Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2507.17837

Parametric Resonance in the Non-Autonomous Sine-Gordon Model

Tomasz Dobrowolski·Jacek Gatlik·Zofia Bryłowska·Panayotis G. Kevrekidis
View original source ↗Read source document ↗Export metadata ↓

Abstract

The sine-Gordon model is studied with model parameters that depend on both space and time. An effective model with one degree of freedom is constructed, allowing the description of the kink movement in both a temporally non-autonomous and spatially inhomogeneous setting. As a highly demanding test of the reduced model, the case of a temporal drive leading to a parametric instability is considered. Here, the boundaries of the instability region in the form of Arnold tongues were examined in both the simplified effective model and the full field model. Good agreement was obtained as concerns the regions of stability. It was observed that only in the bottom parts of Arnold's tongues is the dynamics of the field model more complicated than in the approximate model.

Explore related subjects

nlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tomasz Dobrowolski, Jacek Gatlik, Zofia Bryłowska, Panayotis G. Kevrekidis. 2025-07-23. Parametric Resonance in the Non-Autonomous Sine-Gordon Model. https://doi.org/10.1063/5.0296470

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rolls and Snaking in a Swift-Hohenberg Equation with Non-smooth Nonlinearity

We study rolls and homoclinic snaking in a variation of the one-dimensional Swift-Hohenberg equation, whose standard forms are prototypical order-parameter models for pattern formation in the sciences. Motivated by classes of differential equation models that involve continuous non-smooth low order nonlinear terms, we replace the standard quadratic-cubic nonlinearity by $ν|u|^α-u^3$, $α\in [1,2]$ with $ν> 0$. In the vicinity of zero, for $α<2$ this nonlinearity falls outside the scope of classical Taylor expansion and bifurcation analysis. Our partially analytical and partially numerical results highlight that the non-smooth term modifies the criticality of pattern-forming bifurcations and alters the associated branches of solutions. In particular, $α\in(1,2)$ implies subcriticality of roll bifurcations for any $ν>0$. At $α=1$ differentiability is lost, which has a strong impact on the bifurcations of sign-changing rolls including the disappearance of homoclinic snaking. Homoclinic snaking thus emerges non-smoothly as $α$ increases from $α=1$, and persists when retaining an additional, e.g., quadratic term.

nlin.PS↗
arXiv2026

Vegetation Pattern Formation with an Energy-Mismatch Variational Closure

We study a vegetation-water model motivated by a canopy energy mismatch. Vegetation follows the gradient of a score that rewards biomass and penalizes the squared mismatch, while water obeys a quasi-steady balance. For a fixed interaction kernel, the linear growth rate about positive uniform vegetation splits into a fixed-water term and a water-feedback term. The fixed-water term is real, even in wavenumber, and equal to a constant minus a squared modulus, including when the kernel is asymmetric. All linear phase propagation enters through water feedback. A second-order expansion of the kernel gives a fourth-order vegetation equation, for which we derive finite-wavenumber growth criteria and corrections from state-dependent spatial coefficients. The short-wave damping and fastest-growing scales of this truncated model require separate justification as approximations to a specified kernel. A numerical dispersion example illustrates the growth and phase velocity of its linear modes. The model creates biomass on bare ground throughout the rainfall range illustrated here, including zero rainfall. At some smooth nonnegative states, the vegetation growth rate is negative where biomass vanishes. These features limit its ecological interpretation.

nlin.PS↗
arXiv2026

Two-Parameter Family of Nonlinear Dirac Equations With Scalar-Scalar plus Vector-Vector Interactions

We obtain exact solutions of the nonlinear Dirac equation in $1+1$ dimensions of the form $ψ(x,t) = e^{-iωt} ψ(x)$ for scalar-scalar (SS) plus vector-vector (VV) interaction with interaction Lagrangian given by $$L_{I} = \frac{g^2}{κ+1}[(\barψ ψ)^{κ+1} +\frac{1}{p} (\barψ γ_μψ\barψ γ^μ ψ)^{κ+1}]$$ where $p > 0$ but arbitrary otherwise. We look for solutions with $0 < ω< m$ where $ω, m$ are frequency and mass, respectively. We find solutions for all values of $ω$ in this range. We compute the charge $Q$ and the energy $E$ for each of the solitary wave solutions and explore the region in the ($p, κ$) parameter space in which solitary wave bound states exist (i.e., for which $E/Q < m$). We show that for all the cases while both $E$ and $Q$ depend on the coupling constant $g$, their ratio $E/Q$ is independent of $g$. We further find that in case $pκ\le 1$, the charge density for all the solitary waves have only single hump while for $p κ> 1$ there is a transition from double to single hump and we determine it as a function of $ω/m$. We notice that for all $p$ there is a transition at $κ=2$ in the behavior of $E/Q$ as a function of $ω$ which we speculate is related to the onset of instability of the solutions at $κ=2$. We obtain the nonrelativistic reduction of the two-parameter family to a non-relativistic modified nonlinear Schrödinger equation (NLSE) and discuss the stability of the single hump olitary waves in the domain of validity of the modified NLSE.

nlin.PS↗