Pseudopotentials and Symmetries for Generalized Nonlinear Schrödinger Equations

Abstract

Applying the method of differential ideals and prolongations to equations of the type

$$ iZ_t + Z_{xx} = f\left( {Z,Z*} \right), $$

where f(Z,Z*) is any smooth function, we arrive at eight types of equations admitting “pseudopotentials” (in the sense of Wahlquist and Estabrook [1976]). For five of these, the integrability conditions are equivalent to conservation laws. For the remaining three, including the usual cubically non-linear Schrodinger equation, there are non-trivial pseudo-potentials, which may be interpreted as defining Bácklund transformations or linear scattering equations. For the two cases other than the standard one, however, there is no parameter identifiable as an eigenvalue and no Lie symmetry (other than translation invariance) to generate such a parameter. For the standard case, we show that the real and imaginary parts of the parameter occuring in the Backlund transformation (or scattering equation) may be generated by composing a given transformation with the 2-parameter Lie symmetry group consisting of Galilean boosts and dilations.