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 the cubic defocusing Schrödinger equation in two space dimen15 sions and prove that if the initial data are in H1/2, then there exists a global solution in time. The proof combines the argument from [5] with some new correlation estimates 17 for the Schrödinger equation.

A construction is given for a planar rational Pythagorean hodograph spiral which interpolates any two-point G Hermite data that a spiral can match. When the curvature at one of the points is zero, the construction gives the unique interpolant that is the involute of a rational Pythagorean hodograph curve of the form cubic over linear. Otherwise, the spiral comprises an involute of a Tschirnhaus...

We study the Isomorphism of Polynomial (IP2S) problem with m = 2 homogeneous quadratic polynomials of n variables over a finite field of odd characteristic: given two quadratic polynomials (a,b) on n variables, we find two bijective linear maps (s, t) such that b = t◦a◦s. We give an algorithm computing s and t in time complexity Õ(n) for all instances. The IP2S problem was introduced in cryptog...

ABSTRACT An accurate closed-form solution for the quintic Duffing equation is obtained using a cubication method. In this method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude. The replacement of the origi...

We derive rigorously, for both R2 and [−L, L]×2, the cubic nonlinear Schrödinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the...