# Nonlinear Sciences - Pattern Formation and Solitons Publications (50)

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## Nonlinear Sciences - Pattern Formation and Solitons Publications

This paper explores the classification of parameter spaces for reaction-diffusion systems of two chemical species on stationary domains. The dynamics of the system are explored both in the absence and presence of diffusion. The parameter space is fully classified in terms of the types and stability of the uniform steady state. Read More

Assume a lower-dimensional solitonic structure embedded in a higher dimensional space, e.g., a 1D dark soliton embedded in 2D space, a ring dark soliton in 2D space, a spherical shell soliton in 3D space etc. Read More

Directed information transmission is a paramount requirement for many social, physical, and biological systems. For neural systems, scientists have studied this problem under the paradigm of feedforward networks for decades. In most models of feedforward networks, activity is exclusively driven by excitatory neurons and the wiring patterns between them while inhibitory neurons play only a stabilizing role for the network dynamics. Read More

In this work, we provide two complementary perspectives for the (spectral) stability of traveling waves in Hamiltonian nonlinear dynamical lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical examples. One is as an eigenvalue problem for a stationary solution in a co-traveling frame, while the other is as a periodic orbit modulo shifts. We connect the eigenvalues of the former with the Floquet multipliers of the latter and based on this formulation derive an energy-based spectral stability criterion. Read More

A variable-coefficient forced Korteweg-de Vries equation with spacial inhomogeneity is investigated in this paper. Under constraints, this equation is transformed into its bilinear form, and multi-soliton solutions are derived. Effects of spacial inhomogeneity for soliton velocity, width and background are discussed. Read More

We consider longitudinal nonlinear atomic vibrations in uniformly strained carbon chains with the cumulene structure ($=C=C=)_{n}$. With the aid of ab initio simulations, based on the density functional theory, we have revealed the phenomenon of the $\pi$-mode softening in a certain range of its amplitude for the strain above the critical value $\eta_{c}\approx 11\,{\%}$. Condensation of this soft mode induces the structural transformation of the carbon chain with doubling of its unit cell. Read More

We consider the dynamics of message passing for spatially coupled codes and, in particular, the set of density evolution equations that tracks the profile of decoding errors along the spatial direction of coupling. It is known that, for suitable boundary conditions and after a transient phase, the error profile exhibits a "solitonic behavior". Namely, a uniquely-shaped wavelike solution develops, that propagates with constant velocity. Read More

The examples of the classical liquids confined by rotating helical boundaries are considered and these examples are compared with rotating helical reservoir filled by ultracold bosonic ensemble. From the point of view of observer who co-rotates with classical liquid trapped by reservoir the quantum fluid will move translationally alongside rotation axis while in laboratory frame the quantum fluid will stay in rest. This behavior of quantum ensemble which is exactly opposite to the classical case might be interpreted as a helical analog of Hess-Fairbank effect. Read More

We consider the statics and dynamics of a stable, mobile three-dimensional (3D) spatiotemporal light bullet in a cubic-quintic nonlinear medium with a focusing cubic nonlinearity above a critical value and any defocusing quintic nonlinearity. The 3D light bullet can propagate with a constant velocity in any direction. Stability of the light bullet under a small perturbation is established numerically. Read More

In this paper, we study transverse linear stability of line solitary waves to the $2$-dimensional Benney-Luke equation which arises in the study of small amplitude long water waves in $3$D. In the case where the surface tension is weak or negligible, we find a curve of resonant continuous eigenvalues near $0$. Time evolution of these resonant continuous eigenmodes is described by a $1$D damped wave equation in the transverse variable and it gives a linear approximation of the local phase shifts of modulating line solitary waves. Read More

We investigate modulational instability (MI) in asymmetric dual-core nonlinear directional couplers incorporating effects of differences in effective mode areas and group velocity dispersions, as well as phase- and group- velocity mismatches. Using coupled-mode equations for this system, we identify MI conditions from the linearization with respect to small perturbations. First, we compare the MI spectra of the asymmetric system and its symmetric counterpart in the case of the anomalous group-velocity dispersion (GVD). Read More

We explore the dynamics of strongly localized periodic solutions (discrete solitons, or discrete breathers) in a finite one-dimensional chain of asymmetric vibro-impact oscillators. The model involves a parabolic on-site potential with asymmetric rigid constraints (the displacement domain of each particle is finite), and a linear nearest-neighbor coupling. When the particle approaches the constraint, it undergoes an impact (not necessarily elastic), that satisfies Newton impact law. Read More

We demonstrate a way to generate a two-dimensional rogue waves in two types of broad area nonlinear optical systems subject to time-delayed feedback: in the generic Lugiato-Lefever model and in model of a broad-area surface-emitting laser with saturable absorber. The delayed feedback is found to induce a spontaneous formation of rogue waves. In the absence of delayed feedback, spatial pulses are stationary. Read More

In this work, we study optical self-focusing that leads to collapse events for the time-independent model of co-propagating beams with different wavelengths. We show that collapse events depend on the combined critical power of two beams for both fundamental, vortex and mixed configurations as well as on the ratio of their individual powers. Read More

We investigate the quantum Zeno dynamics of the rogue waves that are encountered in optics and quantum mechanics. Considering their usage in modeling rogue wave dynamics, we analyze the quantum Zeno dynamics of the Akhmediev breathers, Peregrine and Akhmediev-Peregrine soliton solutions of the nonlinear Schrodinger equation. We show that frequent measurements of the wave inhibits its movement in the observation domain for each of these solutions. Read More

It is shown that any two cellular automata (CA) in rule space can be connected by a continuous path parameterized by a real number $\kappa \in (0, \infty)$, each point in the path corresponding to a coupled map lattice (CML). In the limits $\kappa \to 0$ and $\kappa \to \infty$ the CML becomes each of the two CA entering in the connection. A mean-field, reduced model is obtained from the connection and allows to gain insight in those parameter regimes at intermediate $\kappa$ where the dynamics is approximately homogeneous within each neighborhood. Read More

It is known that multidimensional complex potentials obeying $\mathcal{PT}$-symmetry may possess all real spectra and continuous families of solitons. Recently it was shown that for multi-dimensional systems these features can persist when the parity symmetry condition is relaxed so that the potential is invariant under reflection in only a single spatial direction. We examine the existence, stability and dynamical properties of localized modes within the cubic nonlinear Schr\"odinger equation in such a scenario of partially $\mathcal{PT}$-symmetric potential. Read More

We generalize the Vicsek model to describe the collective behaviour of polar circle swimmers with local alignment interactions. While the phase transition leading to collective motion in 2D (flocking) occurs at the same interaction to noise ratio as for linear swimmers, as we show, circular motion enhances the polarization in the ordered phase (enhanced flocking) and induces secondary instabilities leading to structure formation. Slow rotations result in phase separation whereas fast rotations generate patterns which consist of phase synchronized microflocks of controllable self-limited size. Read More

Recently, numerical simulations of a stochastic model have shown that the density of vessel tips in tumor induced angiogenesis adopts a soliton-like profile [Sci. Rep. 6, 31296 (2016)]. Read More

We study a subclass of the May-Leonard stochastic model with an arbitrary even number of species, leading to the arising of two competing partnerships where individuals are indistinguishable. By carrying out a series of accurate numerical stochastic simulations, we show that alliances compete each other forming spatial domains bounded by interfaces of empty sites. We solve numerically the mean field equations associated to the stochastic model in one and two spatial dimensions. Read More

**Affiliations:**

^{1}IIFMdP, Arg.,

^{2}IFCA, Spain,

^{3}IIFMdP, Arg.,

^{4}Augsburg U., Germany

The performance of a ring of linearly coupled, monostable nonlinear oscillators is optimized towards its goal of acting as energy harvester---through piezoelectric transduction---of mesoscopic fluctuations, which are modeled as Ornstein--Uhlenbeck noises. For a single oscillator, the maximum output voltage and overall efficiency are attained for a soft piecewise-linear potential (providing a weak attractive constant force) but they are still fairly large for a harmonic potential. When several harmonic springs are linearly and bidirectionally coupled to form a ring, it is found that counter-phase coupling can largely improve the performance while in-phase coupling worsens it. Read More

The dynamics of dark bright solitons beyond the mean-field approximation is investigated. We first examine the case of a single dark-bright soliton and its oscillations within a parabolic trap. Subsequently, we move to the setting of collisions, comparing the mean-field approximation to that involving multiple orbitals in both the dark and the bright component. Read More

We investigate fundamental nonlinear dynamics of ferrofluidic Taylor-Couette flow - flow confined between two concentric independently rotating cylinders - consider small aspect ratio by solving the ferrohydrodynamical equations, carrying out systematic bifurcation analysis. Without magnetic field, we find steady flow patterns, previously observed with a simple fluid, such as those containing normal one- or two vortex cells, as well as anomalous one-cell and twin-cell flow states. However, when a symmetry-breaking transverse magnetic field is present, all flow states exhibit stimulated, finite two-fold mode. Read More

For field theories in one time and one space dimensions we propose an efficient method to compute the vacuum polarization energy of static field configurations that do not allow a decomposition into symmetric and anti--symmetric channels. The method also applies to scenarios in which the masses of the quantum fluctuations at positive and negative spatial infinity are different. As an example we compute the vacuum polarization energy of the kink soliton in the $\phi^6$ model. Read More

Many car-following models of traffic flow admit the possibility of absolute stability, a situation in which uniform traffic flow at any spacing is linearly stable. Near the threshold of absolute stability, these models can often be reduced to a modified Korteweg-deVries (mKdV) equation plus small corrections. The hyperbolic-tangent "kink" solutions of the mKdV equation are usually of particular interest, as they represent transition zones between regions of different traffic spacings. Read More

In this work we employ a recently proposed bifurcation analysis technique, the deflated continuation algorithm, to compute steady-state solitary waveforms in a one-component, two dimensional nonlinear Schr\"odinger equation with a parabolic trap and repulsive interactions. Despite the fact that this system has been studied extensively, we discover a wide variety of previously unknown branches of solutions. We analyze the stability of the newly discovered branches and discuss the bifurcations that relate them to known solutions both in the near linear (Cartesian, as well as polar) and in the highly nonlinear regimes. Read More

The absence of ionization and observation of white continuum in the initial moment of filamentation of powerful femtosecond laser pulses, propagating in silica glasses, as well as the filamentation without plasma channels observed in the experiments in air, forced us to look for other nonlinear mechanisms of description the above mentioned effects. For this reason we present in this paper new parametric conversion mechanism for asymmetric spectrum broadening of femtosecond laser pulses towards the higher frequencies in isotropic media. This mechanism includes cascade generation with THz spectral shift for solids and GHz spectral delay for gases. Read More

We discuss propagation of traveling waves in a blood filled elastic artery with an axially symmetric dilatation (an idealized aneurysm). The processes in the injured artery are modelled by equations for the motion of the wall of the artery and by equation for the motion of the fluid (the blood). For the case when long-wave approximation holds the model equations are reduced to a version of the perturbed Korteweg-deVries equation. Read More

The well-known (1+1D) nonlinear Schr\"odinger equation (NSE) governs the propagation of narrow-band pulses in optical fibers and others one-dimensional structures. For exploration the evolution of broad-band optical pulses (femtosecond and attosecond) it is necessary to use the more general nonlinear amplitude equation (GNAE) which differs from NSE with two additional non-paraxial terms. That is way, it is important to make clear the difference between the solutions of these two equations. Read More

Predicting the large-amplitude deformations of thin elastic sheets is difficult due to the complications of self-contact, geometric nonlinearities, and a multitude of low-lying energy states. We study a simple two-dimensional setting where an annular polymer sheet floating on an air-water interface is subjected to different tensions on the inner and outer rims. The sheet folds and wrinkles into many distinct morphologies that break axisymmetry. Read More

We study solitary wave propagation in 1D granular crystals with Hertz-like interaction potentials. We consider interfaces between media with different exponents in the interaction potential. For an interface with increasing interaction potential exponent along the propagation direction we obtain mainly transmission with delayed secondary transmitted and reflected pulses. Read More

**Affiliations:**

^{1}U of Auckland,

^{2}U of Auckland

Oscillons are spatially stationary, quasi-periodic solutions of nonlinear field theories seen in settings ranging from granular systems, low temperature condensates and early universe cosmology. We describe a new class of oscillon in which the spatial envelope can have "off centre" maxima and pulsate on timescales much longer than the fundamental frequency. These are exact solutions of the 1-D sine-Gordon equation and we demonstrate numerically that similar solutions exist in up to three dimensions for a range of potentials. Read More

We present a novel compact dual-comb source based on a monolithic optical crystalline MgF$_2$ multi-resonator stack. The coherent soliton combs generated in two microresonators of the stack with the repetition rate of 12.1 GHz and difference of 1. Read More

We show that media with inhomogeneous defocusing cubic nonlinearity growing toward the periphery can support a variety of stable vortex clusters nested in a common localized envelope. Nonrotating symmetric clusters are built of an even number of vortices with opposite topological charges, located at equal distances from the origin. Rotation makes the clusters strongly asymmetric, as the centrifugal force shifts some vortices to the periphery, while others approach the origin, depending on the topological charge. Read More

Traveling wave solutions to Kawahara equation (KE), transmission line (TL), and Korteweg-de Vries (KdV) equation are found by using a modified elliptic function method which is a generalization of the $\mathrm{tanh}$-method. This works by assuming that a polynomial ansatz satisfies a Weierstrass equation, and has two advantages: first, it reduces the number of terms in the ansatz by an order of two, and second, it uses Weierstrass functions which satisfy the elliptic equation for the dependent variable instead of the hyperbolic tangent functions which only satisfy the Riccati equation with constant coefficients. When the polynomial ansatz in the traveling wave variable is of first order, the equation reduces to KdV equation with only a cubic dispersion term, while for the KE equation which includes a fifth order dispersion term the polynomial ansatz must necessary be of quadratic type. Read More

Considering a two-dimensional Bose-Hubbard spinor lattice with weak nearest neighbour interactions and no particle transfer between sites, we theoretically study the transport of energy from one initially excited dimer, to the rest of the lattice. Beyond a critical interaction strength, low energy on-site excitations are quickly dispersed throughout the array, while stronger excitations are self trapped, resulting in localized energy breathers and solitons. These structures are quasiparticle analogues to the discrete 2D solitons in photonic lattices. Read More

We show that nonlinear imaging is possible in periodic waveguide configurations provided that we use two different segments of nonlinear media with opposite signs of the Kerr nonlinearity with, in general, no other restriction about their magnitudes. The second medium is used to implement effective "reverse propagation". A main ingredient in achieving nonlinear imaging is the control of the sign and the amplitude of the coupling coefficient. Read More

In the present work, we revisit two-component Bose-Einstein condensates in their fully three-dimensional (3d) form. Motivated by earlier studies of dark-bright solitons in the 1d case, we explore the stability of these structures in their fully 3d form in two variants. In one the dark soliton is planar and trapping a planar bright (disk) soliton. Read More

The investigation of dynamics of intense solitary wave groups of collinear surface waves is performed by means of numerical simulations of the Euler equations and laboratory experiments. The processes of solitary wave generation, reflection from a wall and collisions are considered. Steep solitary wave groups with characteristic steepness up to $kA_{cr} = 0. Read More

In nonlinear plasmonics, the switching threshold of optical bistability is limited by the weak nonlinear responses from the conventional Kerr dielectric media. Considering the giant nonlinear susceptibility of graphene, here we develop a nonlinear scattering model under the mean field approximation and study the bistable scattering in graphene-coated dielectric nanowires based on the semi-analytical solutions. We find that the switching intensities of bistable scattering can be smaller than 1MW/cm2 at the working frequency. Read More

We consider an optical nonlinear interferometric setup based on Young's double-slit configuration where a nonlinear material is placed exactly after one of the two slits. We examine the effects of Kerr nonlinearity and multi-photon absorption in the resulting interference pattern. The presence of nonlinearity breaks the transverse spatial symmetry of the system, resulting to a modified intensity pattern at the observation plane as a function of the incident intensity. Read More

Many biological and chemical systems could be modeled by a population of oscillators coupled indirectly via a dynamical environment. Essentially, the environment by which the individual elements communicate is heterogeneous. Nevertheless, most of previous works considered the homogeneous case only. Read More

When an interaction quench by a factor of four is applied to an attractive Bose-Einstein condensate, a higher-order quantum bright soliton exhibiting robust oscillations is predicted in the semiclassical limit by the Gross-Pitaevskii equation. Combining matrix-product state simulations of the Bose-Hubbard Hamiltonian with analytical treatment via the Lieb-Liniger model and the eigenstate thermalization hypothesis, we show these oscillations are absent. Instead, one obtains a large stationary soliton core with a small thermal cloud, a smoking-gun signal for non-semiclassical behavior on macroscopic scales and therefore a fully quantum emergent phenomenon. Read More

We present a weakly coupled map lattice model for patterning that explores the effects exerted by weakening the local dynamic rules on model biological and artificial networks composed of two-state building blocks (cells). To this end, we use two cellular automata models based on: (i) a smooth majority rule (model I) and (ii) a set of rules similar to those of Conway's Game of Life (model II). The normal and abnormal cell states evolve according with local rules that are modulated by a parameter $\kappa$. Read More

We characterize the collective modes of a soliton train in a quasi-one-dimensional Fermi superfluid, using a mean-field formalism. In addition to the expected Goldstone and Higgs modes, we find novel long-lived gapped modes associated with oscillations of the soliton cores. The soliton train has an instability that depends strongly on the interaction strength and the spacing of solitons. Read More

Stationary expansion shocks have been recently identified as a new type of solution to hyperbolic conservation laws regularized by non-local dispersive terms that naturally arise in shallow-water theory. These expansion shocks were studied in (El, Hoefer, Shearer 2016) for the Benjamin-Bona-Mahony equation using matched asymptotic expansions. In this paper, we extend the analysis of (El, Hoefer, Shearer 2016) to the regularized Boussinesq system by using Riemann invariants of the underlying dispersionless shallow water equations. Read More

The study of granular crystals, metamaterials that consist of closely packed arrays of particles that interact elastically, is a vibrant area of research that combines ideas from disciplines such as materials science, nonlinear dynamics, and condensed-matter physics. Granular crystals, a type of nonlinear metamaterial, exploit geometrical nonlinearities in their constitutive microstructure to produce properties (such as tunability and energy localization) that are not conventional to engineering materials and linear devices. In this topical review, we focus on recent experimental, computational, and theoretical results on nonlinear coherent structures in granular crystals. Read More

Ordered polarity alignment of a cell population plays a vital role in biology, such as in hair follicle alignment and asymmetric cell division. Here, we propose a theoretical framework for the understanding of generic dynamical properties of polarity alignment in interacting cellular units, where each cell is described by a reaction-diffusion system and the cells further interact with one another through their proximal surfaces. The system behavior is shown to be strongly dependent on geometric properties such as cell alignment and cell shape. Read More

**Authors:**Aakash A. Sahai

A model for the excitation of a non-linear ion-wake mode by a train of plasma electron oscillations in the non-linear time-asymmetric regime is developed using analytical theory and particle-in-cell based computational solutions. The ion-wake is shown to be a driven non-linear ion-acoustic wave in the form of a cylindrical ion-soliton. The near-void and radially-outwards propagating ion-wake channel of a few plasma skin-depth radius, is explored for application to "Crunch-in" regime of positron acceleration. Read More

We consider the strength-duration relationship in one-dimensional spatially extended excitable media. In a previous study~\cite{idris2008analytical} set out to separate initial (or boundary) conditions leading to propagation wave solutions from those leading to decay solutions, an analytical criterion based on an approximation of the (center-)stable manifold of a certain critical solution was presented. The theoretical prediction in the case of strength-extent curve was later on extended to cover a wider class of excitable systems including multicomponent reaction-diffusion systems, systems with non-self-adjoint linearized operators and in particular, systems with moving critical solutions (critical fronts and critical pulses) [Bezekci et al. Read More