Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schr{\"o}dinger equations on $h\mathbb{Z}$

We consider the discrete nonlinear Schr{\"o}dinger equations on a one dimensional lattice of mesh h, with a cubic focusing or defocusing nonlinearity. We prove a polynomial bound on the growth of the discrete Sobolev norms, uniformly with respect to the stepsize of the grid. This bound is based on a construction of higher modified energies.


Introduction
We consider the cubic discrete nonlinear Schödinger equation (called DNLS) on a grid hZ of stepsize h ą 0. This equation is a differential equation on C hZ defined by (see [11] and the references therein for details about its derivation) (1) @g P hZ, iB t u g " p∆ h uq g`ν | u g | 2 u g , where ν P t´1, 1u is a parameter and ∆ h u is the discrete seconde derivative of u. It is defined by @g P hZ, p∆ h uq g " u g`h´2 u g`ug´h h 2 .
We consider both the focusing and the defocusing equations. They correspond respectively to the choices ν " 1 and ν "´1.
DNLS is a popular model in numerical analysis for the spatial discretization of the cubic nonlinear Schrödinger equation (NLS), given by: (2) iB t u " B 2 x u`ν|u| 2 u, see, for example, [3], [4], [5], [9], [10], [11]. Motivated by the approximations properties of NLS by DNLS, we consider the discrete model near its continuous limit i.e. when h goes to 0. So, we introduce norms consistent with the usual continuous norms and we pay attention to establish estimates uniform with respect to h.
We introduce the discrete L 2 space. It is defined by , . -.
This space is natural to solve DNLS. Indeed, since (1) is invariant by gauge transform, as a consequence of the Noether Theorem, the discrete L 2 norm is a constant of the motion of DNLS. Consequently, applying Cauchy Lipschitz Theorem (as L 2 phZq is a Banach algebra) we can prove that DNLS is globally well posed in L 2 phZq.
We introduce the homogeneous discrete Sobolev norms by analogy with respect to the continuous homogeneous Sobolev norms. If n P N is an integer and u P L 2 phZq, its discrete homogeneous Sobolev norm of order n is defined by H n phZq " xp´∆ h q n u, uy L 2 phZq . For example, if u P L 2 phZq, its discrete homogeneous Sobolev norm of order 1 is .
Naturally, we define as usual the non homogeneous discrete Sobolev norms by Applying the triangle inequality we can easily prove that all these norms are controlled by the discrete L 2 norm (4) @ u P L 2 phZq, } u } 9 H n phZq ďˆ2 h˙n } u } L 2 phZq .
So, since the discrete L 2 norm is a constant of the motion of DNLS, any discrete Sobolev norm of a solution of DNLS is globally bounded. However, this bound is not uniform with respect to the stepsize h. Consequently, these estimates are trivial when we consider the continuous limit.
An uniform control of these norms with respect to h may be crucial to establish aliasing or consistency estimates. For example, in [5], the existence and the stability of traveling waves is studied near the continuous limit of the focusing DNLS. The discrete Sobolev norms are used to control an aliasing error generated by the variations of the momentum. It is proven that if for all n P N, the discrete Sobolev norm of order n of the solutions of the focusing DNLS can be bounded by t αn , uniformly with respect to h, then DNLS admits solutions whose behavior is similar to traveling waves for times of order h´β, with β " lim sup n n αn . There is a huge literature about the growth of the Sobolev norms for continuous Schrödinger equations. Since, we are focusing on the continuous limit of DNLS, it is natural to try to adapt the methods used for these equations. If we focus on the continuous Schrödinger equations on R, it seems that there are three families of methods and results.
‚ First, there is the cubic nonlinear Schrödinger equation. This equation is known to be completely integrable. In particular, it admits a sequence of constants of the motion coercive in H n pRq. Consequently, all the Sobolev norms are globally bounded (see, for example, [13]). ‚ Second, there is the linear Schrödinger equation with a potential smooth with respect to t and x. In such case, for all ε ą 0 there is a control of the growth by t ε (see [6]). ‚ Third, in the other cases there are methods using dispersion and/or higher modified energy. With these methods, there is a control of the growth of the H n norm by t αn`β for some α, β P R. For example, in [13], Sohinger proves a control of the H s norm by t A priori, DNLS is not a completely integrable equation, so we can not control its Sobolev norms as for its continuous limit (for a completely integrable spatial discretization of NLS, we can refer to the Ablowitz-Ladik model, see [1]). In this paper, we adapt the last method to the discrete nonlinear Schrödinger equation. In [14], Stefanov and Kevrekidis proved that the dispersion is weaker for the linear discrete Schrödinger equation than for the continuous equation. They got a L 8 decay of the form t´1 2`p htq´1 3 (see also [9]). Using dispersive arguments in our setting seems thus more difficult than in the continuous case and does not seem to strengthen significantly the results. However, the method of constructing modified energies can be applied and turns out to yield results comparable to the continuous case. With our construction, we get the following bound. Theorem 1.1. For all n P N˚, there exists C ą 0, such that for all h ą 0 and all ν P t´1, 1u, if u P C 1 pR; L 2 phZqq is a solution of DNLS then for all t P R where M up0q " } up0q} 9 H 1 phZq`} up0q} 3 L 2 phZq . This theorem is the main result of this paper, it will be proven in the third section. The second section is devoted to the introduction of tools and notations useful to prove it.
We conclude this introduction with some remarks about estimate (5).
‚ If n " 1 then the discrete H 1 norm is globally bounded, uniformly with respect to h. It is a consequence of the conservation of the Hamiltonian of DNLS and its coercivity in H 1 phZq. In the focusing case this argument is specific to the dimension 1. It is based on a discrete Gagliardo-Nirenberg inequality. For the defocusing case, the coercivity is straighforward and can be extended to higher dimensions and with other nonlinearities. ‚ The factor associated to the growing term t up0q . So the growth of the high Sobolev norms is controlled by the size of the initial condition with respect to the low Sobolev norms. ‚ The estimate (5) is homogeneous. More precisely, DNLS is invariant by dilatation in the sense that if u is a solution of DNLS then pt, gq Þ Ñ λ u λg pλ 2 tq is a solution of DNLS with stepsize hλ´1. Estimate (5) is invariant by this transformation (as can be seen from the exponents of M up0q ). Consequently, to prove Theorem 1.1, we just have to prove it with h " 1.

Shannon interpolation
In order to use classical analysis tools, it is very useful to identify sequences of L 2 pZq with functions defined on the real line through an interpolation method. Here, we choose the Shannon interpolation (this choice is quite natural, see [9] or [5]). More precisely, it is the usual interpolation we realize when we identify a sequence on Z with a function whose support in Fourier in a subset of r´π, πs.
In this section, we introduce this interpolation and we give some of its classical properties useful to prove Theorem 1.1. For details or proofs of these classical properties the reader can refer to [5] or [12].
First we need to define the discrete Fourier transform and the Fourier Plancherel transform F : where the right integral is defined by extending the operator defined on L 1 pRq X L 2 pRq. We also use the notation p u " F u.
Now, we define the Shannon interpolation , denoted I, through the following diagram where 1 p´π,πq : R Ñ R the characteristic function of p´π, πq and F´1 is the invert of the Fourier Plancherel transform.
In the following proposition, we give some properties of this interpolation useful to prove Theorem 1.1 .

Proposition 1.
(see for example [12] for details) ‚ The image of I is the set of functions whose Fourier support is a subset of r´π, πs. It is denoted by ‚ If u P L 2 pZq then I u is an entire function whose u is the restriction on Z, i.e. @g P Z, pI uq pgq " u g .
Now, we focus on properties more specific to the discrete Sobolev norms.
Proposition 2. (see [5]) Let u P L 2 pZq be a sequence and let u " I u denote its Shannon interpolation. Then we have for almost all ω P p´π, πq where˚is the usual convolution product.
We deduce two important direct corollaries of this proposition. In the first one we identify the differential equation satisfied by the Shannon interpolation of a solution of DNLS.
Corollary 1. Let u P C 1 pR; L 2 pZqq be a solution of DNLS and let u " I u P C 1 pR; BL 2 q denote its Shannon interpolation, then for all t P R and almost all ω P p´π, πq, In the second corollary, we identify the discrete Sobolev norms.
Corollary 2. Let u P L 2 pZq be a sequence, let u " I u denote its Shannon interpolation. If n P N˚then Consequently, the continuous and discrete homogeneous Sobolev norms are equivalents, i.e.

Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. The idea is to construct some higher modified energies controlling H n phZq norms and whose growth can be controlled by H n´1 phZq norms. The construction of higher modified energies to study growth of Sobolev norms is a classical trick (see [7] or [13]).
As explained at the end of the introduction, since inequality (5) of Theorem 1.1 is homogeneous, without loss of generality, we just need to prove it when h " 1.

3.1.
Construction of the modified energies. DNLS is an Hamiltonian differential equation (see [5]) whose Hamiltonian (i.e. its energy) is defined on L 2 by So H DNLS puq is a constant of the motion (it can be proven directly computing the discrete L 2 inner product of (1) and uptq).
If u P BL 2 is the Shannon interpolation of a sequence u P L 2 pZq this Hamiltonian can be written as a function of p u (it is a consequence of Proposition 2) The principle of the construction of the modified energies is to change the weights of these integrals to get a control of high Sobolev norms. To explain this construction, we need to adopt more compact notations. Some of them are classical for NLS (see [13]).
and we equip it with its natural measure, denoted dw, induced by the canonical Lesbegue measure of R 2m .
If µ P L 8 pV m q and if v P L 2 pRq is supported by r´π, πs, we define Λ m pµ, vq by To prove that Λ m is well defined, we just need to pay attention to the support of µpwq ś m j"1 vpw j qvpw´j q and to apply a convolution Young inequality (see Lemma 3.3 for details).
For example, with this notation, we have a more compact expression of (8) given by (9) 2π H DNLS puq " Then, we define a transformation S m : where pe k q kP ´m,m zt0u is the canonical basis of R ´m,m zt0u .
We define an other transformation D m : L 8 pRq Ñ L 8 pV m q by We say that a function µ P L 8 pV m q is 2π periodic with respect to each one of its variables, and we denote it by µ P L 8 per pV m q, if @k P ´m, m zt0u, µpw`2πe k q " µpwq, w a.e.
The following algebraic lemma explains why these notations are well suited to DNLS.
Lemma 3.1. If m P N˚, µ P L 8 per pV m q and u P C 1 pR; L 2 pZqq is a solution of DNLS whose Shannon interpolation is denoted u, then we have (10) iB t Λ m pµ, p uq " 2Λ m pµD m cos, p uq`νΛ m`1 pS m µ, p uq.
Proof. By definition, the quantity to identify can expanded as follow Now, we have to expand I k and I´k using the definition of DNLS. Applying Proposition 2 we get @w k P p´π, πq, iB t p upw k q " p2 cos w k´2 qp upw k q`ν ÿ ℓPZ z |u| 2 upw k`2 πℓq.
So, since µ is 2π periodic the direction e k , we deduce However, since for all ω P R, p upωq " p up´ωq, we have, for all w k P R, So, realizing the change of variable w k Ð r w k , we get ż Vm µpwqp upw´kq z |u| 2 upw k q ź j‰k p upw j qp upw´jqdw " Λ m`1 pµpw`e k pw m`1´wm`1 qq, p uq.
Similarly, we could prove that I´k "´Λ m pp2 cos w´k´2qµ, p uq´νΛ m`1 pµpw´e´kpw m`1´wm`1 qq, p uq.
So, finally, we get µpw`e k pw m`1´wm`1 qq´µpw´e´kpw m`1´wm`1 qq, p u" 2Λ m pµD m cos, p uq`νΛ m`1 pS m µ, p uq.
Corollary 3. Let f P L 8 pRq, let u P C 1 pR; L 2 pZqq be a solution of DNLS and let u be its Shannon interpolation. Then, we have Proof. The result only involves values of f for ω P p´π, πq. So we can assume that f is a 2π periodic function. Now, we observe that, by definition, we have ż f pωq|p upωq| 2 dω " Λ 1 pf pw 1 q, p uq.
Since 2π periodic functions clearly belong to the D 1 kernel, the first term is zero. So we just need to identify the second term. Indeed, paying attention to its symmetries and remembering that we have assumed that f is 2π periodic function, we get Λ 2 pS 2 rf pw 1 qs , p uq " Λ 2 pf pw 1`w2´w´2 q´f pw 1 q, p uq " Λ 2 pf pw´1q´f pw 1 q, p uq "´1 2 Λ 2 pf pw 1 q`f pw 2 q´f pw´1q´f pw´2q, p uq.
With these notations and results we can explain more precisely the construction of our higher modified energies . But first, we explain why it is natural to introduce correction terms in the construction of our modified energy.
In order to control the discrete 9 H n norm, it would seem natural to control its derivative. Indeed, if u is a solution of DNLS and if u is its Shannon interpolation, applying Corollary 3 (and Corollary 2), we have So a direct estimation of this derivative would naturally lead to (see Lemma 3.3 for a proof of this estimate)ˇˇˇB Then assuming that the discrete homogeneous H 1 norm can be controlled uniformly on time by M up0q (see Theorem 5 for the definition of M up0q and the next subsection for a proof) and applying Grönwall's inequality, we would get an universal constant C ą 0 such that, for all t ě 0, If we proceed by homogeneity to get a result depending of the stepsize h, we would get Such a control is better than the trivial estimate (4) only for times shorter than´n C logphq. So it is quite weak, if we compare it with the estimate of Theorem 5 because this later gives a non trivial control of } uptq} 9 H n pZq for times shorter than h´2 n n´1 .
So to improve this exponential bound, the idea of modified energy is to add a corrector term to } u } 2 9 H n pZq in order to cancel its time derivative (11). However, it turns out that there is an algebraic obstruction to this construction as shows in Lemma 3.2 below. For this reason, we consider another functional ş f n pωq|p upωq| 2 dω, where f n is a real function and such that this last quantity is equivalent to the square of the 9 H n pZq norm. More precisely, observing the formula of the Hamiltonian (see (9)), we consider a modified energy E n given by where µ n P L 8 pV 2 q is a function.
Applying Lemma 3.1 and its Corollary 3, if we want the correction term to cancel the derivative of ş f n pωq|p upωq| 2 dω then µ n has to solve the equation Furthermore, if µ n is a solution of (12), we would have B t E n puq "´iνΛ 3 pS 2 µ n , p uq.
With this construction, we will be able to prove Theorem 1.1 by induction because we will prove that Λ 2 pµ n , p uq and Λ 3 pS 2 µ n , p uq are controlled by the square of the 9 H n´1 pZq norm.
Of course, we would like to iterate this processus cancelling the derivative of E n puq adding a new term to our modified energy. However, such a construction involve major algebraic issues and we do not know if it is possible (we should find some criteria of divisibility by D 3 cos on the ring of trigonometric polynomials on V 3 ).
To realize this strategy, we need, on the one hand, to design a function µ n satisfying (12) without any singularity and, on the other hand, we need to control Λ 2 pµ n , p uq and Λ 3 pS 2 µ n , p uq by the square of the 9 H n´1 pZq norm. The two following lemmas treat each one of these issues.
Opω 2n q , where n P N˚, and if f´f p π 2 q is an even function in 0 and an odd function in π 2 then there exists C ą 0 such that we have Proof. Since f´f p π 2 q is even in 0 and odd in π 2 , it is a 2π periodic function whose Fourier series is Furthermore, since f is a C 8 function, for all m P N˚, there exists C m ą 0 such that ÿ kPN |β k |p2k`1q m ď C m .
To get compact notations, we define the function cos k (and similarly sin k ) by @ω P R, cos k ω :" cospp2k`1qωq.
If we assume that w 1`w2 " w´1`w´2`2πj, with j P N then we have But since 2k`1 is odd, we have p´1q j cos kˆw´1´w´2 2˙" cos kˆw´1´w´2 2`π j˙.
To conclude this proof, we just need to improve (13) when w is small enough. In this case, we can forget the alising terms because if max jPt˘1,˘2u |w j | ă π 2 then w 1`w2 " w´1`w´2. Now we realize the following change of variable Then we define F pX, Y, Z, Hq " D 2 f pwq.
Previously, we have proven that, for all X, Y, Z P R, Consequently, we have F p0, Y, Z, 0q " 0 and B X F pX, 0, Z, 0q " 0.
We can write this inequality with the variables w 1 , w 2 , w´1, w´2. So, since norms are equivalents on tw P R t˘1,˘2u , w 1`w2 " w´1`w´2u, there exists C P p0, π 2 q such that if w P V 2 satisfies max jPt˘1,˘2u |w j | ă C´1 then (14) |D 2 f pwq| ď C|D 2 cospwq| ÿ jPt˘1,˘2u Finally, to prove the lemma we just need to use (14) when w is small enough and (13) when it is large. Lemma 3.3. Let n, m P N, m ě 2. There exists K ą 0 such that for all u P BL 2 , we have Proof. This lemma is somehow a discrete integration by part. By linearity, we just need to prove that Since, supp |p u| Ă r´π, πs we have So, applying Jensen's inequality to x Þ Ñ x n , we get The first term can be estimated by an elementary Young convolution inequality to get The second term is an aliasing term. If k " 0, this term is 0, so we can assume k ‰ 0. Now observe that if the sum of 2m numbers, all smaller than 1 is larger than 2 then at least 2 of them are larger to 1 2m´1 . Consequently, applying the same Young convolution inequality, we have ż ř m j"1 w j´w´j "2kπ To conclude rigorously this proof, we just need to control classically }p u} 2 L 1 pRq by }u} L 2 pRq }B x u} L 2 pRq . Indeed, if v P H 1 pRq, using Cauchy Schwarz inequality, we get So, optimizing this inequality with respect to λ through the transformation v Ð vpλxq, we get }p v} L 1 pRq ď ? 8π}v} L 2 pRq }B x v} L 2 pRq .

3.2.
Proof of Theorem 1.1 by induction. With all these tools, now, we prove Theorem 1.1. As explained at the beginning of this section, we just need to focus on the case h " 1.
We are going to proceed by induction.
‚ We focus on the case n " 1. Let u P C 1 pR; L 2 pZqq be a solution of DNLS. Since H DNLS is a constant of the motion of DNLS, for all t P R, we have Since } u } 2 L 2 pZq is also a constant of the motion, we have Let u be the Shannon interpolation of u. Since u |Z " u (see Proposition 1), we have } uptq} 2 L 8 pZq ď }uptq} 2 L 8 pRq ď c}B x uptq} L 2 pRq }uptq} L 2 pRq , where c is an universal constant associated to the classical Sobolev embedding. Since Shannon interpolation is an isometry we have proven that Now applying the estimate of Corollary (2), we get a discrete Gagliardo-Nirenberg inequality (for a sharper version of this inequality see [8]) Applying this inequality to (15), we get H 1 pZq . Consequently, we have proven that with C " maxp1, c π q. ‚ Let n ě 2, let u P C 1 pR; L 2 pZqq be a solution of DNLS satisfying for all t P R where u is the Shannon interpolation of u and Here, it is easier to work with an inequality on u instead of u but applying the estimate of Corollary (2), (16) is equivalent to the inequality of Theorem 1.1.
First, we are going to construct our modified energy with Lemma 3.2. So we have to choose our function f n . This function has to satisfy some criteria. First, we want ş f n |p upωq| 2 dω to be equivalent to square of the homogeneous H n norm of u. So we are looking for a regular function f n such that (17) @ω P p´π, πq, αω 2n ď f n pωq ď α´1ω 2n .
Second, we want f n´fn p π 2 q to be even in 0 and odd in π 2 . So we cannot choose f n pωq " ω 2n or f n pωq "`2 sinp ω 2 q˘2 n . To satisfy these symmetries it is natural to look for f n as a trigonometric polynomial.
By performing an analysis involving elementary linear algebra, we find that f n defined by (18) f n pωq :" 1´cospωq n´1 ÿ is the trigonometric polynomial of minimal degree (and such f p π 2 q " 1) satisfying the previous hypothesis. Indeed, by construction, f n´1 is even in 0 and odd in π 2 . Furthermore, in R X (i.e. formally), we have (see for example [2]) Since, for all ω P p´π 2 , π 2 q, cos ω " a 1´psin ωq 2 , we deduce that f n pωq " cospωq ÿ kěn C k 2k 4 k psin ωq 2k .
Consequently, we get f n ą 0 on ω P p0, π 2 q and f n pωq " ωÑ0 C n 2n 4 n ω 2n . So, using the symmetries of f n , we deduce that there exists α ą 0 such that (17) is satisfied.
Then we define on V 2 a function µ n P L 8 pV 2 q by µ n " ν 4 In Lemma 3.2, we have proven that µ n is well defined as a L 8 pV 2 q function (in fact, we could have proven that it is a regular function). Furthermore, we have proven that for all w P V 2 , we have (19) |µ n pwq| ď C n ÿ jPt˘1,˘2u where C n depends only of n.
To conclude the induction step we have to control each one of these terms.
-First, we focus on ż t 0 |B s E n pupsqq| ds .
However, as we have proven at the initial step, there exists an universal constant c ą 0 such that @t P R, }B x uptq} 2 L 2 pRq }uptq} 2 L 2 pRq ď cM 8 3 u 0 . So, from the induction hypothesis (see (16)), we get