Numerical methods for nonlinear Schrödinger equations

Nonlinear evolution equations are fundamental in the mathematical description of dynamical processes in natural sciences, engineering, medicine, and economy. In particular, various physical phenomena are modeled by nonlinear partial differential equations. Prominent examples from quantum physics are nonlinear Schrödinger equations. Especially for such problems, the preservation of certain physically relevant quantities is a fundamental property of the equation. For the numerical solution of complex problems arising in practical applications, the use of computers is indispensable. Though, primarily in connection with simulations over long times, the actual result is adulterated by the influence of the discretisation, rounding errors, and inaccuracies in the data. This raises the question how to interpret the results of such computations. In this context, the field of Numerical Analysis contributes with the theoretical study of the stability and error behaviour of the employed numerical approximation. The main objective of the present project is the theoretical analysis of advanced integration methods for time- independent and time-dependent nonlinear Schrödinger equations. In particular, this includes the investigation of the convergence behaviour of the considered numerical discretisations.Primarily, we are concerned with accurate and efficient methods for the numerical solution of the Gross-Pitaevskii equation which arises in quantum physics for the description of Bose-Einstein condensation. For this purpose, space and time discretisations relying on Hermite and Fourier pseudo-spectral methods as well as high-order exponential operator splitting methods are promising.

Nonlinear partial differential equations are fundamental in the mathematical modelling of dynamical processes in natural sciences, engineering, medicine, and economy. Prominent examples from quantum physics include time-dependent nonlinear Schrödinger equations such as Gross-Pitaevskii systems, which arise in the description of multi-component BoseEinstein condensates.For the numerical solution of complex problems arising in practical applications, the use of computers is indispensable. Though, primarily in connection with simulations over longer times, the accumulation of rounding errors is inevitable and the actual result is in?uenced by the chosen discretisation method. The ?eld of Numerical Mathematics is concerned with the development of novel discretisation methods, well adapted to the considered partial differential equations, and the theoretical study of their stability and error behaviour.Within the research project Numerical methods for nonlinear Schrödinger equations advanced numerical methods for time-independent and time-dependent nonlinear Schrödinger equations were developed and their convergence behaviour was analysed. The focus was on space and time discretisation methods based on pseudo-spectral methods and high-order exponential operator splitting methods. In addition, different approaches for the construction of adaptive time-splitting methods enhancing reliability and ef?ciency of the numerical computations were investigated.The favourable behaviour of the considered discretisation methods was con?rmed by numerical experiments for nonlinear Schrödinger equations such as Gross-Pitaevskii equations.