We characterize the nonlinear stage of modulational instability (MI) by studying the longtime asymptotics of the focusing nonlinear Schrödinger (NLS) equation on the infinite line with initial ...conditions tending to constant values at infinity. Asymptotically in time, the spatial domain divides into three regions: a far left and a far right field, in which the solution is approximately equal to its initial value, and a central region in which the solution has oscillatory behavior described by slow modulations of the periodic traveling wave solutions of the focusing NLS equation. These results demonstrate that the asymptotic stage of MI is universal since the behavior of a large class of perturbations characterized by a continuous spectrum is described by the same asymptotic state.
The long-time asymptotic behavior of solutions to the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of ...initial conditions that allow for the presence of discrete spectrum. The results of the analysis provide the first rigorous characterization of the nonlinear interactions between solitons and the coherent oscillating structures produced by localized perturbations in a modulationally unstable medium. The study makes crucial use of the inverse scattering transform for the focusing NLS equation with nonzero boundary conditions, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems. Previously, it was shown that in the absence of discrete spectrum the
xt
-plane decomposes asymptotically in time into two types of regions: a left far-field region and a right far-field region, where to leading order the solution equals the condition at infinity up to a phase shift, and a central region where the asymptotic behavior is described by slowly modulated periodic oscillations. Here, it is shown that in the presence of a conjugate pair of discrete eigenvalues in the spectrum a similar coherent oscillatory structure emerges but, in addition, three different interaction outcomes can arise depending on the precise location of the eigenvalues: (i) soliton transmission, (ii) soliton trapping, and (iii) a mixed regime in which the soliton transmission or trapping is accompanied by the formation of an additional, nondispersive localized structure akin to a soliton-generated wake. The soliton-induced position and phase shifts of the oscillatory structure are computed, and the analytical results are validated by a set of accurate numerical simulations.
A polynomial-in-time growth bound is established for global Sobolev
H
s
(
T
)
solutions to the derivative nonlinear Schrödinger equation on the circle with
s
>
1
. These bounds are derived as a ...consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution.
Peakon traveling wave solutions, both on the line and on the circle, are derived for a novel ab-family of nonlocal evolution equations with cubic nonlinearities. At least two members of this ...ab-family, namely the Fokas-Olver-Rosenau-Qiao equation and the Novikov equation, are known to be integrable. Furthermore, a generalization of the ab-family with nonlinearities of order k\in \mathbb{N}, k\geqslant 2, is considered and its multi-peakon on the line is obtained.
THE NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE FOKAS, ATHANASSIOS S.; HIMONAS, A. ALEXANDROU; MANTZAVINOS, DIONYSSIOS
Transactions of the American Mathematical Society,
01/2017, Volume:
369, Issue:
1
Journal Article
Peer reviewed
The initial-boundary value problem (ibvp) for the cubic nonlinear Schrödinger (NLS) equation on the half-line with data in Sobolev spaces is analysed via the formula obtained through the unified ...transform method and a contraction mapping approach. First, the linear Schrödinger (LS) ibvp with initial and boundary data in Sobolev spaces is solved and the basic space and time estimates of the solution are derived. Then, the forced LS ibvp is solved for data in Sobolev spaces on the half-line 0, ∞) for the spatial variable and on an interval 0, 𝑇, 0 < 𝑇 < ∞, for the temporal variable by decomposing it into a free ibvp and a forced ibvp with zero data, and its solution is estimated appropriately. Furthermore, using these estimates, well-posedness of the NLS ibvp with data (𝑢(𝑥, 0), 𝑢(0, 𝑡)) in
H
x
s
(
0
,
∞
)
×
H
t
(
2
s
+
1
)
/
4
(
0
,
T
)
, 𝑠 > 1/2, is established via a contraction mapping argument. In addition, this work places Fokas’ unified transform method for evolution equations into the broader Sobolev spaces framework.
The initial-boundary value problem for the nonlinear Schrödinger equation on the half-line with initial data in Sobolev spaces
H
s
(
0
,
∞
)
,
1
/
2
<
s
⩽
5
/
2
,
s
≠
3
/
2
, and Robin boundary data ...of appropriate regularity is shown to be locally well-posed in the sense of Hadamard. The proof is through a contraction mapping argument and hence relies crucially on certain estimates for the forced linear counterpart of the nonlinear problem. In particular, the essence of the analysis lies in the pure linear initial-boundary value problem, which corresponds to the case of zero forcing, zero initial data, and nonzero boundary data. This problem, which is studied by taking advantage of the solution formula derived via the unified transform of Fokas, holds an instrumental role in the overall analysis as it reveals the correct function space for the Robin boundary data.
Considered here is a class of Boussinesq systems of Nwogu type. Such systems describe propagation of nonlinear and dispersive water waves of significant interest such as solitary and tsunami waves. ...The initial-boundary value problem on a finite interval for this family of systems is studied both theoretically and numerically. First, the linearization of a certain generalized Nwogu system is solved analytically via the unified transform of Fokas. The corresponding analysis reveals two types of admissible boundary conditions, thereby suggesting appropriate boundary conditions for the nonlinear Nwogu system on a finite interval. Then, well-posedness is established, both in the weak and in the classical sense, for a regularized Nwogu system in the context of an initial-boundary value problem that describes the dynamics of water waves in a basin with wall-boundary conditions. In addition, a new modified Galerkin method is suggested for the numerical discretization of this regularized system in time, and its convergence is proved along with optimal error estimates. Finally, numerical experiments illustrating the effect of the boundary conditions on the reflection of solitary waves by a vertical wall are also provided.
On considère ici une classe de systèmes Boussinesq de type Nwogu. De tels systèmes décrivent la propagation d'ondes d'eau non linéaires et dispersives d'intérêt significatif telles que les ondes solitaires et tsunami. Le problème des valeurs aux limites initiales sur un intervalle fini pour cette famille de systèmes est étudié à la fois théoriquement et numériquement. Tout d'abord, la liéarisation d'un certain système de Nwogu généralisé est résolue analytiquement via la transformée unifiée de Fokas. L'analyse correspondante révèle deux types de conditions aux limites admissibles, suggérant ainsi des conditions aux limites appropriées pour le système de Nwogu non linéaire sur un intervalle fini. Ensuite, le bien-posé est établi, à la fois au sens faible et au sens classique, pour un système de Nwogu régularisé dans le contexte d'un problème de valeur aux limites initiales qui décrit la dynamique des vagues d'eau dans un bassin avec des conditions aux limites des parois. De plus, une nouvelle méthode de Galerkin modifiée est suggérée pour la discrétisation numérique de ce système régularisé dans le temps, et sa convergence est prouvée avec des estimations d'erreur optimales. Enfin, des expériences numériques illustrant l'effet des conditions aux limites sur la réflexion d'ondes solitaires par une paroi verticale sont également fournies.
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