Journal ArticleDOI
Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. I
Harvey Segur,Mark J. Ablowitz +1 more
TLDR
The dominant asymptotic behavior of the solution of the nonlinear Schrodinger equation when there is one soliton and decaying oscillations has been shown in this paper, where the method of solution uses the conservation laws, rather than the integral equations.Abstract:
We find the dominant asymptotic behavior of the solution of the nonlinear Schrodinger equation when there is one soliton and decaying oscillations. The solution behaves like the soliton near the soliton, and like the solution found in the preceding paper (I) elsewhere. The method of solution uses the conservation laws, rather than the integral equations.read more
Citations
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Journal ArticleDOI
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations
TL;DR: In this paper, the authors study defocusing analogues of these equations, namely defocusing nonlinear Schrodinger, defocusing modified Korteweg-de Vries (mKdV), and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable.
Journal ArticleDOI
Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation
TL;DR: Inverse spectral theory is used to derive an invariant representation of the modulational equations for the slow modulations of N-phase wave trains for the Korteweg-de Vries equation.
Journal ArticleDOI
On the evolution of packets of water waves
Mark J. Ablowitz,Harvey Segur +1 more
TL;DR: In this paper, the authors consider the evolution of water waves that travel predominantly in one direction, but in which the wave amplitudes are modulated slowly in both horizontal directions, and they find that the two-dimensional evolution of the wave packets depends fundamentally on the dimensionless surface tension and fluid depth.
Journal ArticleDOI
Blow-up in Nonlinear Schroedinger Equations-I A General Review
TL;DR: The general properties of nonlinear Schroedinger equations: iut + p:∇∇u + f(|u|2)u = 0 are reviewed in this paper.
Journal ArticleDOI
Asymptotic Solutions of the Korteweg-deVries Equation
Mark J. Ablowitz,Harvey Segur +1 more
TL;DR: The long-time asymptotic solution of the Korteweg-deVries equation, corresponding to initial data which decay rapidly as |x|∞ and produce no solitons, is found to be considerably more complicated than previously reported.
References
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BookDOI
Dynamical systems, theory and applications
TL;DR: In this article, the authors studied the integrability of nonlinear wave equations with constant coefficients on the torus, and the existence of heteroclinic orbits, with applications to the hopf bifurcation problem.
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