In this article the generalized Lotka-Volterra model of ensemble of four excitory or inhibitory coupled elements are studied. It is shown that in the phase space of the model there exist heteroclinic network: a connected union of two or more heteroclinic cycles. A partition of the plane of coupling parameters into sets of existence of various heteroclinic networks is constructed.

In this work, we present and analyze a general framework for vegetation dynamics in arid and semi-arid ecosystems in which non-local interactions are purely competitive. The generality of the formulation enables a systematic search for ecological mechanisms that may lead to self-organized patterns. We identify two distinct mechanisms generating Turing instabilities across a broad class of models. The first mechanism arises from intensified competition in the areas between vegetated patches due to the cumulative pressure from their surroundings, and is well-documented in the literature. The second mechanism is novel and occurs when local growth outpaces competitive susceptibility near the uniform equilibrium. The analytical findings are complemented by numerical simulations of two benchmark models, both exhibiting a supercritical Turing bifurcation that leads to the formation of stable and robust vegetation patterns.

Standard physics-informed neural network implementations have produced large error rates when using these models to solve the regularized long wave (RLW) equation. Two improved PINN approaches were developed in this research: an adaptive approach with self-adaptive loss weighting and a conservative approach enforcing explicit conservation laws. Three benchmark tests were used to demonstrate how effective PINN's are as they relate to the type of problem being solved (i.e., time dependent RLW equation). The first was a single soliton traveling along a line (propagation), the second was the interaction between two solitons, and the third was the evolution of an undular bore over the course of $t=250$. The results demonstrated that the effectiveness of PINNs are problem specific. The adaptive PINN was significantly better than both the conservative PINN and the standard PINN at solving problems involving complex nonlinear interactions such as colliding two solitons. The conservative approach was significantly better at solving problems involving long term behavior of single solitons and undular bores. However, the most important finding from this research is that explicitly enforcing conservation laws may be harmful to optimizing the solution of highly nonlinear systems of equations and therefore requires special training methods. The results from our adaptive and conservative approaches were within $O(10^{-5})$ of established numerical solutions for the same problem, thus demonstrating that PINNs can provide accurate solutions to complex systems of partial differential equations without the need for a discretization of space or time (mesh free). Moreover, the finding from this research challenges the assumptions that conservation enforcement will always improve the performance of a PINN and provides researchers with guidelines for designing PINNs for use on specific types of problems.

Optical solitons are self-sustained wave packets that propagate without distortion due to a balance between dispersion and nonlinearity. Their unique stability underpins key photonic applications while also playing a central role in nonlinear wave physics. However, real-time control over soliton dynamics in non-dissipative systems remains a major challenge, limiting their practical applications in photonic systems. Here, we introduce a fiber-based platform for soliton manipulation, by creating programmable external potentials through synchronous arbitrary phase modulation in a recirculating optical fiber loop. We demonstrate precise soliton trapping, parametric excitation, and coupled multi-soliton interactions, revealing particle-like behavior in excellent agreement with a Hamiltonian description in which solitons are treated as interacting classical particles. The strong analogy with matter-wave solitons in Bose-Einstein condensates highlights the broader implications of our approach, which provides a versatile experimental tool for the study of nonlinear wave dynamics and engineered soliton manipulation.

This research investigates the complex spatiotemporal behaviors of Chialvo neuron maps under the influence of Levy noise on three different network topologies that is a ring network, a two dimensional lattice affected by electromagnetic flux, and a delayed coupled lattice. On the ring structure, we show that adding non uniform Levy noise induces the formation of new collective dynamics like standing and traveling waves. The frequency and type of these emergent patterns depend sensitively on the intrinsic excitability parameter and the noise intensity, revealing new pathways to control synchronization behavior through noise modulation. In the 2D lattice network, we show that electromagnetic flux and noise together induce a diverse range of behaviors, from synchronized waves to desynchronized states. Most strikingly, spiral wave chimeras emerge under moderate noise, with coherent and incoherent regions coexisting, highlighting the fine balance between external forcing and stochastic perturbations. Finally, upon introducing delay in the lattice structure, the system displays a rich variety of dynamical regimes such as labyrinth patterns, rotating spirals, and target waves whose stability and transitions are greatly affected by both delay and coupling strength.

This paper presents a comprehensive and systematic study of the possible connection between thermalization of cubic nonlinear lattices with nearest-neighbor coupling and the structure of the mixing tensor that arises due to the presence of nonlinearities. The approach is based on rewriting the underlying lattice system as a nonlinear evolution equation governing the dynamics of the modal amplitudes (or projection coefficients). In this formulation, the linear coupling become diagonalizable, whereas all cubic nonlinear terms transform into a combinatorial sum over a product of three modal amplitudes weighted by a fourth-order mixing tensor. We have identified the exact structure of several mixing tensors (corresponding to different types of cubic nonlinearities) and tied their symmetry properties (quasi-Hermiticity and permutation symmetries associated with two lattice conservation laws) with thermalization or lack thereof. Furthermore, we have observed through direct numerical simulations that the modal occupancies of lattices preserving these tensorial symmetries approach a Rayleigh-Jeans distribution at thermal equilibrium. In addition, we provided few examples that indicate that cubic lattices with broken tensorial symmetries end up not to equilibrate to a Rayleigh-Jeans distribution. Finally, an inverse approach to the study of thermalization of cubic nonlinear lattices is developed. It establishes a duality property between lattices in local and modal bases. The idea is to establish a trade-off between the type of nonlinearities in local base and their respective interactions in supermode base. With this at hand, we were able to identify a large class of nonlinear lattices that are embedded in the modal space and admit a simple form that can be used to shed more light on the role that localization (or delocalization) of the supermodes play in thermalization processes.

Generalised Hydrodynamics (GHD) describes the large-scale inhomogeneous dynamics of integrable (or close to integrable) systems in one dimension of space, based on a central equation for the fluid density or quasi-particle density: the GHD equation. We consider a new, general form of the GHD equation: we allow for spatially extended interaction kernels, generalising previous constructions. We show that the GHD equation, in our general form and hence also in its conventional form, is Hamiltonian. This holds also including force terms representing inhomogeneous external potentials coupled to conserved densities. To this end, we introduce a new Poisson bracket on functionals of the fluid density, which is seen as our dynamical field variable. The total energy is the Hamiltonian whose flow under this Poisson bracket generates the GHD equation. The fluid density depends on two (real and spectral) variables so the GHD equation can be seen as a $2+1$-dimensional classical field theory. In its $1+1$-dimensional reduction corresponding to the case without external forces, we further show the system admits an infinite set of conserved quantities that are in involution for our Poisson bracket, hinting at integrability of this field theory.

A model of population growth and dispersal is considered where the spatial habitat is a lattice and reproduction occurs generationally. The resulting discrete dynamical systems exhibits velocity locking, where rational speed invasion fronts are observed to persist as parameters are varied. In this article, we construct locked fronts for a particular piecewise linear reproduction function. These fronts are shown to be linear combinations of exponentially decaying solutions to the linear system near the unstable state. Based upon these front solutions, we then derive expressions for the boundary of locking regions in parameter space. We obtain leading order expansions for the locking regions in the limit as the migration parameter tends to zero. Strict spectral stability in exponentially weighted spaces is also established.

Light propagation in semiconductors is the cornerstone of emerging disruptive technologies holding considerable potential to revolutionize telecommunications, sensors, quantum engineering, healthcare, and artificial intelligence. Sky-high optical nonlinearities make these materials ideal platforms for photonic integrated circuits. The fabrication of such complex devices could greatly benefit from in-volume ultrafast laser writing for monolithic and contactless integration. Ironically, as exemplified for Si, nonlinearities act as an efficient immune system self-protecting the material from internal permanent modifications that ultrashort laser pulses could potentially produce. While nonlinear propagation of high-intensity ultrashort laser pulses has been extensively investigated in Si, other semiconductors remain uncharted. In this work, we demonstrate that filamentation universally dictates ultrashort laser pulse propagation in various semiconductors. The effective key nonlinear parameters obtained strongly differ from standard measurements with low-intensity pulses. Furthermore, the temporal scaling laws for these key parameters are extracted. Temporal-spectral shaping is finally proposed to optimize energy deposition inside semiconductors. The whole set of results lays the foundations for future improvements, up to the point where semiconductors can be selectively tailored internally by ultrafast laser writing, thus leading to countless applications for in-chip processing and functionalization, and opening new markets in various sectors including technology, photonics, and semiconductors.

We study the back-reaction of fermion fields on the kink solution in one space and one time dimension. We employ a variational procedure to determine an upper limit for the minimum of the total energy. This energy has three contributions: the classical kink energy, the energy of valence fermions and the fermion vacuum polarization energy. The latter arises from the interaction of the kink with the Dirac sea and is required for consistency of the semi-classical expansion for the fermions. Earlier studies only considered the valence part and observed a substantial back-reaction. This was reflected by a sizable distortion of the kink profile. We find that this distortion is strongly mitigated when the Dirac sea is properly accounted for. As a result the back-reaction merely produces a slight squeeze or stretch of the kink profile.

We study simultaneous collisions of two, three, and four kinks and antikinks of the $\phi^6$ model at the same spatial point. Unlike the $\phi^4$ kinks, the $\phi^6$ kinks are asymmetric and this enriches the variety of the collision scenarios. In our numerical simulations we observe both reflection and bound state formation depending on the number of kinks and on their spatial ordering in the initial configuration. We also analyze the extreme values of the energy densities and the field gradient observed during the collisions. Our results suggest that very high energy densities can be produced in multi-kink collisions in a controllable manner. Appearance of high energy density spots in multi-kink collisions can be important in various physical applications of the Klein-Gordon model.

We report that an instability boundary of a single-mode state in Kerr ring microresonators with ultrahigh quality factors breaks the parameter space span by the pump laser power and frequency into a sequence of narrow in frequency and broad in power resonance domains - Arnold tongues. Arnold resonances are located between the Lugiato-Lefever (lower) and universal (higher) thresholds. Pump power estimates corresponding to the universal threshold are elaborated in details. RF-spectra generated within the tongues reveal a transition between the repetition-rate locked and unlocked regimes of the side-band generation.

We investigate theoretically the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The long-term statistical properties of the noise-induced MI have been previously observed in experiments and in simulations but have not been explained so far. In the framework of inverse scattering transform (IST), we propose a model of the asymptotic stage of the noise-induced MI based on $N$-soliton solutions ($N$-SS) of the integrable focusing one-dimensional nonlinear Schr\"odinger equation (1D-NLSE). These $N$-SS are bound states of strongly interacting solitons having a specific distribution of the IST eigenvalues together with random phases. We use a special approach to construct ensembles of multi-soliton solutions with statistically large number of solitons $N\sim100$. Our investigation demonstrates complete agreement in spectral (Fourier) and statistical properties between the long-term evolution of the condensate perturbed by noise and the constructed multi-soliton bound states. Our results can be generalised to a broad class of integrable turbulence problems in the cases when the wave field dynamics is strongly nonlinear and driven by solitons.

We report a new system of artificial life called Lenia (from Latin lenis "smooth"), a two-dimensional cellular automaton with continuous space-time-state and generalized local rule. Computer simulations show that Lenia supports a great diversity of complex autonomous patterns or "lifeforms" bearing resemblance to real-world microscopic organisms. More than 400 species in 18 families have been identified, many discovered via interactive evolutionary computation. They differ from other cellular automata patterns in being geometric, metameric, fuzzy, resilient, adaptive, and rule-generic. We present basic observations of the system regarding the properties of space-time and basic settings. We provide a broad survey of the lifeforms, categorize them into a hierarchical taxonomy, and map their distribution in the parameter hyperspace. We describe their morphological structures and behavioral dynamics, propose possible mechanisms of their self-propulsion, self-organization and plasticity. Finally, we discuss how the study of Lenia would be related to biology, artificial life, and artificial intelligence.

The nonlinear Schr\"odinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays a prominent role in the modeling and understanding of the wave phenomena relevant to many fields of nonlinear physics. The question of random input problems in the one-dimensional and integrable NLSE enters within the framework of integrable turbulence, and the specific question of the formation of rogue waves (RWs) has been recently extensively studied in this context. The determination of exact analytic solutions of the focusing 1D-NLSE prototyping RW events of statistical relevance is now considered as the problem of central importance. Here we address this question from the perspective of the inverse scattering transform (IST) method that relies on the integrable nature of the wave equation. We develop a conceptually new approach to the RW classification in which appropriate, locally coherent structures are specifically isolated from a globally incoherent wave train to be subsequently analyzed by implementing a numerical IST procedure relying on a spatial periodization of the object under consideration. Using this approach we extend the existing classifications of the prototypes of RWs from standard breathers and their collisions to more general nonlinear modes characterized by their nonlinear spectra.

We present an experimental study on the perturbed evolution of Korteweg-deVries soliton gases in a weakly dissipative nonlinear electrical transmission line. The system's dynamics reveal that an initially dense, fully randomized, soliton gas evolves into a coherent macroscopic state identified as a soliton condensate through nonlinear spectral analysis. The emergence of the soliton condensate is driven by the spatial rearrangement of the systems's eigenmodes and by the proliferation of new solitonic states due to nonadiabatic effects, a phenomenon not accounted for by the existing hydrodynamic theories.

The Landau-Lifshitz-Gilbert (LLG) equation is a fascinating nonlinear evolution equation both from mathematical and physical points of view. It is related to the dynamics of several important physical systems such as ferromagnets, vortex filaments, moving space curves, etc. and has intimate connections with many of the well known integrable soliton equations, including nonlinear Schr\"odinger and sine-Gordon equations. It can admit very many dynamical structures including spin waves, elliptic function waves, solitons, dromions, vortices, spatio-temporal patterns, chaos, etc. depending on the physical and spin dimensions and the nature of interactions. An exciting recent development is that the spin torque effect in nanoferromagnets is described by a generalization of the LLG equation which forms a basic dynamical equation in the field of spintronics. This article will briefly review these developments as a tribute to Robin Bullough who was a great admirer of the LLG equation.

We review the study of rogue waves and related instabilities in optical and oceanic environments, with particular focus on recent experimental developments. In optics, we emphasize results arising from the use of real-time measurement techniques, whilst in oceanography we consider insights obtained from analysis of real-world ocean wave data and controlled experiments in wave tanks. Although significant progress in understanding rogue waves has been made based on an analogy between wave dynamics in optics and hydrodynamics, these comparisons have predominantly focused on one-dimensional nonlinear propagation scenarios. As a result, there remains significant debate about the dominant physical mechanisms driving the generation of ocean rogue waves in the complex environment of the open sea. Here, we review state-of-the-art of rogue wave studies in optics and hydrodynamics, aiming to clearly identify similarities and differences between the results obtained in the two fields. In hydrodynamics, we take care to review results that support both nonlinear and linear interpretations of ocean rogue wave formation, and in optics, we also summarise results from an emerging area of research applying the measurement techniques developed for the study of rogue waves to dissipative soliton systems. We conclude with a discussion of important future research directions.

We report water wave experiments performed in a long tank where we consider the evolution of nonlinear deep-water surface gravity waves with the envelope in the form of a large-scale rectangular barrier. Our experiments reveal that, for a range of initial parameters, the nonlinear wave packet is not disintegrated by the Benjamin-Feir instability but exhibits a specific, strongly nonlinear modulation, which propagates from the edges of the wavepacket towards the center with finite speed. Using numerical tools of nonlinear spectral analysis of experimental data we identify the observed envelope wave structures with focusing dispersive dam break flows, a peculiar type of dispersive shock waves recently described in the framework of the semi-classical limit of the 1D focusing nonlinear Schrodinger equation (1D-NLSE). Our experimental results are shown to be in a good quantitative agreement with the predictions of the semi-classical 1D-NLSE theory. This is the first observation of the persisting dispersive shock wave dynamics in a modulationally unstable water wave system.

Active fluids exhibit spontaneous flows with complex spatiotemporal structure, which have been observed in bacterial suspensions, sperm cells, cytoskeletal suspensions, self-propelled colloids, and cell tissues. Despite occurring in the absence of inertia, chaotic active flows are reminiscent of inertial turbulence, and hence they are known as active turbulence. Here, we survey the field, providing a unified perspective over different classes of active turbulence. To this end, we divide our review in sections for systems with either polar or nematic order, and with or without momentum conservation (wet/dry). Comparing to inertial turbulence, we highlight the emergence of power-law scaling with either universal or non-universal exponents. We also contrast scenarios for the transition from steady to chaotic flows, and we discuss the absence of energy cascades. We link this feature to both the existence of intrinsic length scales and the self-organized nature of energy injection in active turbulence, which are fundamental differences with inertial turbulence. We close by outlining the emerging picture, remaining challenges, and future directions.