Geometric Breathers of the mKdV Equation
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Geometric Breathers of the mKdV Equation Miguel A. Alejo
Received: 19 August 2011 / Accepted: 16 February 2012 / Published online: 2 March 2012 © Springer Science+Business Media B.V. 2012
Abstract In this paper we present some numerical results about the existence of a new family of solutions of the geometric mKdV equation. This equation is characterized by the fact that the curvature evolves according to the focusing mKdV equation. The new type of solutions are similar to the breathers found by M. Wadati in 1973. We exhibit curves with and without self-intersections, and also some examples of initially simple closed curves that have self-crossings at later times. Keywords mKdV equation · Breather · Nonvanishing condition · Closed curve Mathematics Subject Classification (2000) Primary 37K15 · 35Q53 · Secondary 35Q51 · 37K10
1 Introduction In this work we are interested in a geometric PDE that is related to the focusing modified Korteweg-de Vries equation (shortly mKdV) ∂k 3 ∂k ∂k + 3 + k2 = 0, ∂t ∂s 2 ∂s
(s, t) ∈ R × R.
(1.1)
If k(s, t) ∈ R denotes the curvature of a curve and s is the arclength parameter, the above PDE is nothing but the evolution of the curvature of the geometric flow of planar curves z(s, t) = x(s, t) + iy(s, t) ∈ C, which satisfies the following Initial Value Problem (IVP in short) ⎧ ∂z ∂k 1 2 ∂z ⎪ ⎨ ∂t (s, t) = (−i ∂s (s, t) − 2 k(s, t) ) ∂s (s, t), (1.2) z(s, 0) = z0 , ⎪ ⎩ 2 |zs | = 1. M.A. Alejo () Department of Theoretical Biology, University of Bonn, 53115 Bonn, Germany e-mail: [email protected]
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M.A. Alejo
This geometric flow was first introduced by G. Lamb in [1] and by R. Goldstein and D. Petrich in [2]. The mathematical motivation comes in both cases from a celebrated paper by H. Hasimoto [3]. In that work he proved that the one dimensional cubic non-linear Schrödinger equation describes the evolution of the curvature and the torsion of a vortex filament according to the so called Localized Induction Approximation. This approximation is based on two steps. First, a suitable truncation of the Biot-Savart integral is done, and then a Taylor expansion is made to conclude that the local behavior of the velocity of the filament flows in the direction of the binormal vector with a modulus proportional to the curvature. A similar procedure is used by R. Goldstein and D. Petrich in [2] (see also the work of J. Langer [4]), this time with the aim to describe locally the evolution of the boundary of a vortex patch in the plane that moves according to the Euler equations. In this case a complete Taylor expansion can be computed explicitly. Each one of the terms of the sum gives a corresponding geometric flow, that if it is described by using the curvature as unknown, it is nothing but the hierarchy of the focusing mKdV. All these geometric flows have the property that they preserve the enclosed area. This is a natural conservation law for the vortex patch problem. However, they also preserve the total length of the curve and this falls into contradiction with the observed phen
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