Factorization of the nonlinear Schrödinger equation and applications

Abstract

We consider factorizations of the stationary and non-stationary Schrödinger equation in which are based on appropriate Dirac operators. These factorizations lead to a Miura transform which is an analogue of the classical one-dimensional Miura transform but also closely related to the Riccati equation. In fact, the Miura transform is a nonlinear Dirac equation. We give an iterative procedure which is based on fix-point principles to solve this nonlinear Dirac equation. The relationship to nonlinear Schrödinger equations like the Gross–Pitaevskii equation is highlighted.

§Dedicated to Richard Delanghe on the occasion of his 65th birthday.

Keywords:

• Clifford analysis
• Schrödinger equation
• Nonlinear Dirac equation

Keywords:

• Mathematics Subject Classifications 2000: Primary 30G35
• Secondary 35J10
• 35F30
• 35Q55

Notes

§Dedicated to Richard Delanghe on the occasion of his 65th birthday.