Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1408.2358

Interaction of sine-Gordon kinks and breathers with a $\mathcal{PT}$-symmetric defect

Danial Saadatmand·Sergey V. Dmitriev·Denis I. Borisov·P. G. Kevrekidis
View original source ↗Read source document ↗Export metadata ↓

Abstract

The scattering of kinks and low-frequency breathers of the nonlinear sine-Gordon (SG) equation on a spatially localized $\mathcal{PT}$-symmetric perturbation (defect) with a balanced gain and loss is investigated numerically. It is demonstrated that if a kink passes the defect, it always restores its initial momentum and energy and the only effect of the interaction with the defect is a phase shift of the kink. A kink approaching the defect from the gain side always passes, while in the opposite case it must have sufficiently large initial momentum to pass through the defect instead of being trapped in the loss region. The kink phase shift and critical velocity are calculated with the use of the collective variable method. Kink-kink (kink-antikink) collisions at the defect are also briefly considered, showing how their pairwise repulsive (respectively, attractive) interaction can modify the collisional outcome of a single kink within the pair with the defect. For the breather, the result of its interaction with the defect strongly depends on the breather parameters (velocity, frequency and initial phase) and on the defect parameters. The breather can gain some energy from the defect and as a result potentially even split into a kink-antikink pair or it can lose a part of its energy. Interestingly, the breather translational mode is very weakly affected by the dissipative perturbation, so that a breather penetrates more easily through the defect when it comes from the lossy side, than a kink. In all studied soliton-defect interactions the energy loss to radiation of small-amplitude extended waves is negligible.

Explore related subjects

nlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Danial Saadatmand, Sergey V. Dmitriev, Denis I. Borisov, P. G. Kevrekidis. 2014-08-11. Interaction of sine-Gordon kinks and breathers with a $\mathcal{PT}$-symmetric defect. https://doi.org/10.1103/physreve.90.052902

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↗