N OETHER'S Theorem and Higher Conservation Laws in Ultrashort Pulse Propagation

Abstract LAMB has constructed a number of local conservation laws for the differential equation στ = sin σ describing ultrashort pulse propagation in a resonant medium. On the other hand it is known that from any solution of this equation others may be found by BÄCKLUND transformation. In this paper we introduce extended BÄCKLUND transformations Ba and compose them to form infinitesimal invariance transformations Ba+ε B from which via NOETHER'S theorem a system of conservation laws follows. Some of them are recognized as LAMB'S conservation laws. Another conservation law results from a scale invariance called LIE'S transformation.