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The nonlinear Schrodinger equation on Z and R with bounded initial data: examples and conjectures

We study the nonlinear Schr\"odinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially bounded in time for any bounded data. In the continuum, local existence is proved for real analytic data by a Newton iteration scheme. Global existence for NLS with a regularized nonlinearity follows by analyzing a local energy norm.


Introduction
The aim of this note is to study nonlinear Schrödinger type equations on Z and on R with initial data that are bounded, and do not vanish at infinity. Examples include periodic, quasi-periodic and random data. We present some modest results describing the dynamics for such data. Although we shall phrase our results in one dimension, most of our methods can be adapted to higher dimensions.
There is an extensive literature on space periodic data, with bounds on the Sobolev norms as a function of time [Bou93,Bou94,Bou00]. More recently, localized perturbations of periodic initial data have been studied [KST17,CHKP17]. The periodic case appears naturally in nonlinear optics, describing periodic signal propagating through a fiber, and local perturbations would correspond to noise and modulation due to information carrying signal.
Much less is known about quasi-periodic and random cases. The quasi-periodic data arises when there is another periodic signal, not commensurate with the periodic signal. The nonlinearity then will produce a solution with arbitrary number of frequencies. In the case of the KdV equation, Damanik and Goldstein and Binder at al. [DG16,BDGL18] have proved the global existence of uniformly bounded almost periodic solutions for certain small amplitude quasi-periodic initial data. This remarkable result uses the integrability of KdV and the fact that the corresponding Schrödinger operator has only absolutely continuous spectrum. The case of global existence of large initial data has not yet been understood. For non-integrable NLS, T. Oh [Oh15b], proved the existence of solutions for quasi-periodic data for short time. He also proved that solutions exist for all time for limit-periodic data, [Oh15a]. The global existence of special space-time quasi-periodic solutions was recently established by W. Wang [Wan19] by using Bourgain's semi-algebraic set methods together with a Newton iteration scheme. It is generally believed that typical solutions to non-integrable equations are not uniformly bounded in time unless the maximum is controlled by a conservation law. Note that in one dimension, the maximum of a solution of NLS with periodic data is bounded by the energy.
In this paper we consider the nonlinear Schrödinger equation (1.1) iψ t + ∂ xx ψ = |ψ| 2 ψ, ψ : R 1+1 → C, ψ : R × Z → C, ψ(0, x) = ψ 0 (x), 1 with data which are locally in H s , with uniform bounds over the reals. The case when ψ : R×Z → C corresponds to a nonlinear Schrödinger equation on the one dimensional lattice with ∂ xx given by the discrete Laplacian. The case when ψ : R × R → C corresponds to a nonlinear Schrödinger equation in the continuum. What makes the NLS hard to analyze on R is the lack of a finite propagation speed. The speed of a signal is proportional to the derivative of the solution. In contrast to the nonlinear Schrödinger equation, the nonlinear wave equation, has a finite propagation speed. This means that at time t the solution u(t, x) depends only on data in the backward light cone {x ′ : |x ′ − x| ≤ t}. The rest of the data can be set to 0. Hence the solution exists for all time by standard arguments. The following Proposition shows how finite propagation speed enables one to get bounds on the time evolution in one dimension.
Proposition 1. If u(t, x) is a solution to the cubic nonlinear wave equation in one dimension, (1.2) with initial conditions which are uniformly C 1 (R) × C 0 (R), then By direct computation, . Then by the Sobolev embedding theorem, . This proves the proposition.
Remark: If the nonlinearity in (1.2) is replaced by u 2p+1 , we would observe a growth rate bounded by For the solution to the NLS on the lattice, ψ : R × Z → C, it is easy to prove that global solutions exist for all uniformly bounded initial data and that the solution grows at most like Ct 1 2 . The nonlinear Schrödinger equation on the lattice may be thought of as having approximate finite propagation speed since derivatives are uniformly bounded. The bound Ct 1 2 may be proved for the nonlinear Schrödinger equation on the lattice with either focusing or defocusing nonlinearity. However, observe that the proof of Proposition 1 uses the fact that the wave equation is defocusing. For the defocusing, nonlinear Schrödinger equation on the lattice, we may improve the bounds to Ct Remark: The solution to the 1D linear Schrödinger with initial data ψ 0 (x) = j a j e −(x−j) 2 , j ∈ Z is given by It is easy to show that if the |a j | are uniformly bounded, then the sup norm of this solution is bounded by Ct 1/2 . The phases in the exponential can be cancelled by the a j so this is the best one can do. On the other hand, if the a j are independent complex random variables such that E|a j | 2 = 1, then E|e it∆ ψ 0 | 2 (x) ≤ C. On the lattice, the upper bound t 1/2 follows from Proposition 2 and a t 1/2 lower bound is proved in the Appendix for particular a j .
We also study a simplified model of NLS in which the nonlinearity is regularized. The Hamiltonian of the nonlinear interaction we consider has the form |u φ (x, t)| 4 dx where u φ denotes the convolution of u with a smooth positive function φ(x) of compact support. In this case we prove that solutions exist for all time and are polynomially bounded for uniformly smooth initial data. See Theorem 2 in Sec 3.
The global existence for NLS with bounded smooth initial data is still open. However in the last section we extend the local time results of T. Oh for quasi-periodic initial to the real analytic setting by developing a Newton iteration for the short time evolution. This is partly inspired by work of Greene and Jacobowitz [GJ71] on analytic embedding. In particular we can include data of the form j a j e −(x−j) 2 where j ∈ Z and |a j | ≤ 1. The a j may be random.
Remark: Besides global existence a natural question would be to prove that the time average of the energy per unit volume of space is uniformly bounded and then to consider the possibility that the limit of the average exists as time goes to infinity. Partial answer is given in Section two, for the case of the NLS on the lattice.
1.1. Invariant measures for NLS equations. In many cases there exists an invariant measure which helps to control and describe the time evolution. The work of Lebowitz, Rose and Speer [LRS88] on the Gibbs measure for the focussing NLS on the circle is the foundational paper in this field. The construction of invariant measures for the NLS is directly related to the statistical mechanics problem with the NLS energy functional defining the theory. This equilibrium measure may also describe a soliton gas of NLS. This work was extended by Bourgain [Bou94,Bou96] who also studied the time evolution for rough data in the support of this measure. Recent works of Lebowitz, Mounaix and Wang [LMW13] as well as Carlen, Fröhlich and Lebowitz [CFL16] describe the rate of equilibration to the Gibbs measure when a suitable noise perturbation is added. More closely related to this note is Bourgain's work on NLS in the defocussing dynamics in a periodic box as the period goes to infinity [Bou00]. He proves that weak limits of solutions as L goes to infinity converge to a unique distributional solution in C(H s ), for s < 1 2 , which depends continuously on initial data in compact space time regions.
Let us now consider the NLS on a periodic lattice of length L. There is an equilibrium Gibbs measure given by The + and -sign are the focussing and defocussing cases respectively. Let · be the translation invariant expectation and define φ(t, j, ω) to be the solution of the periodic NLS with random initial data distributed by the Gibbs measure. Then since the total energy is conserved and |φ(t, j)| 4 is independent of j, we conclude that |φ(t, 0)| 4 is uniformly bounded independently of L. This shows that averaging gives good control of the time evolution. We speculate that the time average of the local energy at 0 is bounded. In addition one would like to describe its time fluctuations. But this is far beyond the scope of this note. In [LS11], Lukkarinen and Spohn investigated the dynamics of the lattice NLS with initial data governed by the infinite volume Gibbs measure. They proved that as the coupling goes to 0, the time rescaled solution to NLS obeys a kinetic equation. Recent work of Mendl and Spohn [MS15] describes equilibrium time correlations on a one dimensional lattice. On the 3 dimensional lattice, Chatterjee and Kirkpatrick, [CK12], studied the statistical mechanics with the focussing non linearity. They prove that as the density is varied, a first order phase transition occurs corresponding to soliton collapse.
Remark: In [TZ10] the detailed dynamics of a gas of solitons of the Ginzburg-Landau equation is studied. In particular, it is shown that there are solutions in which pairs of coupled solitons and separated from others is a possible class of (chaotic) solutions. The approach in this paper is based on the analysis of the infinite system of coupled ODE's corresponding to the internal degrees of freedom of the soliton pairs (center of mass, relative phase, amplitude). It should be pointed out that this system is not Hamiltonian, as it corresponds to the complex NLS.

Dynamics of NLS on the lattice
The Schrödinger equation on the lattice has properties similar to the nonlinear wave equation. In this section let −∆ = ∂ * ∂ be the finite difference Laplacian on Z. Here The lattice NLS is given by Proposition 2. If |ψ(0, x)| ≤ A then there exists a constant C such that for any x 0 ∈ Z and t 0 ≥ 1, Proof: The proof of local existence and uniqueness is an application of Picard iteration on the space l ∞ (Z) since the linear operator ∆ is bounded on l ∞ (Z), the Taylor series for e it∆ is uniformly bounded for times 0 ≤ t ≤ 1.
To get a bound on the solution at arbitrary time t 0 near x 0 , where t 0 is possibly large, we define with R ≥ 1, and let the local mass be given by By using ∂ * is the adjoint of ∂, and summation by parts, we have The first term inside the braces is positive. To estimate the last term note that we get the inequality for t 0 ≥ t , and Proposition 2 follows from Taking R = 1 completes the proof.
If we replace the local mass by the local energy |∇ψ| 2 (x) + |ψ(x)| 4 in (2.5) we can get a improved estimate on the time growth t 1/4 , compatible with Proposition 2. This result only holds for the defocussing case, and would not hold in the focussing case, unlike the proof of Proposition 2.
Remark: This argument could be generalized to the defocussing equation In that case |ψ(t, x)| t 1 p+2 .

Regularized nonlinearity
Next, we study the regularized nonlinear Schrödinger equation, where φ is a real valued, symmetric Schwartz function, and * indicates the usual convolution Remark: The operator f → φ * f is a smoothing operator. One important example of such an operator is the Fourier truncation operator. Since the solution to iu t + u xx = 0 travels at velocity ξ at frequency ξ, truncating the nonlinear term in Fourier space allows us to treat (3.1) using a finite propagation speed argument. A solution to (3.1) conserves the quantities mass, and energy, Remark: Both quantities could be infinite in this section.
3.1. Local well-posedness. Local well-posedness may be proved using perturbative arguments, for data lying in a large function space.
The solution u is bounded and uniformly continuous on [0, T ] × R.
Proof: This proof would work equally well in the focussing or defocussing cases. We first prove by using stationary phase. Without loss of generality, it suffices to show that e it∆ u 0 is bounded at the origin by the right hand side of (3.5). Using the stationary phase kernel of e it∆ , Let χ be as in (1.4). Integrating by parts, Making another integration by parts argument shows that the second term on the right hand side of (3.7) is also bounded by the right hand side of (3.5).
Now then, by the fundamental theorem of calculus, Since χ(y) is smooth and compactly supported, χ(y)u 0 (0) H 1 u 0 L ∞ , and therefore by the Sobolev embedding theorem and the fact that e it∆ is a unitary operator for L 2 -based Sobolev spaces, Finally, as in (3.7), (3.10) The right hand side of (3.10) is also bounded by the right hand side of (3.5). This finally proves that (3.5) holds.
A similar argument may also be made for the Duhamel term Since φ * u is a smoothing operator, Theorem 1 also implies Therefore, Theorem 1 holds by straightforward Picard iteration. To see this, let where C(φ) also depends on the implicit constant appearing in (3.5) when T ≤ 1. Then define the operator Then by (3.5) and (3.12), if u ∈ X, Then for T sufficiently small, Moreover, , which proves that for T sufficiently small, Φ is a contraction.

3.2.
Global well-posedness. The proof of global well-posedness for (3.1) relies on mass and energy conservation laws. Although it is not in general true that a function in L ∞ (R) will lie in L 2 (R), it is true that such embeddings do hold on any bounded set. We therefore use local conservation laws to prove global well-posedness.
Theorem 2. If u 0 lies in C 4 (R), then (3.1) has a global solution whose local energy (see (3.18) with R = t 8 ) is a quantity bounded by Ct 8 when t is large. Moreover, we show that |u(t, x)| t 8/3 . Proof: Define the local energy Remark: Observe that we rely heavily on the fact that the local energy is positive definite. In the case of finite mass, (3.3) < ∞, we could prove global well-posedness for both the focussing and defocussing problems. Here, we only prove global well-posedness for the defocussing problem.
We may compute Here we are using the inner product and the commutator is given by Using (3.1) and the fact that u t , iu t = u t , −u xx + N (u) = 0, Observe that if χ( x−x0 R ) were replaced by 1, the local energy (3.18) would be exactly equal to the mass in 1 2 × (3.3) plus the energy in (3.4). It is also clear that in the case that χ = 1, χ ′ (x) = 0 and [φ, χ 2 ] = 0, so the last two terms in (3.23) would drop out. The first term in (3.23) would also be zero due to the conservation of mass calculations. For the local energy, we will exploit the fact that a 1 R appears in − 2 , so terms depending on the size of u itself will grow as R increases. We will show that for any 0 < R < ∞, we are able to win the tug of war that develops between 1 R and the nonlinear terms for a time that increases as R increases. Taking R → ∞ we shall prove global well-posedness and |u(t, x)| t 8/3 .
Integrating by parts, (3.25) By the definition of χ, Therefore, Next, (3.28) The last equality follows from the fact that To analyze the commutator [φ, χ 2 ], first consider |x − x 0 | > 4R. In that case, Since φ is a Schwartz function, For |x − x 0 | ≤ 4R observe that by the fundamental theorem of calculus, Then, Once again, since φ is a Schwartz function, by Young's inequality, Therefore,

Decompose
(3.38) Using (3.5) and the fact that ∂ x commutes with e it∂xx , Next observe that by Duhamel's formula, Next, Now then, by Strichartz estimates and the fact that φ is a Schwartz function, and thus in L 1 , Meanwhile, computing the kernel for a generic function f , Since χ ′ ( x−x0 R ) is supported on |x − x 0 | ≤ 2R and (1 − χ( z−x0 4R )) is supported on |z − x 0 | > 4R, |x − z| > 2R in (3.43). In the region where |y − z| > |x − y|, using the fact that φ is a Schwartz function and Young's inequality, Next, consider the case when |x − y| > |y − z|. In this case we use the fact that Then let ψ be a smooth, compactly supported function, ψ(x) = 1 for |x| ≤ 1, and ψ supported on |x| ≤ 2. Using (3.45) and integrating by parts, Applying (3.45) and integrating by parts three more times, and then making the argument in (3.42)-(3.44) with x − y and y − z reversed, implies that Then consider We can compute (3.49) To simplify notation let f = |φ * u| 3 . Since φ is a Schwartz function, (3.50) Term C: For this term as well, split u t = iu xx − iN (u).

Remark:
If we assumed smooth initial data, we could combine the previous proof with Gronwall's inequality to prove bounds on higher order derivatives of the solution.

Real analytic local well-posedness
In this section we prove a local result for the one dimensional, nonlinear Schrödinger equation Such an equation has been studied by Tadahiro Oh for initial data that is of the form (4.2) u 0 (x) = cos(x) + cos( √ 2x).
Such an initial value problem cannot be solved directly using the usual methods of Strichartz estimates combined with the fact that the initial data is in an L 2 -based space. Moreover, the fact that the data (4.2) consists of two periodic functions whose periods are not rational multiples of one another prevents the study of (4.1) on a torus.
In this note, we study (4.1) for initial data that is real analytic in a strip of width 3, that is, the set For example, the initial data (4.4) u 0 (x) = n∈Z a n e −(x−n) 2 , |a n | ≤ 1 satisfies this condition. In fact, (4.4) is an entire function since if z = x + iy, (4.5) u 0 (z) = n∈Z a n e −(z−n) 2 = e y 2 n∈Z a n e −(x−n) 2 e −2iy(x−n) , so the sum (4.5) converges uniformly in any compact subset of C, proving that u 0 (x) is an entire function.
We prove the local well-posedness result for real analytic data. We suspect that Gevrey or very smooth data will suffice to get the existence of local solutions by using Moser's regularization [Mos61]. See Hörmander [H85] for an abstract version of the Newton-Nash implicit function. Proof: Define the formal iteration as follows. First let (4.6) ψ 1 (t, x) = e it∆ u 0 .
The rapid convergence of the Newton iteration relies on the fact that R n is quadratic in ξ n . For any n > 1, arguing by induction, Therefore, if we can show that ξ n → 0 and R n → 0 in a suitable Banach space as n → ∞, we are done.
It is convenient to rewrite (4.11) and (4.12) in matrix notation. Let (4.14) Then (4.11) has the form Let Φ(t, t 1 ) be the fundamental solution operator to this equation, Formally, Φ n (t, t 1 ) can be expressed as a time ordered integral Then the solution to (4.15) is given by Indeed, under (4.18), u n+1 (0) = 0, and Lemma 1 (Properties of Φ). Let Then, Proof: Φ n is the solution operator to (4.15) with b n = 0. Standard local well-posedness arguments show that (4.15) has a solution. Next, (4.21) holds for the interval of existence via Gronwall's inequality. The estimates (4.20) and (4.21) combine to show that Φ n (t, t 1 ) is well-defined for all t, t 1 ∈ R.
Proof: Without loss of generality suppose t 1 = 0 and t 2 = 1. Then for any N , Therefore, by direct computation, Then compute, Similarly, This proves the lemma.
By taking the time derivative of Φ(t) −1 Φ(t) = Id, Next, if f is a real analytic function, making a Taylor expansion, Therefore, we have the identity, where Φ n,a is the fundamental solution for M 0 + V n (t, x + a).
The norm that will be used to control the iteration is given by The following general lemma relates different norms.
Lemma 3. Let r n+1 < r n and set δ n = r n − r n+1 . Then, (4.32) f (r n+1 , p) p f (r n , 0)δ −p n . Proof: We consider the simplest form of the estimate The inequality follows since the factor in brackets is uniformly bounded by δ −1 n . The case for general p may proved the same way.
Remark: It may be possible to refine the estimates to bound the implicit constant in (4.32) to q!, which would agree with Cauchy integral formula. However, it is not too important to determine the exact constant here.
The proof of the convergence of the Newton iteration relies on the bound for the operator Φ n (t, s), where |t|, |s| ≤ 1.
Lemma 4. For |t| ≤ 1 and any r n+1 > 0, Proof: As in the previous section we have This takes care of the case when V n ≡ 0.
When V n is not identically equal to zero, we use the commutator estimates in Lemma 2. Let χ j be a partition of unity, where j χ(x − j) = j χ j (x) = 1 for any x ∈ R, and set (4.37) Therefore, (1 + (x − j)) −2 g n,j (r n , 0) ≤ sup j g n,j (r n , 0).
The L 2 norm of the last term is bounded by (4.45) e t|Vn(t)| V n (r n+1 , 3) 2 f L 2 .
The other remaining terms, Finally, using the Sobolev embedding theorem, (4.48) g n,j L ∞ g n,j L 2 + ∇g n,j L 2 , completes the proof of the lemma.
Newton iteration: The induction argument make use of the fact that if b n is real analytic, then u n+1 , given by (4.18), is also real analytic. Additionally, a product of two real analytic functions is real analytic, as well as the complex conjugate of a real analytic function is real analytic.

Appendix: Linear Schrödinger time evolution on Z
Let ψ(t, n) = e it∆ ψ 0 (n) where ψ 0 = j a j δ j and ∆ is the finite difference Laplacian.