Almost periodic solutions of the KdV equation

Journal Article · Thu Jul 01 04:00:00 EDT 1976 · SIAM Rev.; (United States)

DOI:https://doi.org/10.1137/1018074· OSTI ID:7125846

Lax, P D

The almost periodic behavior in time of space periodic solutions of the KdV equation, u/sub t/ + uu/sub x/ + u/sub xxx/ = 0, is discussed. A new proof, based on a recursion relation of Lenart, is given for the existence of an infinite sequence of conserved functionals, F/sub n/(u), of form ..integral..P/sub n/(u)dx, P/sub n/ a polynomial in u and its derivatives. The following result is reviewed and extended: the functions u minimizing F/sub N+1/(u) subject to the constraints F/sub j/(u) = A/sub j/, j = 0, ..., N, form N-dimensional tori which are invariant under the KdV flow. The extension consists of showing that for certain ranges of the constraining parameters A/sub j/ the functional F/sub N+1/(u) has minimax stationary points; these too form invariant N-tori. The Hamiltonian structure of the KdV equation, which is used in these studies, is described briefly. In an Appendix, numerical studies of the stability of some invariant 2-tori for the KdV flow are described; the numerical evidence points to stability.

• Research Organization: New York Univ.
• OSTI ID: 7125846
• Journal Information: SIAM Rev.; (United States), Journal Name: SIAM Rev.; (United States) Vol. 18:3; ISSN SIREA
• Country of Publication: United States
• Language: English

Related Subjects

658000* -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
99 GENERAL AND MISCELLANEOUS
990200 -- Mathematics & Computers
DIFFERENTIAL EQUATIONS
ELECTROMAGNETIC RADIATION
EQUATIONS
FUNCTIONALS
FUNCTIONS
HYDROGEN COMPOUNDS
KORTEWEG-DE VRIES EQUATION
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
OPTICS
OXYGEN COMPOUNDS
RADIATIONS
STABILITY
TIME DEPENDENCE
VISIBLE RADIATION
WATER
WAVE PROPAGATION