Justification of the Nonlinear Schrödinger Approximation for the Water Wave Problem with Surface Tension

• Applicant: Professor Dr. Wolf-Patrick Düll
• Subject Area: Mathematics
• Term: since 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 329302595

In 1968 Zakharov derived a Nonlinear Schrödinger (NLS) equation as a formal approximation equation to describe the evolution of the envelopes of oscillating wave packet-like solutions to the water wave problem. The mathematically rigorous justification of the validity of this approximation has been an open problem for more than four decades. In recent years the NLS approximation has been justified for the water wave problem without surface tension. In the first part of this project the problem of the mathematically rigorous justification of the Nonlinear Schrödinger approximation for the 2-dimensional water wave problem with surface tension has been solved by proving error estimates between exact wave packet-like solutions to the water wave problem and their formal approximations obtained via the 1-dimensional cubic Nonlinear Schrödinger equation in the physical relevant scales. The obtained error estimates are even valid for the cases with and without surface tension and uniform with respect to the strength of the surface tension as the surface tension and the height of the wave packet go to zero.The objective of the second part of this project is to prove analogous justification results for Nonlinear Schrödinger approximations for the 3-dimensional water wave problem by transferring and generalizing the methods developed in the first part of the project.

• DFG Programme: Research Grants