Steady solutions for the Schrödinger map equation

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 obtain by S. Kida in 1981, and therefore they are also related, thanks to the so-called Hasimoto transformation, to traveling wave solutions of the 1d cubic non-linear Schrödinger equation.

KEYWORDS:

• Vortex filaments
• Shrödinger map equation
• bifurcation theory

Disclosure statement

The authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest, or non-financial interest in the subject matter or materials discussed in this manuscript. Moreover, data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Additional information

Funding

C.G. has been supported by RYC2022-035967-I (MCIU/AEI/10.13039/501100011033 and FSE+), and partially by the ERC-StG-852741 (CAPA), by Grants PID2022-140494NA-I00 and PID2022-137228OB-I00 funded by MCIN/AEI/10.13039/501100011033/FEDER, UE, by Grant C-EXP-265-UGR23 funded by Consejeria de Universidad, Investigacion e Innovacion & ERDF/EU Andalusia Program, and by Modeling Nature Research Unit, project QUAL21-011, and L. Vega is funded in part by MICINN (Spain) projects Severo Ochoa CEX2021-001142, and PID2021-126813NB-I00 (ERDF A way of making Europe). This work was developed during the semester program Hamiltonian Methods in Dispersive and Wave Evolution Equations at ICERM.