2008/12/12 by P. G. Grinevich, Grinevich, P. G., K. V. Kaipa +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.0812.2494
arxiv created 2009/04/23 · arxiv updated 2009/12/01
The most basic characteristic of x-quasiperiodic solutions u(x,t) of the sine-Gordon equation utt-uxx+sin u=0 is the topological charge density denoted n. The real finite-gap solutions u(x,t) are expressed in terms of the Riemann theta-functions of a non-singular hyperelliptic curve Γ and a positive generic divisor D of degree g on Γ, where the spectral data (Γ, D) must satisfy some reality conditions. The problem addressed in note is: to calculate n directly from the theta-functional expressions for the solution u(x,t). The problem is solved here by introducing what we call the multiscale or elliptic limit of real finite-gap sine-Gordon solutions. We deform the spectral curve to a singular curve, for which the calculation of topological charges reduces to two special easier cases.