We apply Painlevé test to the most general variable coefficient nonlinear Schrödinger (VCNLS) equations as an attempt to identify integrable classes and compare our results versus those obtained by the use of other tools like group-theoretical approach and the Lax pairs technique of the soliton theory. We present explicit transformation formulae that can be used to generate new analytic solutions of VCNLS equations from those of the integrable NLS equation.

We use Lie point symmetries of the 2+1-dimensional cubic Schrödinger equation to obtain new analytic solutions in a systematic manner. We present an analysis of the reduced ODEs, and in particular show that although the original equation is not integrable they typically can belong to the class of Painlevé type equations. *

We consider a class of generalized Kuznetsov-Zabolotskaya-Khokhlov (gKZK) equations and determine its equivalence group, which is then used to give a complete symmetry classification of this class. The infinite-dimensional symmetry is used to reduce such equations to (1 + 1)-dimensional PDEs. Special attention is paid to group-theoretical properties of a class of generalized dispersionless KP (gdKP) or Zabolotskaya-Khokhlov equations as a subclass of gKZK equations. The conditions are determined under which a gdKP equation is invariant under a Lie algebra containing the Virasoro algebra as a subalgebra. This occurs if and only if this equation is completely integrable. A similar connection is shown to hold for generalized KP equations.

Güngör, Faruk (Dogus Author) -- Hasanov, Mahir H. (Dogus Author)A canonical variable coefficient nonlinear Schrödinger equation with a four-dimensional symmetry group containing SL(2, ℝ) group as a subgroup is considered. This typical invariance is then used to transform by a symmetry transformation a known solution that can be derived by truncating its Painlevé expansion and study blow-ups of these solutions in the L p-norm for p > 2, L ∞-norm and in the sense of distributions

Analytical solutions of variable coefficient nonlinear Schrödinger equations having four-dimensional symmetry groups which are in fact the next closest to the integrable ones occurring only when the Lie symmetry group is five-dimensional are obtained using two different tools. The first tool is to use one dimensional subgroups of the full symmetry group to generate solutions from those of the reduced ODEs (Ordinary Differential Equations), namely group invariant solutions. The other is by truncation in their Painlevé expansions.