Sobolev norm growths for Schrödinger equations:
Some numerical examples
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1. NLS on the two-dimensional torus
We consider the cubic nonlinear Schrödinger equation (NLS)
set on the two-dimensional torus:
In Fourier this
equation is written
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
We can prove that this
set of indices is made of quadruples forming rectangles on ℤ2.
It has been proven by Bourgain
that the Sobolev norms of the solutions of NLS satisfy bounds of
for some coefficient α(s) > 0. On the other hand, Colliander,
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)
This result was refined
in particular with a control of the time T with respect to ε and M.
In a similar direction, Carles
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
of only 5 modes. In the next simulations, we consider this
particular initial value, with ε
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.
We represent the
evolution of the Fourier modes in logarithmic scale and in 2D.
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.
Click on the image below to see the corresponding movie:
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
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
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.
potential in one dimension
consider the equation
the linear Schrödinger equation set on the one
dimensional torus with a potential that depend on the time.
Let us first consider the potential
the initial condition the analytic function
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.
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
where βN is a 2πN periodic
potential defined as follows:
where PN(t) is a polynomial of degree 3 such
extend βN to by 2πN periodicity.
Hence β(t) is C1, with
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.
Sobolev norms look as follows:
make some comments: the previous simulations do not contradict
the existing bounds of J.
, (see for instance Jean-Marc's paper here
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!