Adaptivity with moving grids

Adaptivity with moving grids 109
5.2.2. Focusing solutions of the nonlinear Schrödinger equation
The nonlinear Schrödinger equation described in Section 5.2.1,
iu t +∆u + u|u| 2 =0, x ∈ R n , (5.28)
is a model for the modulational instability of water waves and plasma waves,
and is important in studies of nonlinear optics where the refractive index
of a material depends on the intensity of a laser beam. In all dimensions it
has the conserved quantities
∫
|u| 2 dx and
∫ (|∇u| 2 − 1 2 |u|4 )
dx,
corresponding to mass and energy respectively. In one dimension (5.28) is
integrable and can give rise to soliton-type solutions. More generally, it is
an example of a Hamiltonian PDE. Many numerical (usually non-adaptive)
methods have been derived to take advantage of this integrability (Budd
and Piggott 2005, McLachlan 1994). If posed in n dimensions, where n ≥ 2,
then the PDE (5.28) is no longer integrable and it may admit singular
(focusing) solutions for certain initial data. A review of these is given by
Sulem and Sulem (1999), who also describe some numerical computations
using a moving mesh method based on Winslow’s algorithm. Numerically
this is a very difficult problem, as high resolution in time and space is
required to capture the strong self-focusing of the solution, to deal with the
highly oscillatory nature of the solution ‘tail’, and to compute over a large
domain to avoid boundary effects (Budd et al. 1999a, Ceniceros 2002). In
such singular solutions both the maximum value of the solution modulus,
and its phase, blow up in a finite time T . The precise form of this blowup
and the initial conditions that lead to blow-up are the subject of much
investigation (see the review in Sulem and Sulem (1999)), and much remains
unresolved. Two significant open questions are: (1) What is the exact
nature of the blow-up profile in two dimensions (where it is known to be
approximately self-similar but the precise form is still unclear), and (2) Do
there exist radially symmetric self-similar solutions in three dimensions? In
the latter case these are conjectured to take the form
u(r, t) =
1
√
(T − t)
e ia log(T −t) Q(y), r = √ T − ty, r= |x|,
where a is an unknown constant. The existence of such a solution can be
addressed by a scale-invariant method applied to the (one-dimensional) class
of radially symmetric solutions. In this we take the monitor function
M = |u(r, t)| 2 ,
derived in Section 5.2.1, and use a collocation-based discretization with
N = 81 points. In such a calculation we would expect to observe a discrete