107 research outputs found
On the asymptotic stability of N-soliton solutions of the defocusing nonlinear Schr\uf6dinger equation
We consider the Cauchy problem for the defocusing nonlinear Schr\uf6dinger
(NLS) equation for finite density type initial data. Using the dbar generalization of the nonlinear
steepest descent method of Deift and Zhou, we derive the leading order approximation
to the solution of NLS for large times in the solitonic region of space\u2013time,
|x| < 2t, and we provide bounds for the error which decay as t \u2192 1efor a general
class of initial data whose difference from the non vanishing background possesses a
fixed number of finite moments and derivatives. Using properties of the scattering map
of NLS we derive, as a corollary, an asymptotic stability result for initial data that are
sufficiently close to the N-dark soliton solutions of NLS
On critical behaviour in systems of Hamiltonian partial differential equations
We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture
Review on the Stability of the Peregrine and Related Breathers
In this note, we review stability properties in energy spaces of three important nonlinear Schrödinger breathers: Peregrine, Kuznetsov-Ma, and Akhmediev. More precisely, we show that these breathers are unstable according to a standard definition of stability. Suitable Lyapunov functionals are described, as well as their underlying spectral properties. As an immediate consequence of the first variation of these functionals, we also present the corresponding nonlinear ODEs fulfilled by these nonlinear Schrödinger breathers. The notion of global stability for each breather mentioned above is finally discussed. Some open questions are also briefly mentioned
Long-time asymptotic analysis for defocusing Ablowitz-Ladik system with initial value in lower regularity
Recently, we have given the bijectivity for defocusing Ablowitz-Ladik
systems in the discrete Sobolev space by inverse spectral method.
Based on these results, the goal of this article is to investigate the
long-time asymptotic property for the initial-valued problem of the defocusing
Ablowitz-Ladik system with initial potential in lower regularity. The main idea
is to perform proper deformations and analysis to the corespondent
Riemann-Hilbert problem with the unit circle as the jump contour . As a
result, we show that when , the solution admits
Zakharov-Manakov type formula, and when , the solution
decays fast to zero
On a fourth order nonlinear Helmholtz equation
In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz
equation in for positive, bounded and -periodic functions . Using
the dual method of Evequoz and Weth, we find solutions to this equation and
establish some of their qualitative properties
On Maxwell-Bloch systems with inhomogeneous broadening and one-sided nonzero background
The inverse scattering transform is developed to solve the Maxwell-Bloch
system of equations that describes two-level systems with inhomogeneous
broadening, in the case of optical pulses that do not vanish at infinity in the
future. The direct problem, which is formulated in terms of a suitably-defined
uniformization variable, combines features of the formalism with decaying as
well as non-decaying fields. The inverse problem is formulated in terms of a
matrix Riemann-Hilbert problem. A novel aspect of the problem is
that no reflectionless solutions can exist, and solitons are always accompanied
by radiation. At the same time, it is also shown that, when the medium is
initially in the ground state, the radiative components of the solutions decay
upon propagation into the medium, giving rise to an asymptotically
reflectionless states. Like what happens when the optical pulse decays rapidly
in the distant past and the distant future, a medium that is initially excited
decays to the stable ground state as and for sufficiently large
propagation distances. Finally, the asymptotic state of the medium and certain
features of the optical pulse inside the medium are considered, and the
emergence of a transition region upon propagation in the medium is briefly
discussed.Comment: submitted to Comm. Math. Phy
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