Steady solutions for the Schr\"odinger map equation
Claudia Garc\'ia, Luis Vega
TL;DR
This paper constructs a new family of steady solutions for the Schr"odinger map equation using bifurcation methods, complementing previous solutions and linking to traveling wave solutions of the 1D cubic NLS via the Hasimoto transformation.
Contribution
It introduces a novel family of steady solutions for the Schr"odinger map equation, expanding the understanding of vortex filament dynamics.
Findings
New steady solutions for the binormal flow are constructed.
Solutions are related to traveling waves of the 1D cubic nonlinear Schr"odinger equation.
The work complements and extends previous solutions by Kida (1981).
Abstract
In this paper we use bifurcation methods to construct a new family of solutions of the binormal flow, also known as the vortex filament equation, which do not change their form. Our examples are complementary to those obtained by S. Kida in 1981, and therefore they are also related, thanks to the so-called Hasimoto transformation, to travelling wave solutions of the 1d cubic non-linear Schr\"odinger equation.
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Taxonomy
TopicsAdvanced Mathematical Physics Problems · Nonlinear Waves and Solitons · Numerical methods for differential equations