Sobolev norm growths for Schrödinger equations:
Some numerical examples



This page reflects works in progress with
Radoin Belaouar. Do not hesitate to contact us if you have any comment or suggestion.

1. NLS on the two-dimensional torus

We consider the cubic nonlinear Schrödinger equation (NLS) set on the two-dimensional torus:
2 2 i∂tu = - Δu
            + |u| u, u(0,x) = u0(x), x ∈ T .
In Fourier this equation is written
2 2 ∑ j ∈ ℤ ,
            i∂tuj = |j| uj + uku ℓ¯um. j=k+ℓ-m
This equation is resonant in the sense that the eigenvalues of the linear part are all integers. A normal form analysis shows that for small initial values, the nonlinear energy exchanges between the Fourier modes are driven by the resonant modulus
2 4 2 2 2 2 K =
            {(k,ℓ,m, j) ∈ (ℤ ) | k + ℓ = m + j and |k| + |ℓ| = |m| + |j|
            }.
We can prove that this set of indices is made of quadruples forming rectangles on 2.

It has been proven by
Bourgain that the Sobolev norms of the solutions of NLS satisfy bounds of the form
α(s) ∥u(t)∥ Hs ≤
            Ct
for some coefficient α(s) > 0. On the other hand, Colliander, Keel, Staffilani, Takaoka and Tao proved the following result (see the paper Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 2010): for given ε,M > 0 and s > 1, there exists T and a solution u(t), t (0,T) such that
∥u(0)∥ Hs ≤ ε,
            and ∥u(T )∥ Hs ≥ M.
This result was refined by Kaloshin and Guardia, see arXiv:1205.5188 in particular with a control of the time T with respect to ε and M.

In a similar direction, Carles and Faou proved (see the paper Energy cascades for NLS on the torus, Discrete Contin. Dyn. Syst. 2012) the existence of an energy cascade for the particular initial datum


u
                    = ε(1 + 2cos(x) + 2cos(y)), 0

made of only 5 modes. In the next simulations, we consider this particular initial value, with ε = 0.5.The numerical scheme used is based on a splitting algorithm between the linear and nonlinear parts, that both can be solved exactly in Fourier and (x,y) variables respectively. This algorithm can be implemented in an efficient way using the Fast Fourier Transform algorithm. The main issues with this method are aliasing problems due to the space discretization, and step-size resonances due to the time integration. We refer to the book Geometric numerical integration and Schrödinger equations, by E. Faou, European Math. Soc. 2012 for discussions and convergence proofs of this numerical scheme.
To avoid this numerical resonances problem, we systematically use stepsizes satisfying a Courant-Friedrichs-Lewy condition of the form τN2 < c, where τ is the time step. Henre,  N = 32,64,128 and 256 is the number of modes in each direction.

We represent the evolution of the Fourier modes in logarithmic scale and in 2D.
Click on the image below to see the corresponding movie:

movie32.png movie64.png
movie128.png movie256.png

The dynamics of the modes in Fourier is as follows: starting from five modes, the energy propagates to high frequencies. The frequency support increases with the time. The following plots show the dynamics at different times, starting from the time t = 0, for which only five modes exist, we then observe a propagation of energy to the high modes. When the energy of the modes near the boundary is significantly large, aliasing phenomena come into play, which indicates that the simulation can become wrong. That is why to observe the dynamics of the exact NLS equation, we need a large number of modes, which in turn requires the use of smaller and smaller time steps to avoid stepsize resonances.

c1.png c2.png
c3.png c4.png


We consider now the evolution of Sobolev norms of indices 4 and 6. We plot the evolution of the norms for various numbers of points on the grid
N = 32,64,128, 256 and 512 in each direction. We observe a drift in all cases.

H4norm.jpg

Note that due to the CFL (Courant-Friedrichs-Lewy) restriction, the CPU time required for each simulation dramatically increases with the number of modes N = 32,64,128, 256 and 512. The simulation with 512 points lasted 1.5 month on a computer at the CMAP.
 
H6norm.jpg



2. Time dependent potential in one dimension

We consider the equation
i∂tu(t,x) = - Δu (t,x) + V(t,x)u(t,x ), x ∈ T1,
that is the linear Schrödinger equation set on the one dimensional torus with a potential that depend on the time. Let us first consider the potential
V(t,x) =
            cos(x) sin(t2)
and as the initial condition the analytic function
u0(x) =
            εexp (- icos(x)) with ε = 0.1.
Note that the time derivatives of the potential are not bounded, but it is smooth in the space variable. We first show a simulation using a splitting method with 512 Fourier modes, and a time step δt = 10-6 so that the CFL number is of order 0.3. In Figure 1 we plot the evolution of the Fourier modes uj of the solution, in logarithmic scale.
modesgr

We observe a nice energy cascade, which in seems to converge towards a solution with constants actions. The stretching in time of the potential introduces a kind of scattering effect. In a second simulation, we take the potential
V (x,t)
            = βN (t)cos(x)
where βN is a 2πN periodic potential defined as follows:
β (t) =
            sin(t2) t ∈ (0,πN ) β (t) = sin(t2)PN (t) t ∈ (πN, 2πN )
where PN(t) is a polynomial of degree 3 such that
PN(πN )
            = 1, P ′(πN ) = P(2πN ) = P ′(2πN ) = 0.
We then extend βN to ℝ by 2πN periodicity. Hence β(t) is C1, with
′ ′ β(0)
            = β (0) = β(2πN ) = β (2πN ) = 0.
Moreover, we have
∥β (t)∥
            = 1 and ∥β ′∥ ≤ CN. L∞ L∞
We take N = 10. We observe growths of the Sobolev norms in ts, if s denotes the Sobolev index. This growth seems to be due to several compositions of the scattering operator observed above.

modesgr

 
The Sobolev norms look as follows:

modesgr modesgr
modesgr

Let us make some comments: the previous simulations do not contradict the existing bounds of J. Bourgain, W.-M. Wang or J.-M. Delort, (see for instance Jean-Marc's paper here), essentially because the potential we consider has a large derivative with respect to the time (though is is bounded and smooth in the space variable). However, the nonlinear effect of the oscillating potential can be observed in a relatively short time. And it gives nice pictures!