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

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## Nonlinear Sciences - Pattern Formation and Solitons Publications

We study the scattering of a long longitudinal radiating bulk strain solitary wave in the delaminated area of a two-layered elastic structure with soft (`imperfect') bonding between the layers within the scope of the coupled Boussinesq equations. The direct numerical modelling of this and similar problems is challenging and has natural limitations. We develop a semi-analytical approach, based on the use of several matched asymptotic multiple-scale expansions and averaging with respect to the fast space variable, leading to the coupled Ostrovsky equations in bonded regions and uncoupled Korteweg-de Vries equations in the delaminated region. Read More

We study a heretofore ignored class of spiral patterns for oscillatory media as characterized by the complex Landau-Ginzburg model. These spirals emerge from modulating the growth rate as a function of $r$, thereby turning off the instability. These spirals are uniquely determined by matching to those outer conditions, lifting a degeneracy in the set of steady-state solutions of the original equations. Read More

We present theoretical and experimental studies on pattern formation with bistable dynamical units coupled in a star network configuration. By applying a localized perturbation to the central or the peripheral elements, we demonstrate the subsequent spreading, pinning, or retraction of the activations; such analysis enables the characterization of the formation of stationary patterns of localized activity. The results are interpreted with a theoretical analysis of a simplified bistable reaction-diffusion model. Read More

We study rogue wave excitation pattern in a two-component Bose-Einstein condensate with pair-transition effects. The results indicate that rogue wave excitation can exist on a stripe phase background for which there are cosine and sine wave background in the two components respectively. The rogue wave peak can be much lower than the ones of scalar matter wave rogue waves, and varies with the wave period changing. Read More

Motivated by the recent theoretical study of (bright) soliton diode effects in systems with multiple scatterers, as well as by experimental investigations of soliton-impurity interactions, we consider some prototypical case examples of interactions of dark solitons with a pair of scatterers. In a way fundamentally opposite to the case of bright solitons (but consonant to their "anti-particle character"), we find that dark solitons accelerate as they pass the first barrier and hence cannot be trapped by a second equal-height barrier. A pair of unequal barriers may lead to reflection from the second one, however trapping in the inter-barrier region cannot occur. Read More

The particular type of four-kink multi-solitons (or quadrons) adiabatic dynamics of the sine-Gordon equation in a model with two identical point attracting impurities has been studied. This model can be used for describing magnetization localized waves in multilayer ferromagnet. The quadrons structure and properties has been numerically investigated. Read More

We introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent precision over very long periods. Its major advantage is that it is extremely simple (it is basically a centered box scheme) while remaining locally well defined. Read More

We present a novel approximation method which can predict the number of solitons asymptotically appearing under arbitrary rapidly decreasing initial conditions. The number of solitons can be estimated without integration of original soliton equations. As an example, we take the one-dimensional nonlinear Schrodinger equation and estimate the behaviors of scattering amplitude in detail. Read More

The spectral renormalization method was introduced by Ablowitz and Musslimani in 2005, [Opt. Lett. {\bf 30}, pp. Read More

We have investigated mixed-gap vector solitons involving incoherently coupled fundamental and dipole components in a parity-time (PT) symmetric lattice with saturable nonlinearity. For the focusing case, vector solitons emerge from the semi-infinite and the first finite gaps, while for the defocusing case, vector solitons emerge from the first finite and the second finite gaps. For both cases, we find that stronger saturable nonlinearity is relative to sharper increase/decrease of soliton power with propagation constant and to narrower existence domain of vector solitons. Read More

This paper investigates cells proliferation dynamics in small tumor cell aggregates using an individual based model (IBM). The simulation model is designed to study the morphology of the cell population and of the cell lineages as well as the impact of the orientation of the division plane on this morphology. Our IBM model is based on the hypothesis that cells are incompressible objects that grow in size and divide once a threshold size is reached, and that newly born cell adhere to the existing cell cluster. Read More

**Authors:**Shin-itiro Goto

It is shown that Maxwell's equations in media without source can be written as a contact Hamiltonian vector field restricted to a Legendre submanifold, where this submanifold is in a fiber space of a bundle and is generated by either electromagnetic energy functional or co-energy functional. Then, it turns out that Legendre duality for this system gives the induction oriented formulation of Maxwell's equations and field intensity oriented one. Also, information geometry of the Maxwell fields is introduced and discussed. Read More

We demonstrate that pulses of linear physical systems, weakly perturbed by nonlinear dissipation, exhibit soliton-like behavior in fast collisions. The behavior is demonstrated for linear waveguides with weak cubic loss and for systems described by linear diffusion-advection models with weak quadratic loss. We show that in both systems, the expressions for the collision-induced amplitude shifts due to the nonlinear loss have the same form as the expression for the amplitude shift in a fast collision between two optical solitons in the presence of weak cubic loss. Read More

We consider a cubic nonlinear wave equation on a network and show that inspecting the normal modes of the graph, we can immediately identify which ones extend into nonlinear periodic orbits. Two main classes of nonlinear periodic orbits exist: modes without soft nodes and others. For the former which are the Goldstone and the bivalent modes, the linearized equations decouple. Read More

We analyze the stability of a non-Hermitian coupler with respect to modulational inhomogeneous perturbations in the presence of unbalanced gain and loss. At the parity-time (PT) symmetry point the coupler is unstable. Suitable symmetry breakings lead to an asymmetric coupler, which hosts nonlinear supermodes. Read More

We numerically investigate and experimentally demonstrate an in-situ topological band transition in a highly tunable mechanical system made of cylindrical granular particles. This system allows us to tune its inter-particle stiffness in a controllable way, simply by changing the contact angles between the cylinders. The spatial variation of particles' stiffness results in an in-situ transition of the system's topology. Read More

Nonlocality is a key feature of many physical systems since it prevents a catastrophic collapse and a symmetry-breaking azimuthal instability of intense wave beams in a bulk self-focusing nonlinear media. This opens up an intriguing perspective for stabilization of complex topological structures such as higher-order solitons, vortex rings and vortex ring-on-line complexes. Using direct numerical simulations, we find a class of cylindrically-symmetric $n$-th order spatial solitons having the intensity distribution with a central bright spot surrounded by $n$ bright rings of varying size. Read More

We consider a system of equations for the description of nonlinear waves in a liquid with gas bubbles. Taking into account high order terms with respect to a small parameter, we derive a new nonlinear partial differential equation for the description of density perturbations of mixture in the two-dimensional case. We investigate integrability of this equation using the Painlev\'e approach. Read More

**Affiliations:**

^{1}LAMA,

^{2}University of Colorado

Some effects of surface tension on fully-nonlinear, long, surface water waves are studied by numerical means. The differences between various solitary waves and their interactions in subcritical and supercritical surface tension regimes are presented. Analytical expressions for new peaked travelling wave solutions are presented in the case of critical surface tension. Read More

In the present work, we explore the existence, stability and dynamics of single and multiple vortex ring states that can arise in Bose-Einstein condensates. Earlier works have illustrated the bifurcation of such states, in the vicinity of the linear limit, for isotropic or anisotropic three-dimensional harmonic traps. Here, we extend these states to the regime of large chemical potentials, the so-called Thomas-Fermi limit, and explore their properties such as equilibrium radii and inter-ring distance, for multi-ring states, as well as their vibrational spectra and possible instabilities. Read More

The stationary Gross-Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the PT (parity-time reversal) symmetry. Under rather general assumptions on the potentials we prove bifurcations of PT-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrodinger operator with a complex PT-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrodinger equation. Read More

We combine experiments, large scale simulations and continuum models to study the emergence of coherent structures in a suspension of magnetically driven microrollers sedimented near a floor. Collective hydrodynamic effects are predominant in this system, leading to strong density-velocity coupling. We characterize a uniform suspension and show that density waves propagate freely in all directions in a dispersive fashion. Read More

We study time-independent radially symmetric BPS solitons in a CP(2) model interacting with an Abelian gauge field whose dynamics is controlled by the usual Maxwell term. In this sense, we develop a consistent self-dual framework verifying the existence of a well-defined lower bound for the corresponding energy. We saturate such a lower bound by focusing on those solutions satisfying a particular set of coupled first-order differential equations, the BPS ones. Read More

We investigate the dynamics of the localized nonlinear matter wave in spin-1 Bose-Einstein condensates with trapping potentials and nonlinearities dependent on time and space. We solve the three coupled Gross-Pitaevskii equation by similarity transformation and obtain two families of exact matter wave solutions in terms of Jacobi elliptic functions and Mathieu equation. The localized states of the spinor matter wave describe the dynamics of vector breathing solitons, moving breathing solitons, quasibreathing solitons and resonant solitons. Read More

Faraday waves are a classic example of a system in which an extended pattern emerges under spatially uniform external forcing. Motivated by systems in which uniform excitation is not plausible, we study both experimentally and theoretically the effect of inhomogeneous forcing on Faraday waves. Our experiments show that vibrations restricted to finite regions lead to the formation of localized subharmonic wave patterns, whose localization depends on the extent of the excited region. Read More

We investigate the formation of optical localized nonlinear structures, evolving upon a non-zero background plane wave, in a dispersive quadratic medium. We show the existence of quadratic Akhmediev breathers and Peregrine solitary waves, in the regime of cascading second-harmonic generation. This finding opens a novel path for the excitation of extreme rogue waves and for the description of modulation instability in quadratic nonlinear optics. Read More

A normal form approximation for the evolution of a reaction-diffusion system hosted on a directed graph is derived, in the vicinity of a supercritical Hopf bifurcation. Weak diffusive couplings are assumed to hold between adjacent nodes. Under this working assumption, a Complex Ginzburg-Landau equation (CGLE) is obtained, whose coefficients depend on the parameters of the model and the topological characteristics of the underlying network. Read More

Chirality is ubiquitous in nature and chiral objects in condensed matter are often excited states protected by system's topology. The use of chiral topological excitations to carry information has been demonstrated, where the information is robust against external perturbations. For instance, reading, writing, and transfer of binary information are demonstrated with chiral topological excitations in magnetic systems, skyrmions, for spintronic devices. Read More

Random network models play a prominent role in modeling, analyzing and understanding complex phenomena on real-life networks. However, a key property of networks is often neglected: many real-world networks exhibit spatial structure, the tendency of a node to select neighbors with a probability depending on physical distance. Here, we introduce a class of random spatial networks (RSNs) which generalizes many existing random network models but adds spatial structure. Read More

Spatially distributed limited-cycle oscillators are seen in various physical and biological systems. In internal organs, mechanical motions are induced by the stimuli of spatially distributed limit-cycle oscillators. We study several mechanical motions by limit-cycle oscillators using simple model equations. Read More

Passive Kerr cavities driven by coherent laser fields display a rich landscape of nonlinear physics, including bistability, pattern formation, and localised dissipative structures (solitons). Their conceptual simplicity has for several decades offered an unprecedented window into nonlinear cavity dynamics, providing insights into numerous systems and applications ranging from all-optical memory devices to microresonator frequency combs. Yet despite the decades of study, a recent theoretical study has surprisingly alluded to an entirely new and unexplored paradigm in the regime where nonlinearly tilted cavity resonances overlap with one another [T. Read More

Domain walls are minimizers of energy for coupled one-dimensional Gross--Pitaevskii systems with nontrivial boundary conditions at infinity. It has been shown that these solutions are orbitally stable in the space of complex $\dot{H}^1$ functions with the same limits at infinity. In the present work we adopt a new weighted $H^1$ space to control perturbations of the domain walls and thus to obtain an improved orbital stability result. Read More

We present the study of the dark soliton dynamics in an inhomogenous fiber by means of a variable coefficient modified nonlinear Schr\"{o}dinger equation (Vc-MNLSE) with distributed dispersion, self-phase modulation, self-steepening and linear gain/loss. The ultrashort dark soliton pulse evolution and interaction is studied by using the Hirota bilinear (HB) method. In particular, we give much insight into the effect of self-steepening (SS) on the dark soliton dynamics. Read More

We refer the appearance of three-dimensional (3D) multiheaded chimera states that display cascades of self-organized spatiotemporal patterns of coexisting coherence and incoherence. We demonstrate that number of the incoherent chimera's domains can grow additively under appropriate varying the system parameters generating thereby head-adding cascades of the scroll wave chimeras. The phenomenon is derived for Kuramoto model of N identical phase oscillators placed in the 3D cube with periodic boundary conditions, parameters are coupling radius $r$ and phase lag $\alpha$. Read More

Canards are special solutions to ordinary differential equations that follow invariant repelling slow manifolds for long time intervals. In realistic biophysical single cell models, canards are responsible for several complex neural rhythms observed experimentally, but their existence and role in spatially-extended systems is largely unexplored. We describe a novel type of coherent structure in which a spatial pattern displays temporal canard behaviour. Read More

We examine integrable turbulence in the framework of dispersive hydrodynamics by realizing an optical fiber experiment in which the defocusing Kerr nonlinearity strongly dominates linear dispersive effects. In this context, Riemann invariants of the asymptotic nonlinear geometric optics system are shown to represent appropriate observable quantities that provide new insight into the understanding of statistical features of the initial stage of development of integrable turbulence. The real-time observation of Riemann invariants in optics is achieved by combining heterodyne and time-division multiplexing techniques in a fast detection setup. Read More

Spatially extended systems can support local transient excitations in which just a part of the system is excited. The mechanisms reported so far are local excitability and excitation of a localized structure. Here we introduce an alternative mechanism based on the coexistence of two homogeneous stable states and spatial coupling. Read More

**Authors:**A. Tikan, C. Billet, G. El, A. Tovbis, M. Bertola, T. Sylvestre, F. Gustave, S. Randoux, G. Genty, P. Suret, J. M. Dudley

We present experimental evidence of the universal emergence of the Peregrine soliton predicted in the semi-classical (zero-dispersion) limit of the focusing nonlinear Schr\"{o}dinger equation [Comm. Pure Appl. Math. Read More

We analyze the gKdV equation, a generalized version of Korteweg-de Vries with an arbitrary function $f(u)$. In general, for a function $f(u)$ the Lie algebra of symmetries of gKdV is the $2$-dimensional Lie algebra of translations of the plane $xt$. This implies the existence of plane wave solutions. Read More

Exact bright, dark, antikink solitary waves and Jacobi elliptic function solutions of the generalized Benjamin-Bona-Mahony equation with arbitrary power-law nonlinearity will be constructed in this work. The method used to carry out the integration is the F-expansion method. Solutions obtained have fractional and integer negative or positive power-law nonlinearities. Read More

During embryo development, patterns of protein concentration appear in response to morphogen gradients. The archetypal pattern is the French flag composed of three chemically-distinct zones separated by sharp borders. Chemical concentration serves the underlying cells both as an indicator of location and as a fate selector. Read More

In the present work, we aim at taking a step towards the spectral stability analysis of Peregrine solitons, i.e., wave structures that are used to emulate extreme wave events. Read More

The Ablowitz-Ladik equation is a very important model in the nonlinear mathematical physics. In this paper, the hyperbolic function solitary wave solutions, the trigonometric function periodic wave solutions and the rational wave solutions with more arbitrary parameters of 2-dimensional Ablowitz-Ladik equation are derived by using the GG-expansion method, and the effect of the parameters (including the coupling constant and other parameters) on the linear stability of the exact solutions is analysed and numerically simulated. Read More

Three (2+1)-dimensional equations, they are KP equation, cylindrical KP equation and spherical KP equation, have been reduced to the same KdV equation by different transformation of variables respectively. Since the single solitary wave solution and 2-solitary wave solution of the KdV equation have been known already, substituting the solutions of the KdV equation into the corresponding transformation of variables respectively, the single and 2-solitary wave solutions of the three (2+1)-dimensional equations can be obtained successfully. Read More

In this paper, the famous Klein-Gordon-Zakharov equations are firstly generalized, the new special types of Klein-Gordon-Zakharov equations with the positive fractional power terms (gKGZE) are presented. In order to derive the exact solutions of new special gKGZE, the subsidiary higher order ordinary differential equations (sub-ODEs) with the positive fractional power terms are introduced, and with the aids of the Sub-ODE, the exact solutions of three special types of the gKGZE are derived, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of gKGZE satisfy certain constraint conditions. Read More

We study dynamics of Dirac solitons in prototypical networks modeling them by the nonlinear Dirac equation on metric graphs. Soliton solutions of the nonlinear Dirac equation on simple metric graphs are obtained. It is shown that these solutions provide reflectionless vertex transmission of the Dirac solitons under suitable conditions. Read More

Chimera states have been studied in 1D arrays, and a variety of different chimera states have been found using different models. Research has recently been extended to 2D arrays but only to phase models of them. Here, we extend it to a nonphase model of 2D arrays of neurons and focus on the influence of nonlocal coupling. Read More

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