Several theories for weakly damped free-surface flows have been formulated. In this paper we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli’s equation), but also to the kinematic boundary condition. The nonlin...

We prove local existence and uniqueness of solutions for the one-dimensional nonlinear Schrödinger (NLS) equations iut + uxx ± |u| 2 u = 0 in classes of smooth functions that admit an asymptotic expansion at infinity in decreasing powers of x. We show that an asymptotic solution differs from a genuine solution by a Schwartz class function which solves a generalized version of the NLS equation. ...

The cubic nonlinear Schroedinger equation (NLS) describes the space-time evolution of narrow-banded wave trains in one space and one time (1 + 1) dimensions. The richness of nonlinear wave motions described by NLS is exemplified by the fully nonlinear envelope soliton and “breather” solutions, which are fully understood only in terms of the general solution of the equation as described by the i...

The Benjamin-Feir modulational instability effects the evolution of perturbed planewave solutions of the cubic nonlinear Schrödinger equation (NLS), the modified NLS, and the band-modified NLS. Recent work demonstrates that the BenjaminFeir instability in NLS is “stabilized” when a linear term representing dissipation is added. In this paper, we add a linear term representing dissipation to the...

We analyze a certain class of integral equations associated with Marchenko equations and Gel’fand-Levitan equations. Such integral equations arise through a Fourier transformation on various ordinary differential equations involving a spectral parameter. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution in terms of the unper...

Homoclinic varieties play a crucial role in the dynamics of perturbations of the focusing Nonlinear Schrr odinger equation (NLS). We undertake a Mel'nikov analysis to investigate the possibility of persistence of transversal homoclinic orbits for a conservative perturbation of the NLS.

We consider the modulated harmonic wave in discrete series connected Josephson transmission line (JTL). formulate approach to modulation problems for equations based on calculus. check up by applying it Fermi-Pasta-Ulam-Tsingou type problem. Applying JTL, we obtain equation describing amplitude, which turns out be defocusing nonlinear Schr\"odinger (NLS) equation. compare profile of single soli...

We transpose work by T.Mizumachi to prove smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 1D. As an application we extend to dimension 1D a result on asymptotic stability of ground states of NLS proved by Cuccagna & Mizumachi for dimensions ≥ 3. §

We transpose work by K.Yajima and by T.Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2. §

Using the connection between closed solution curves of the vortex filament flow and quasiperiodic solutions of the nonlinear Schrödinger equation (NLS), we relate the knot types of finite-gap solutions to the Floquet spectra of the corresponding NLS potentials, in the special case of small amplitude curves close to multiply-covered circles.