Nonlinear evolution equations solvable by the inverse spectral transform.—I

• DOI: 10.1007/bf02727634
• Publication Date: 2007-12-18T14:12:53Z
• Authors: F. Calogero; A. Degasperis
• Keywords: 0103 physical sciences; 01 natural sciences

This paper is the second of a series based on a general method to discover and investigate nonlinear evolution equations (NEEs) solvable by the inverse spectral transorm (IST); the results reported are those that obtain by applying this method to the one-dimensional matrix Schrodinger (linear) problem. We give a class of NEEs solvable by IST that is more general than that introduced previously by Wadati and Kamijo, even in the simpler case with only one space variable and constant coefficients; moreover, we introduce classes of NEEs involving more than one space variable and containing coefficients that are not constant. We also introduce and discuss a very general class of Backlund transormations (BTs), and derive a number of results (nonlinear superposition principle, multisoliton, ladder, conserved quantities, generalized resolvent formula) implied by them; and we clarify the relationship between BTs and the IST technique. The most remarkable aspect of the results presented in this paper is the discovery that the solitons associated to these NEEs, although possessing all the stability properties that characterize solitons, generally do not move with constant speed (even in the simpler case with one space variable only, that is, discussed in greater detail). This paper is focused on a general presentation of the approach and on the proofs of the results (some of which had been previously reported without proofs); but we also discuss the simpler NEE belonging to this class, and we present some related novel results, including its Lagrangian and the display of some special cases, that constitute interesting novel examples of fairly simple NEEs solvable by IST.