Infinite-dimensional homogeneous manifolds with translational symmetry and nonlinear partial differential equations

Thesis/Dissertation · Thu Jan 01 00:00:00 EST 1987

OSTI ID:5149008

Szmigielski, J

A construction relating scale invariant, nonlinear partial differential equations and the orbits of the group of translations on infinite-dimensional homogeneous manifolds is proposed. It presupposes that the homogeneous manifold M has translations and scale transformations among its automorphisms. This assumption is made to ensure that the set of orbits of the group of translations is invariant under scale transformations. What is important is that the proposed construction provides a way of inducing scale covariance of derived nonlinear equations, for which the set of orbits can be identified with a set of solutions of these equations. Application to the derivation of the potential KdV, the sine-Gordon, the modified KdV and the nonlinear Schroedinger equation yields as an important intermediate result the construction of their respective Lax pairs out of the commuting vectors tangent to the orbits. In yet another application an infinite dimensional, scale invariant Riccati equation is derived. The latter is shown to be related to the potential KP equation. The orbit leading to the Riccati equation is then computed. Also, the relation to the Zakharov-Shabat dressing method is briefly discussed.

• Research Organization: Georgia Univ., Athens, GA (USA)
• OSTI ID: 5149008
• Resource Relation: Other Information: Thesis (Ph. D.)
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
QUANTUM FIELD THEORY
MATHEMATICAL MANIFOLDS
PARTIAL DIFFERENTIAL EQUATIONS
INVARIANCE PRINCIPLES
NONLINEAR PROBLEMS
ORBITS
SCALING LAWS
SCHROEDINGER EQUATION
SINE-GORDON EQUATION
DIFFERENTIAL EQUATIONS
EQUATIONS
FIELD EQUATIONS
FIELD THEORIES
WAVE EQUATIONS
645400* - High Energy Physics- Field Theory