Pseudopotentials and Lie symmetries for the generalized nonlinear Schrödinger equation

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Abstract
The generalized nonlinear Schrödinger equation zxx+izt = f(z, z*) is analyzed from the point of view of the existence of pseudopotentials, Bäcklund transformations, Lie symmetries, and conservation laws. Applying the Wahlquist–Estabrook method of closed differential ideals we show that eight classes of nontrivial interaction terms f(z,z*) exist for which the equation allows the existence of pseudopotentials. Five of them simply lead to conservation laws, the remaining three to Bäcklund transformations. The usual ’’cubic’’ nonlinear Schrödinger equation with f(z,z*) = εz‖z‖2 is obtained as a special case. It is also the only case for which the Bäcklund transformation contains a free parameter. We show that the real and complex parts of this parameter are generated by the dilation and Galilei invariance of the equation.