With the aid of a direct projective approach, a general transformation solution for the nonautonomous nonlinear Schrödinger (NLS) system is derived. Based on certain known exact solutions of the projective equation, some periodic and localized excitations with novel properties are correspondingly revealed by entrancing appropriate system parameters. The integrable constraint conditions for the ...

The critical nonlinear Schrödinger equation (NLS) on bounded domains models the propagation of cw laser beams in hollow-core fibers. Unlike the NLS on unbounded domains which models propagation in bulk media, the ground-state waveguide solutions are stable and the condition of critical power for singularity formation is generically sharp. © 2001 Elsevier Science B.V. All rights reserved. PACS: ...

The Fourier transforms of the products of two respectively three solutions of the free Schrödinger equation in one space dimension are estimated in mixed and, in the first case, weighted L norms. Inserted into an appropriate variant of the Fourier restriction norm method, these estimates serve to prove local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger (NLS) equation...

The formation of extreme localizations in nonlinear dispersive media can be explained and described within the framework of nonlinear evolution equations, such as the nonlinear Schrödinger equation (NLS). Within the class of exact NLS breather solutions on a finite background, which describe the modulational instability of monochromatic wave trains, the hierarchy of rational solutions localized...

We consider the focusing inhomogeneous biharmonic nonlinear Schrödinger equation in $$H^2(\mathbb {R}^N)$$ , $$\begin{aligned} iu_t + \Delta ^2 u - |x|^{-b}|u|^{\alpha }u=0, \end{aligned}$$ when $$b > 0$$ and $$N \ge 5$$ . first obtain a small data global result $$H^2$$ which, five-dimensional case, improves previous from Pastor second author. In sequel, we show main result, scattering below ma...

A geometrical description of the Heisenberg magnet (HM) equation with classical spins is given in terms of flows on the quotient space G/H+ where G is an infinite dimensional Lie group and H+ is a subgroup of G. It is shown that the HM flows are induced by an action of R2 on G/H+, and that the HM equation can be integrated by solving a Birkhoff factorization problem for G. For the HM flows whic...

We study two resonant Hamiltonian systems on the phase space L2(R → C): the quintic one-dimensional continuous resonant equation, and a cubic resonant system that appears as the modified scattering limit of a specific NLS equation. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scale...

An exact pulse for the parametrically forced nonlinear Schrodinger equation (NLS) is isolated. The equation governs wave envelope propagation in dispersion-managed fiber lines with positive residual dispersion. The pulse is obtained as a ground state of an averaged variational principle associated with the equation governing pulse dynamics. The solutions of the averaged and original equations a...

We present numerical simulations of the defocusing nonlinear Schrödinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig-Merle and Kilip-Visan who considered some energy supercritical wave equations and proved that if the solution is a priori bounded in the critical Sobolev space (i.e. the space whose homogeneous norm is invar...