arXiv subjects

Abba Ramadan

Publications and source records attributed to Abba Ramadan.

6 recordsLinked to original sources

arXiv2026

Minimizers for the Cahn-Hilliard energy functional with the Flory-Huggins potential under strong anchoring conditions

In this paper, we theoretically and numerically study the minimizers for the Cahn-Hilliard energy with the Flory-Huggins potential under the strong anchoring condition, i.e., the Dirichlet boundary condition. We reveal bifurcation phenomena mediated by the boundary condition, the transition layer thickness, and the temperature of the system. Numerical simulations are also presented to approximate the minimizers of this energy by solving a gradient-flow equation, namely the Allen-Cahn equation constrained with strong anchoring conditions and random initial data. The effects of varying the transition layer thickness and temperature are presented to confirm the theoretical analysis.

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arXiv2026

Curve Lengthening Bifurcations in Modally Filtered Nonlinear Schr\"odinger Systems

Extensions of the parametric nonlinear Schr\"odinger equations (PNLS) for phase-sensitive optical resonance are developed that preserve the curve lengthening bifurcation seen in the original system. This bifurcation occurs in sharp interface reductions when the motion of the interface transitions from curvature-driven flow (curve shortening) to motion against curvature regularized by higher order Willmore effects (curve lengthening). We construct a specific class of down-phase self-interaction operators via a spectral transform of the down-up operator. While in the bifurcation regime, the corresponding modally filtered nonlinear Schr\"odinger systems preserve the linear stability of the front, admit the sign flip in the linear term in the normal velocity while preserving the proper sign of the Willmore terms.

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arXiv2023

Curvature and Chaos in the Defocusing Parameteric Nonlinear Schrodinger System

The parametric nonlinear Schrodinger equation models a variety of parametrically forced and damped dispersive waves. For the defocusing regime, we derive a normal velocity for the evolution of curved dark-soliton fronts that represent a $π$-phase shift across a thin interface. We establish that depending upon the strength of parametric term the normal velocity evolution can transition from a curvature driven flow to motion against curvature regularized by surface diffusion of curvature. In the former case interfacial length shrinks, while in the later the interface length generically grows until self-intersection followed by a transition to chaotic motion.

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arXiv2023

On the stability of solitary waves in the NLS system of the third-harmonic generation

We consider the NLS system of the third-harmonic generation, which was introduced by Sammut. Our interest is in solitary wave solutions and their stability properties. The recent work of Oliveira and Pastor, discussed global well-posedness vs. finite time blow-up, as well as other aspects of the dynamics. These authors have also constructed solitary wave solutions, via the method of mountain pass/Nehari manifold, in an appropriate range of parameters. Specifically, the waves exist only in spatial dimensions $n=1,2,3$. They have also established some stability/instability results for these waves. In this work, we systematically build and study solitary waves for this important model. We construct the waves in the largest possible parameter space, and we provide a complete classification of their stability. In dimension one, we show stability, whereas, in $n=2,3$, they are generally spectrally unstable, except for a small region, where they do enjoy an extra pseudo-conformal symmetry. Finally, we discuss instability by a blow-up. In the case $n=3$, and for a more restrictive set of parameters, we use virial identities methods to derive the strong instability, in the spirit of Ohta's approach. In $n=2$, the virial identities reduce matters, via conservation of mass and energy, to the initial data. Our conclusions mirror closely the well-known results for the scalar cubic focusing NLS, while the proofs are much more involved.

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arXiv2021

On the stability of the compacton waves for the degenerate KdV and NLS models

In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-deVries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of $p$. Namely, we show that all waves are spectrally stable for $2 8$. This extends previous work of Germain, Harrop-Griffits and Marzuola, who have established orbital stability for some specific waves, in the range $p<8$.

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arXiv2020

On the standing waves of the Schroedinger equation with concentrated nonlinearity

We study the concentrated NLS on ${\mathbf R^n}$, with power non-linearities, driven by the fractional Laplacian, $(-Δ)^s, s>\frac{n}{2}$. We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space $H^s$. Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable. Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobolev embedding $H^s({\mathbf R^n})\hookrightarrow L^\infty({\mathbf R^n})$, which provides an alternative calculation and justification of the sharp constants in these inequalities.

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