Abstract

It is well known that self-focusing and self-(de)focusing theories are described by the (2 + 1)-dimensional nonlinear Schrodinger equation iu_z + u_s1a1 + u_s2s2 = a₁u|u|², where u(z, S₁, S₂) is the slowly varying field amplitude, z is the propagation coordinate, a₁ is a constant, and S₁ and S₂ are proportional to transverse coordinates. (A similar equation stands for the dispersive self-modulation of waves including one transverse dimension.) Since not integrable, one cannot use the inverse scattering method to solve it. However, particular solutions can be obtained on the basis of symmetry arguments. In fact, one observes that the equation is invariant under translations of coordinates, rotation in the (S₁, S₂)-plane, constant change of phase, simultaneous dilation of coordinates, Galilean boosts and Talanov's lens transformation.¹ By appropriate combinations of these symmetries, one can find an optimal set of invariant quantities that can be used to reduce it to ordinary differential equations. These equations are finally solved, for particular parameter values, by identification to Pain-leve type equations. The norm of the solutions are all expressed in terms of elliptic functions of the variables $p^2=S_1^2+S_2^2$ or tanθ = S₁/S₂. Their diagram representations show similar behavior with solutions obtained from numerical analysis.²

© 1989 Optical Society of America