Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1410.7542

Breathers and black rogue waves of coupled nonlinear Schrödinger equations with dispersion and nonlinearity of opposite signs

Jin Hua Li·Hiu Ning Chan·Kin Seng Chiang·Kwok Wing Chow
View original source ↗Read source document ↗Export metadata ↓

Abstract

Breathers and rogue waves of special coupled nonlinear Schrödinger systems (the Manakov equations) are studied analytically. These systems model the orthogonal polarization modes in an optical fiber with randomly varying birefringence. Unlike the situation in a waveguide with zero birefringence, rogue waves can occur in these Manakov systems with dispersion and nonlinearity of opposite signs, provided that a group velocity mismatch is present. The criterion for the existence of rogue waves correlates exactly with the onset of modulation instability. Theoretically the Hirota bilinear transform is employed and rogue waves are obtained as a long wave limit of breathers. In terms of wave profiles, a black rogue wave (intensity dropping to zero) and the transition to a four-petal configuration are identified. Sufficiently strong modulation instabilities of the background may overwhelm or mask the development of the rogue waves, and such thresholds are correlated to actual physical properties of optical fibers. Numerical simulations on the evolution of breathers are performed to verify the prediction of the analytical formulations.

Explore related subjects

nlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jin Hua Li, Hiu Ning Chan, Kin Seng Chiang, Kwok Wing Chow. 2014-10-28. Breathers and black rogue waves of coupled nonlinear Schrödinger equations with dispersion and nonlinearity of opposite signs. https://arxiv.org/abs/1410.7542

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↗