arXiv subjects

Rahmi Rusin

Publications and source records attributed to Rahmi Rusin.

5 recordsLinked to original sources

arXiv2026

Dynamics and stabilization of topological edge solitons in driven-damped nonlinear SSH lattices

We study topological edge solitons in a nonlinear Su--Schrieffer--Heeger (SSH) lattice subject to parametric driving and linear damping. Starting from a vertically driven pendulum chain, we derive an effective driven--damped nonlinear SSH model and investigate its stationary edge-localized states. Analytical calculations reveal the existence of two phase-locked dissipative edge-soliton families that emerge from the nonlinear continuation of the topological edge mode. Using numerical continuation and spectral stability analysis, we construct the corresponding nonlinear branches and determine their stability properties. We show that parametric driving and damping fundamentally modify the conservative edge-state family by generating two dissipative branches with markedly different stability characteristics: one branch remains predominantly unstable, whereas the other develops substantially larger stability regions and significantly weaker instability growth rates. Direct numerical simulations further demonstrate that the robust branch can remain strongly localized over long time intervals even when weakly unstable. Simulations of the full driven--damped Klein--Gordon pendulum chain confirm the persistence of the edge-localized dynamics predicted by the reduced model. These results identify parametric driving and damping as an effective mechanism for enhancing the robustness and persistence of nonlinear topological localization in active lattice systems.

nlin.PS↗

arXiv2025

Spontaneous symmetry-breaking in the nonlinear Schr\"odinger equation on star graphs with inhomogeneities

We investigate the nonlinear Schr\"odinger equation on a three-edge star graph, where each edge contains a linear localized inhomogeneity in the form of a Dirac delta linear potential. Such systems are of significant interest in studying wave propagation in networked structures, with applications in, e.g., Josephson junctions. By reducing the system to a set of finite-dimensional coupled ordinary differential equations, we derive explicit conditions for the occurrence of a symmetry-breaking bifurcation in a symmetric family of solutions. This bifurcation is shown to be of the transcritical type, and we provide a precise estimate of the bifurcation point as the propagation constant, which is directly related to the solution norm, is varied. In addition to the symmetric states, we explore non-positive definite states that bifurcate from the linear solutions of the system. These states exhibit distinct characteristics and are crucial in understanding solutions of the nonlinear system. Furthermore, we analyze the typical dynamics of unstable solutions, showing their behavior and evolution over time. Our results contribute to a deeper understanding of symmetry-breaking phenomena in nonlinear systems on metric graphs and provide insights into the stability and dynamics of such solutions.

nlin.PS↗

arXiv2019

Symmetry breaking bifurcations in the NLS equation with an asymmetric delta potential

We consider the NLS equation with a linear double well potential. Symmetry breaking, i.e., the localisation of an order parameter in one of the potential wells that can occur when the system is symmetric, has been studied extensively. However, when the wells are asymmetric, only a few analytical works have been reported. Using double Dirac delta potentials, we study rigorously the effect of such asymmetry on the bifurcation type. We show that the standard pitchfork bifurcation becomes broken and instead a saddle-centre type is obtained. Using a geometrical approach, we also establish the instability of the corresponding solutions along each branch in the bifurcation diagram

math.DS↗

arXiv2019

Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation

We study the integrable nonlocal nonlinear Schrödinger equation proposed by Ablowitz and Musslimani, that is considered as a particular example of equations with parity-time ($\mathcal{PT}$) symmetric self-induced potential. We consider dynamics (including collisions) of moving solitons. Analytically we develop a collective coordinate approach based on variational methods and examine its applicability in the system. We show numerically that a single moving soliton can pass the origin and decay or be trapped at the origin and blows up at a finite time. Using a standard soliton ansatz, the variational approximation can capture the dynamics well, including the finite-time blow up, even though the ansatz is relatively far from the actual blowing-up soliton solution. In the case of two solitons moving towards each other, we show that there can be a mass transfer between them, in addition to wave scattering. We also demonstrate that defocusing nonlinearity can support bright solitons.

nlin.PS↗

arXiv2018

Variational approximations using Gaussian ansatz, false instability, and its remedy in nonlinear Schrödinger lattices

We study the fundamental lattice solitons of the discrete nonlinear Schrödinger (DNLS) equation and their stability via a variational method. Using a Gaussian ansatz and comparing the results with numerical computations, we report a novel observation of false instabilities. Comparing with established results and using Vakhitov-Kolokolov criterion, we deduce that the instabilities are due to the ansatz. In the context of using the same type of ansatzs, we provide a remedy by employing multiple Gaussian functions. The results show that the higher the number of Gaussian function used, the better the solution approximation.

nlin.PS↗