What are the solution spaces of Nonlinear Schrödinger equations?

Question

As we know, the solution space of Schrödinger equation is a Hilbert space, however, what about it of Nonlinear Schrödinger equations such as $$i\partial_t\psi=-{1\over 2}\partial^2_x\psi+\kappa|\psi|^2 \psi$$?

Answer 1

This is not a question about physics. As has been stressed numerous times here, solutions of the NLS cannot be interpreted as quantum mechanical wave-functions. Their evolution is not unitary. As a consequence, the solution space has much less physical relevance.

The cubic NLS you wrote down appears in various approximations to nonlinear dispersive waves (including KdV, nonlinear perturbations of Klein-Gordon waves, and water waves); it describes the modulation profile of slowly varying wave packets with small amplitude.

The equation is Hamiltonian, with the ``energy'':

$\frac{1}{2}\int |\nabla \psi|^2\,\mathrm{d}x+\frac{\kappa}{4}\int|\psi|^4\,\mathrm{d}x$

and the mass

$\int |\psi|^2\,\mathrm{d}x$

as conserved quantities. These conserved quantities allow you to solve the equation for all time given initial data in the Sobolev space $H^1$ (this just means that the integrals of $|\nabla \psi|^2$ and $|\psi|^2$ are convergent) in case $\kappa > 0$ or, if $\kappa < 0$, whenever the nonlinearity is weak enough to be controlled by the gradient term for all times. This last condition can be expressed in terms of the Sobolev embedding. In dimension one, it is satisfied for power nonlinearities that are less than quintic. If the nonlinearity is too severe (for example, in dimension greater than 2 for the cubic case you asked about) perfectly nice solutions can blow up in finite time. In that case speaking of ``solution space'' does not make much sense since we cannot associate uniformly a time evolution to every vector.

Mathematicians have expended a lot of effort to solve this equation in various spaces of rough functions. Like $H^1$, most of them happen to be Hilbert spaces, but this has little physical (and no quantum mechanical) relevance. It is just a matter of convenience.

Answer 2

Although,the set of solutions of the nonlinear Schrodinger equation (NLS) is not a Hilbert space and the field $\psi$ cannot be interpreted as a wave function, this does not mean that the NLS cannot be quantized. It can if we interpret $\psi$ as a classical field.

In this case the space of solutions or equivalently, the space of initial conditions (configurations) (by considering a solution of a this PDE as an evolution of an intial condition or configuration) can be interpreted as a classical phase space (It turns out to be an infinite dimensional symplectic manifold).

It is a quite general property that the space of solutions or equivalently,the space of the initial data of a wide class of partial differential equations is a symplectic manifold. This happens in ordinary mechanics. Also in the case of the linear Schrodinger equation or in field theories having linear equations of motion, this symplectic manifold is the projective Hilbert space of (the Hilbert space) of solutions. This point constitutes the main answer to the question and it is mutualto the linear and nonlinear Schrodinger equations.

Not only that,in the case of the NLE, the evolution of the classical configurations is Hamiltonian (i.e., half of the parameters can be interpreted as positions and theother half as momenta). There are choices of the initial parameters which satisfy almost canonical commutation relations such as the inverse scattering parameters. In this case,the quantization can be performed quite straightforwardly.

The only difference between this procedure and the familiar second quantization of the linear Schrodinger field, is that the solutions of the NLE depend nonlinearly on the initial parameters. Of course, it required a great deal of ingenuity to derive these solutions.

This principle has been applied in other cases of quantization of nonlinear field theories such as the Chern Simons theory.