Philippe Gravejat

Travelling Waves for the Gross-Pitaevskii Equation II

The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results (see [3,4,8]) where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.

On the Korteweg–de Vries Long-Wave Approximation of the Gross–Pitaevskii Equation I

The fact that the Korteweg-de Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation was derived several years ago in the physical literature (see e.g. [17]). In this paper, we provide a rigorous proof of this fact, and compute a precise estimate for the error term. Our proof relies on the integrability of both the equations. In particular, we give a relation between the invariants of the two equations, which, we hope, is of independent interest.

On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation II

In this paper, we proceed along our analysis of the Korteweg–de Vries approximation of the Gross–Pitaevskii equation initiated in [6]. At the long-wave limit, we establish that solutions of small amplitude to the one-dimensional Gross–Pitaevskii equation split into two waves with opposite constant speeds , each of which are solutions to a Korteweg–de Vries equation. We also compute an estimate of the error term which is somewhat optimal as long as travelling waves are considered. At the cost of higher regularity of the initial data, this improves our previous estimate in [6].

Construction of the Pauli–Villars-Regulated Dirac Vacuum in Electromagnetic Fields

Using the Pauli–Villars regularization and arguments from convex analysis, we construct solutions to the classical time-independent Maxwell equations in Dirac’s vacuum, in the presence of small external electromagnetic sources. The vacuum is not an empty space, but rather a quantum fluctuating medium which behaves as a nonlinear polarizable material. Its behavior is described by a Dirac equation involving infinitely many particles. The quantum corrections to the usual Maxwell equations are nonlinear and nonlocal. Even if photons are described by a purely classical electromagnetic field, the resulting vacuum polarization coincides to first order with that of full Quantum Electrodynamics.