important features of the above ODE system is the following lemma.
for t ∈ I.
Proof. The proof is straightforward calculation. We only give the proof of (3.3) and (3.5), as the proofs for (3.4) and (3.6) are analogous. For (3.3) we multiply equation (3.1) for ξ = α 1 by 2c α1 and take the imaginary part to get where we use that for {ξ 1 , ξ 3 , ξ 5 } = {β 1 , β 1 , β 2 } and {ξ 2 , ξ 4 } = {α 1 , α 2 }. A similar argument as above but using equation (3.1) for ξ = β 1 yields Collecting the information in (3.7) and (3.8) we can obtain On the other hand, to prove (3.5), we will see that Hence the argument used in above gives which concludes the proof of (3.5). Remark 3.1. The conservation laws in (3.3)-(3.5) are used to the study of properties of the system of (3.1) including the global existence results. In particular, the ODE system (3.1)-(3.2) has global solutions. We will show that the system has periodic orbits in times. Let us set data {a ξ (0)} in (3.2) where By Lemma 3.2 and Remark 3.1, we have that for all t ∈ R, We now write Remark 3.2. From the relation in (3.9) and (3.10), we easily see that The equation (3.1) has a gauge freedom. On writing G ξ (0) = 0, and using (3.11), we have that the equation (3.1) becomes the following equation for d ξ , where I(0) = 0 anḋ Let us discuss the contribution of the right-hand of (3.12).

ODE dynamics
We now use the symplectic polar coordinates where θ ξ (t) ∈ R. From (3.9) and (3.10), we may assume that 0 < I ξ (t) < 1 (see the hypotheses in Corollary 1.1). This simply reflects the constrain on system of (3.16). Substituting these equations into (3.16) gives Remark 4.1. It is possible to consider the following Hamiltonian function associated to (4.1)-(4.2): (since by Lemma 3.3). In particular, we have that the system (4.1)-(4.2) takes the symplectic form By introducing the symplectic change of variables in a similar way to [7], we define θ = t (θ α1 , θ α2 , θ β1 , θ β2 ), I = t (I α1 , I α2 , I β1 , I β2 ), and define new variables Then the Hamiltonian flow in the action-angle coordinates (4.1)-(4.2) satisfies By (3.9)-(3.10) and Remark 4.1, we have As discussed in Remark 4.1, the structure of (4.1) and (4.2) is reminiscent of what the function K = K(t) constraints dynamics of (φ j , J j ) as well as (θ j , I j ). Actually, we will interest in the periodic solutions (θ j , I j ). We achieve this by finding a periodic solution K(t) for the ODE system on (φ 1 , J 1 ). For the purpose of obtaining the solutions (φ 1 , J 1 ), one considers the equations for φ 1 and for J 1J Since from J 1 = I α1 /2 = K/2, we conclude that φ 1 , K satisfy the following system:  Moreover, µT is a constant independent of µ.
We use the rescaling theorem done by t → µt to concrete µT as a constant independent of µ.
Once one has the periodic solution K(t) in Proposition 4.1, one can obtain the following proposition.

Error estimates
In this section, we shall evaluate the residual terms a ξ (t), ξ ∈ Λ for (2.5). Recall M * = max{|M |, |N |}. First we show the conservation laws.

Then we have that
Proof. As we shall see in (1.4) and M 0 = 3/2 by L 2 -norm conservation, one can give Also by (1.5) and via the Sobolev inequality, we have 3) The estimate (5.1) follows from (5.2) and (5.3).
On the other hand, as in Section 3, setting b ξ (t) = a ξ (t)e iΦ(t,ξ) , Notice that by (2.2), we have for We use (5.2) and ℓ 2 ξ ֒→ ℓ 1 ξ to see that To consider the error estimates, we need to introduce the following definition.
We introduce the notation: Now we illustrate how the error term in (5.5) would stay bounded for a reasonable amount of time. More precisely we have the following lemma.
Without loss of generality, we consider only the case t ≥ 0. Thanks to Duhamel's formula of (5.5), one can write the solution a ξ (t) of (5.5) as the integral form: Moreover, if one divides the sum of N 2 into two parts, (ξ 1 , ξ 2 , ξ 3 ) ∈ Λ * * and otherwise, then For this purpose, we have the several lemmas. Let us first consider the term N 2 .
Lemma 5.4. Assume the same hypotheses of Proposition 5.1. Then, Lemma 5.5. Assume the same hypotheses of Proposition 5.1. Then Proof of Lemma 5.5. From the proof of Lemma 5.4, it follows that , which is bounded by 1 100 ε 2 + c λ 2 + |µ| ε 2 t.
It remains to consider the term N 11 .
Using integration by parts, it follows that the contribution of this case to the left-hand side of (5.10) is bounded by By (5.6) and (5.11), we have that the first term of (5.12) has the bound c|λ|M −1−δ * ≤ 1 100 ε, (5.13) where δ < 1 and M * is large enough. To deal with the second term in (5.12), we use the equation (5.5). By symmetry we may only consider the case that the differential operator ∂ s applies to b ξ1 (s), and so the contribution of this case to the second term of (5.12) is bounded by (5.14) Fortunately, except for j = 1 in (5.14), it follows that the contribution of these cases to (5.14) is bounded by where δ < 1 and M * is large enough. For j = 1,Ṅ 1 (s, ξ 1 ) is expressed as Since ξ 1 ∈ Λ and constrain (1.6), we have that at least one of ξ ′ j is elements of Λ, thus we divide the sum into two sub-cases (using symmetry): (i)' two of ξ ′ j are not elements of Λ, (ii)' ξ ′ 2 ∈ Λ, ξ ′ 1 ∈ Λ, ξ ′ 3 ∈ Λ, (iii)' ξ ′ 2 ∈ Λ and ξ ′ j ∈ Λ for j = 1, 3. In the case (i)', it easily follows that the contribution to (5.14) is bounded by cλ 2 M δ * ε 2 t λ 2 εt.