TY - JOUR

T1 - Initial value problems of the Sine-Gordon equation and geometric solutions

AU - Toda, Magdalena

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2005/5

Y1 - 2005/5

N2 - Recent results using inverse scattering techniques interpret every solution φ(x, y) of the sine-Gordon equation as a nonlinear superposition of solutions along the axes x=0 and y=0. This has a well-known geometric interpretation, namely that every weakly regular surface of Gauss curvature K=-1, in arc length asymptotic line parametrization, is uniquely determined by the values φ(x, 0) and φ(0, y) of its coordinate angle along the axes. We introduce a generalized Weierstrass representation of pseudospherical surfaces that depends only on these values, and we explicitely construct the associated family of pseudospherical immersions corresponding to it.

AB - Recent results using inverse scattering techniques interpret every solution φ(x, y) of the sine-Gordon equation as a nonlinear superposition of solutions along the axes x=0 and y=0. This has a well-known geometric interpretation, namely that every weakly regular surface of Gauss curvature K=-1, in arc length asymptotic line parametrization, is uniquely determined by the values φ(x, 0) and φ(0, y) of its coordinate angle along the axes. We introduce a generalized Weierstrass representation of pseudospherical surfaces that depends only on these values, and we explicitely construct the associated family of pseudospherical immersions corresponding to it.

KW - Generalized Weierstrass representation

KW - Loop algebra

KW - Loop group

KW - Pseudospherical surface

UR - http://www.scopus.com/inward/record.url?scp=21644450884&partnerID=8YFLogxK

U2 - 10.1007/s10455-005-1582-9

DO - 10.1007/s10455-005-1582-9

M3 - Article

AN - SCOPUS:21644450884

VL - 27

SP - 257

EP - 271

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

SN - 0232-704X

IS - 3

ER -