arXiv subjects

Vassilios M Rothos

Publications and source records attributed to Vassilios M Rothos.

4 recordsLinked to original sources

arXiv2026

Separatrix Splitting and Chaotic Dynamics in Collective-Coordinate Reductions of Driven $ϕ^4$ Kinks

We investigate the emergence of chaotic dynamics in collective-coordinate reductions of a driven and spatially modulated $ϕ^4$ field describing the motion of topological kinks. Focusing on finite-dimensional effective models, we consider both translation-only and constraint-consistent two--collective--coordinate reductions in the presence of spatial pinning, dissipation, and traveling-wave forcing. Using Melnikov theory, we obtain an explicit analytical characterization of separatrix splitting and derive closed-form criteria for the onset of chaotic dynamics in the reduced phase space. In the two--collective--coordinate framework the Melnikov analysis is formulated in an extended phase space, allowing the distinct roles of translational motion and internal-mode excitation to be identified. Numerical simulations of the reduced systems, including stroboscopic Poincaré sections and Lyapunov exponent computations, confirm the analytical predictions and reveal chaotic layers organized around the unperturbed separatrix.

nlin.PS↗

arXiv2026

Spectral stability and slow--fast structure of traveling waves in a regularized sine--Gordon equation

We investigate the dynamics and spectral stability of traveling kink and antikink solutions in a dissipative sine--Gordon equation with two distinct fourth--order regularization mechanisms: a mixed space--time (inertial) term and a purely spatial (elliptic) term. The model includes damping, bias forcing, and higher--order dissipative effects, and is motivated by refined descriptions of fluxon dynamics in long Josephson junctions. Using a collective--coordinate reduction, we derive a Melnikov--type condition for speed selection, yielding explicit predictions for asymptotic propagation speeds, which are validated by direct numerical simulations of the full partial differential equation. Spectral stability is analyzed using Evans function techniques adapted to the singular slow--fast structure induced by the regularization. By formulating the linearized problem on a consistent--splitting domain, we show that no additional point spectrum bifurcates from the origin. Near the edges of the essential spectrum, a square--root transformation is used to resolve branch singularities and establish analyticity in a lifted spectral variable. Numerical Evans function computations near $λ=0$ and near the essential spectrum edges confirm the analytical results, indicating absence of unstable eigenvalues for both kink and antikink solutions.

nlin.PS↗

arXiv2026

Ground States and Periodic--to--Localized Convergence in Two--Dimensional Saturable Discrete Nonlinear Schrödinger Equations

We study a two--dimensional discrete nonlinear Schrödinger equation with saturable nonlinearity on the lattice $\mathbb Z^2$. Using a variational approach based on the Nehari manifold, we establish the existence of nontrivial periodic ground states on finite lattices and establish the existence of exponentially localized ground states in $\ell^2(\mathbb Z^2)$. A principal result is the rigorous passage from periodic to localized states: we show that, up to lattice translations, periodic ground states converge strongly in $\ell^2(\mathbb Z^2)$ to a localized ground state as the lattice periods tend to infinity. The analysis combines variational methods, spectral properties of the discrete Laplacian, and concentration--compactness techniques adapted to the two--dimensional discrete setting. We further derive qualitative properties of the resulting solutions, including positivity and exponential localization, and establish a conditional orbital stability result within the Grillakis--Shatah--Strauss framework. Numerical computations illustrate the theoretical results and confirm the predicted convergence and localization behavior.

math.AP↗

arXiv2026

Adiabatic perturbation theory for the $F=1$ spinor nonlinear Schrödinger equation with nonvanishing boundary conditions

We develop a systematic adiabatic perturbation theory for the integrable $F=1$ spinor nonlinear Schrödinger equation under nonvanishing boundary conditions, formulated entirely within the framework of the associated Riemann--Hilbert problem. In this setting, localized nonlinear excitations are characterized by discrete spectral data consisting of a complex eigenvalue and an associated polarization vector. For a general class of small perturbations preserving the background, we derive the perturbation-induced evolution of the scattering data directly at the level of the Riemann--Hilbert problem. In the one-soliton sector, this yields a closed finite-dimensional dynamical system governing the slow evolution of the effective soliton parameters, including the spectral variables, the soliton center and phase, the residue amplitude, and the internal polarization state. The latter evolves according to a constrained dynamical equation with no scalar analogue. For localized perturbations, the modulation equations are expressed in explicit integral form in terms of the one-soliton eigenfunctions, providing a fully computable description of the dynamics. In the limit of vanishing boundary conditions, the resulting system reduces to the perturbation theory obtained by E. V. Doktorov, et al, Phys. Rev. A 77 (2008), no. 4, 043617.

nlin.SI↗