Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrodinger equations

We study a dynamics and energy exchanges between a linear oscillator and a nonlinear interaction state for the one dimensional, quintic nonlinear Schrodinger equation. Grebert and Thomann proved that there exist solutions with initial data built on four Fourier modes, that confirms the conservative exchange of wave energy. Captured multi resonance in multiple Fourier modes, we simulate a similar energy exchange in long-period waves.

1. Introduction. In this paper, we consider the defocusing quintic nonlinear Schrödinger equation where L > 0, u = u(t, x) : R × T L → C is a complex-valued function and the spatial domain T L is taken to be a torus of length L, i.e., we assume the periodic boundary condition. In the case when L = 2π, we denote T = T 2π as usual. Our aim of this paper is to consider the long periodic solutions (L 1) to (1), while there are exchanges of resonant energy at particular frequencies. The sign of nonlinearity (+1 for the defocusing case and −1 for the focusing case) will not play the central role in the present discussion. For simplicity, we focus on the situation of the defocusing case.
The equation (1) which impose the constraints on a dynamics of mass density of solutions. We briefly recall known results concerning the Cauchy problem for the quintic NLS. In the non-periodic scenario, i.e., x ∈ R, the equation is called mass-critical or L 2 -critical from the viewpoint of scaling. Indeed, the one-dimensional quintic nonlinear Schrödinger equations with non-periodic boundary condition is left invariant by the scaling u → u λ ; u(t, x) → u λ (t, x) = λ 1/2 u(λ 2 t, λx), λ > 0, which preserves the homogeneous Sobolev normḢ s (R) with s = 0. On R-case, the local well-posedness was proved by Cazenave and Weissler [2] for data in L 2 (see also [8] and [13]). Notice that in [2], the existence time of solution depends on the position of data and not only on its size. One can also prove the global wellposedness in L 2 provided that the initial data in L 2 is sufficiently small. Concerning the local well-posedness theory in fractional Sobolev spaces, we refer to the paper [8]. The global well-posedness and scattering in the threshold space L 2 was obtained by Dodson [6]. More precisely, it is shown that for all u 0 ∈ L 2 (R), there exist a unique time global solution to (1) and u ± ∈ L 2 (R) satisfying that as t → ±∞. It is possible to consider the global well-posedness and scattering for the nonlinear Schrödinger equations with energy sub-critical nonlinearities. This is established in [5] for initial data below the energy norm. We now turn to the case of periodic boundary conditions. The local wellposedness was proved by Bourgain [1] for data in H s (T), with s > 0. This combined with the H 1 -energy conservation law (an a priori estimate for solutions) leads to global well-posedness in H 1 (T). Similar results hold for the equation in the Lperiodic boundary condition case T L for any L > 0, without having to change the proof. The Sobolev H s (T L ) norm is given by In this paper, we want to understand the interaction of mass for each frequency ξ ∈ 2πZ/L. In fact, the L 2 -norm conservation law (2) constrains this object. For the periodic boundary value problem, i.e., x ∈ T (the case of L = 2π) with replacing the nonlinearity |u| 4 u by ν|u| 4 u with ν > 0 i∂ t u + ∂ 2 x u = ν|u| 4 u, (t, x) ∈ R × T, Grébert and Thomann [9] examined the dynamics exhibited by the solution of (4). More precisely, they proved the following theorem.

Remark 1.
Another interesting result is the two-dimensional cubic nonlinear Schrödinger equation. In [4], Colliander, Keel, Staffilani, Takaoka and Tao showed the weak turbulence property for the 2D defocusing cubic nonlinear Schrödinger equation: More precisely, for any s > 1, K 1 ε > 0, there exists a time T 1 such that the initial value problem corresponding to the equation in (5) has a global in time solution u(t) satisfying that This exhibits the H s -norm inflation of solutions to the cubic nonlinear Schrödinger equations, that admits solutions with transferring wave energy from low to high Fourier modes.
In this paper, we proved that there exist solutions of the one-dimensional quintic nonlinear Schrödinger equations with initially excited in multi-frequency modes, where the mass is localized and involves conservative energy exchange between the modes initially excited. Let us now define the wavenumber set consisting of nonlinear resonance interactions in the equation (1). Definition 1.3 (Resonance interaction set). Let k ∈ N/L be fixed. For j ∈ Z, we set α 1,j , α 3,j , α 2,j and α 4,j as follows: With α m,j , we set R m = {α m,j | j ∈ Z, 0 ≤ j < L} , for 1 ≤ m ≤ 4, and R = ∪ 4 m=1 R m . We start by defining the following norms. Definition 1.4. We define the sets N l , N m , N r , N h to be subsets of the set 2πZ/L as follows: For the Sobolev index s ∈ R, let m(ξ) be the multiplier function defined on 2πZ/L to be where τ and τ obey the formula Our main result is the following theorem.
Theorem 1.5. Let s ∈ (1, 3/2), let ν > 0 be a small constant and let L be a natural number. Then there exist a positive number k 0 , so that for k 0 ≤ k ∈ N/L, there exist a smooth global solution u(t) to (1) such that for |t| 1/(L 3 ν 2 ), where u Rm (t) is whose Fourier series coefficients u Rm (t)(ξ), ξ ∈ 2πZ/L are zero for ξ ∈ R m so that and where K(t) = 1 2ν 2 sin arctan Moreover, the error term e(t) associated with the expression in (6) satisfies that Remark 3. In Theorem 1.5, we pick L to be a natural number for simplicity. Only a small modification for the proof is required in the general L > 0. It is not important to assume the natural number L.
Remark 4. The equation (1) has a gauge symmetry. After conducting gauge transformation u → e iθ u with θ = n/L, it is easily to shift α m,j → α m,j + 2πn/L (1 ≤ m ≤ 4) for n ∈ Z and prove the corresponding theorem to Theorem 1.5.

Remark 5.
Choosing k large enough such as k 3/2−s L 6 ν 2 , one can prove by Theorem 1.5 that . Then for small data such that ν L −3/2 , we have that T 0 1 and a solution u(t) to (1) satisfying for s ∈ (1, 3/2), where the right-hand side is positive.
Remark 6. We can expect similar results to hold for the following more general nonlinearities with essentially the same proof: The proof of Theorem 1.5 relies on obtaining the dynamics in a toy model (finite dimensional approximation) of nonlinear Schrödinger equations along with error estimates between finite dimensional model and the full infinite dimensional model.
In Section 2, we present some notation. Section 3 describes the reductions of the equation (1) as an infinite system of ODE's. In Section 4, we construct the appropriate Toy model equation associated with the finite dimensional ODE system of reduced NLS equation, and solve it. In Section 5, we give qualitative estimates for solutions to the finite dimensional ODE system. In Section 6, we prove that the ODE system derived in Section 4 approximates the dynamics of the quintic nonlinear Schrödinger equation (1). Theorem 1.5 is established in Section 7.

2.
Notation. Let us introduce some notation. We prefer to use the notation · = (1 + | · | 2 ) 1/2 . The over dotȧ(t) denotes the derivative of a(t) with respect to time t.
We use c, C to denote various constants. 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.
For an odd natural number n and complex-valued functions f 1 , f 2 , . . . , f n defined on the set 2πZ/L, we write the discrete convolution (convolution sum) where the superscript notation * of * indicates a sum running over the hyperplane set ξ 1 − ξ 2 + ξ 3 − . . . + ξ n = ξ with summation index.
3. Reductions of the equation (1) as an infinite system of ODEs. In this section, we consider the smooth solution to (1). Namely, suppose that u is a smooth global in time solution to (1). Let us start with the ansatz In what follows, we shall omit the time variable t of a ξ (t) in abbreviated form without confusion. With this transform, the equation (1) becomes where the factor φ(ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 , ξ) means the oscillation frequency such that In the next section, we seek the quintic resonant structure in the last term on the right-hand side of (7). 4. Toy model. In this section, we construct a finite dimensional model, in some sense, as the corresponding approximation model of (7). Consequently, it will approximate an exact solution to (7).
Taking into the observation in Lemma 4.2, we have that the cardinal of the pair (j 1 , 1 , j 2 , 2 , j 3 , 3 ) without the case {j 1 , j 2 , j 3 } = { 1 , 2 , 3 }, appearing in the right-hand side of (8) is greater than or equal to 4. Now we define the resonance interaction sets.
By removing non-resonance term in the term on the right-hand side of (7), we propose to a finite dimensional system of ODE, which we call the resonant system corresponding to (7). Consider the initial value problem for the following resonant truncation of (7); for ξ = α m,j ∈ R m , 1 ≤ m ≤ 4, where we denote by res(ξ) the set such that the pair (ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 , ξ) satisfies the resonance condition. We then prove that the approximate solutions of (11).

5.
Dynamics of approximate solutions for some time. In this section, we shall study the resonant truncation ODE system in (11).
The resonant truncation ODE system in (11) retains conserved quantities as follows.
Proof of Lemma 5.1. It will be convenience to raise the frequency representation ξ in (12) to ξ 6 . Namely it suffices to show that Multiplying r ξ6 to (11) and taking the imaginary part, we have where ξ 6 ∈ R. The term on the left-hand side of the above equation will be Then after the summation over ξ 6 ∈ R, we arrive at the following: which is zero, since by symmetrization schemes.
We now turn our attention to the dynamical structures of the resonant truncation ODE system in (11). In a similar way to [4,9,12], we may rewrite (11) to the equation of motion in the Hamiltonian symplectic coordinates.
If we set we can write (11) as a system. Actually, inserting this back into (11), we see that Taking the real part, we get that for ξ = α m,j ∈ R We take the imaginary part to get for ξ = α m,j ∈ R.
The solutions to (14) have the following conserved quantities.
We can also conclude that the estimates in (17) hold by means of similarity computation as above.
Let us proceed to construction of the specific solution to (13) and (14). We define We establish the following lemma.

Averaging property.
It is natural to expect that the average of the L sums by approximates the source of a mass located in frequency space R m . By Lemma 5.4, we may suppose Φ 1,2,1,2,3,4 j1,j1,j2,j2,j3,j3 (t) = π/2 in (13)-(14). In a certain sense that should be taken the average of both the left-and right-hand sides of (14) with respect to 6362 HIDEO TAKAOKA 0 ≤ j < L, we reformulate the ODE system (14) as the following ODE system: A similar argument in Lemmas 5.1 and 5.3 shows that It is easily to see that from (19) which are also the conserved quantities of the system in (19).
For fixed 0 ≤ j < L, use (26) to see that By virtue of the proof of Lemma 5.3, (28) implies provided t L 3 /ν 2 . Thanks to the conservation laws obtained in Lemma 5.3, (20) and (21) along with the initial condition at t = 0, the estimate in (29) implies that We apply the Gronwall inequality to get I α2,j 2 I α2,j 4 I α4,j I α1,j 2 I α1,j 4 I α3,j − I R4 I 2 Using (30) and (31), we arrive at max 1≤m≤4 0≤j<L which completes the proof.

HIDEO TAKAOKA
It follows that from the mass conservation laws (2) and (12), where c > 0 is independent of t ∈ R.
We define a subset of 2πZ/L. Definition 6.1. Introduce some notions. For ξ ∈ 2πZ/L, rewrite the form as where

Remark 10.
In the spirit of the standard local well-posedness theory observed in [2,13] and an ODE technique along with the conservation laws in (2), (3) and (12), we easily see that there exists a unique smooth global in time solution to the initial value problem for the corresponding equations to (7) and (11), respectively. By a slight abuse of notation above, we define the modified H s -energy for the difference between the solutions to (7) and (11) as follows: where δ 1 is a fixed large constant, in fact, as like δ = 10 10 . First, we prepare the following lemma to estimate the convolution sum. Lemma 6.2. Let s > 1, and let (c j,l ) l∈2πZ (1 ≤ j ≤ 6) be sequences of non-negative numbers. Then it follows that and where constants term to the right-hand side are independent of k.
Then for |t| ≤ T , Proof. Denote We note that m(ξ) ∼ 1 for ξ ∈ A r . From the assumption, equations (7), (11) and Lemma 6.2, we see that Proof. It suffices to show that In the case when ξ 6 ∈ A r , we distinguish two cases: • ξ m ∈ R for all 1 ≤ m ≤ 5 in the sum (42), • there is at least one element of ξ m ∈ R (1 ≤ m ≤ 5) such that ξ m ∈ R in the sum (42).
In the second case, from a ξm = e ξm for ξ ∈ R, the contribution of this case to the left-hand side of (42) is bounded by Next we may suppose ξ 6 ∈ A r so that there is at least one element of ξ m (1 ≤ m ≤ 5) such that ξ m ∈ R, a ξm = e ξm and | τ 6 | | τ m |. In this case, it is easy see that m(ξ 6 ) m(ξ m ) so that the contribution of this case to the left-hand side of (42) is bounded by the same bound as in (45), which is acceptable.
A crucial step in the proof of Theorem 1.5 is to establish the following proposition.
In the case when m(ξ 6 ) 1 in the sum of (48), by the above observation, we have the bound of the left-hand side of (48) by which is, applying Lemma 6.3, bounded by In the case when m(ξ 6 ) 1, from (43), we have which implies that at least there exists one element ξ 0 of ξ m (1 ≤ m ≤ 5) in the sum such that ξ 0 ∈ R and m(ξ 6 ) ∼ m(ξ 0 ). Then the contribution of this case to the left-hand side of (48) is bounded by Thus the proof of that result is complete.
Lemma 6.6. Under the hypotheses of Proposition 2, Proof. The proof of this lemma follow easily from the form of R 2 ξ and the fact that R 2 ξ = 0 for ξ ∈ R, so that shall be omitted.
To bound the last factor on the right-hand side of (47), we have the following.
Lemma 6.7. Under the hypotheses of Proposition 2, Proof. The proof of this lemma is a straightforward. The main objective is to use equations (7), (11) and the a priori bound. The contribution to the upper bound in term of | R 4 (t)| is the sums of the following two terms: Here we only give a proof for the first term, because the second term can be handled similarly.
There is at least one a ξm (1 ≤ m ≤ 5) of each quintic nonlinearity a ξ1 a ξ2 a ξ3 a ξ4 a ξ5 in (49) such that m(ξ m ) m(ξ a ). This is easy to prove by writing ξ m as the formula (34). In view of above, in the case that the derivative ∂ t falls into either the function associated to ξ a or ξ b , we have the bound of | R 4 (t)| by On the other hand, in the case that the derivative ∂ t falls into the function associated with neither ξ a or ξ b , we have the same bound as above. Thus the proof is complete.
It remains to estimate the term for non-negative time.
6.1. Several technical lemmas. First we shall prepare several lemmas. We keep the convention of notation expressed at Definition 6.1. For frequencies ξ ∈ 2πZ/L, we use the expression in (33) and (34).
This concludes the proof of the lemma.
Proof. In the case when any two elements of η 1 , η 3 , η 5 do not match with each other, we consider all cases: however η 2 1 + η 2 3 + η 2 5 = η 2 2 + η 2 4 + η 2 6 is no longer satisfied. Therefore, the proof of Lemma 6.8 permits us to conclude the proof of the corollary. 6.2. Contribution of ξ ∈ A r to (50). Let us first consider the contribution of ξ ∈ A r to (50). Note that m(ξ) 1 for ξ ∈ A r . The contribution of this case to (50) is Note that m(ξ) 1 for ξ ∈ A r . By Lemma 6.3, we have that the contribution of this case is bounded by 6.3. Contribution of ξ ∈ A r to (50). Consider the associated function × a ξ1 (t )a ξ2 (t )a ξ3 (t )a ξ4 (t )a ξ5 (t )e ξ6 (t )e −it φ(ξ1,ξ2,ξ3,ξ4,ξ5,ξ6) dt .