Schedule for: 25w5346 - Dynamics of Coherent Structures in Discrete and Continuum Nonlinear Systems

We give a systematic approach to the derivation of gradient flows from "intrinsic'' free energies that characterize the state of a sharp interface solely in terms of its local structure. The Canham-Helfrich free energy is the classic example of an intrinsic sharp interface free energy. We apply the approach to a model energy for brine inclusion in ice in the presence of an antifreeze protein. The effect is an energetic preference for an interface with either a zero or high curvature interface -- leading to a faceted interface. The facet-forming normal velocity inherits a Cahn-Hilliard type structure embedded within the evolution for the second fundamental form of the interface. We construct stationary solutions, and discuss their linear stability.

We study the nonlinear dynamics of perturbed, spectrally stable $T$-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schr\""odinger equation with forcing that arises in nonlinear optics. It is known that for each $N\in\NM$, such a $T$-periodic wave is asymptotically stable against $NT$-periodic, i.e. subharmonic, perturbations, in the sense that initially nearby data will converge at an exponential rate to a (small) spatial translation of the underlying wave. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential rates of decay depend on $N$ and, in fact, tend to zero as $N\to\infty$, leading to a lack of uniformity in the period of the perturbation. Relying upon a Bloch decomposition and recent results on nonlinear stability for localized perturbations, we show how to obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in $N$. This talk is based on joint works with Mat Johnson, Wesley Perkins and Bj\""orn de Rijk.

Optimizing a pairwise potential given by Laplacian's Green's function is a classical problem that occurs in numerous applications, including spike patterns in reaction-diffusion systems, brownian motion in the presence of small traps, and oxygen transport flow. Here, we present results for a periodic rectangle. We classify regular lattice patterns and study their stability up to 200 particles. We also construct an explicit family of hexagonal-type patterns and show their stability.

Α summary of recent results concerning the Adlam-Allen model is attempted. This model was established in 1958 and it originates from plasma physics. The last years an attempt to revisit this model was performed. These studies revealed that this model can support a variety of coherent structures and their existence and dynamical stability is examined. Finally, by using multiscale techniques we were able to establish connections to many prototypical nonlinear wave equations.

We present an abstract framework to establish the existence and spectral stability of multiple front and pulse solutions, along with their periodic extensions, in spatially periodic semilinear evolution problems. We adopt a spatial dynamics approach to construct these solutions near formal superpositions of well-separated primary fronts or pulses by characterizing invertibility through exponential dichotomies. Employing Evans-function techniques, we prove that the spectrum of the system's linearization about such a solution converges to the union of the spectra of the primary fronts or pulses as the distances between them tend to infinity. As an application, we consider to the Gross-Pitaevskii equation with periodic potential, where we construct multi-solitons and soliton trains close to superpositions of bright gap solitons and analyze their stability. In particular, our results provide the first proof of orbital stability for periodic waves in the Gross-Pitaevskii equation. This is joint work with Lukas Bengel.

We present a weakly nonlinear, weakly dispersive Boussinesq system for water waves propagating on a 1D branching channel, namely for studying reflection-transmission on a metric graph. Our graph model uses a new nonlinear compatibility condition at the vertex which improves reflection-transmission properties, and therefore generalizes the well-known Neumann-Kirchhoff condition. The model includes forking-angles in a systematic fashion. Our vertex condition is formulated by looking also at solutions of the 2D (parent) fattened graph model, namely a graph-like domain with a small lateral width. We present numerical simulations comparing solitary-wave-propagation on the 1D (reduced) graph model with the respective results of the 2D model, where a compatibility condition is not needed at the forked region. We will comment on ongoing studies, in particular regarding the importance of including angle-information, a feature not present in most waves-on-graph models. 

We investigate the interaction between two flat-top solitons within the cubic-quintic nonlinear Schrödinger equation framework. Our study results point towards a significant departure of flat-top solitons’ collisional characteristics from the conventional behaviors exhibited in the scattering dynamics of two bright solitons. Our investigation outlines specific regimes corresponding to the dual flat-top solitons’ elastic and inelastic collisions. We determine regimes in the parameter space where exchange in the widths of the interacting flat-top solitons is possible, even within the elastic collision domain. We find a periodic occurrence of completely elastic scattering of flat-top solitons in terms of the solitons’ parameters. Investigating the internal energy transfers between the solitons and the emitted radiation reveals the origin of inelasticity in flat-top soliton collisions. We perform a variational calculation that accounts for the amount of radiation produced by the collision and hence provides further insight into the physics underlying the loss of elasticity of collisions.

We found two stationary solutions of the parametrically driven, damped nonlinear Schrödinger equation with a nonlinear term proportional to $|\psi(x,t)|^{2 \kappa } \psi(x,t)$ for positive values of $\kappa$. By linearizing the equation around these exact solutions, we derived the corresponding Sturm-Liouville problem. Our analysis reveals that one of the stationary solutions is unstable, while the stability of the other solution depends on the amplitude of the parametric force, the damping coefficient, and the nonlinearity parameter $\kappa$. An exceptional change of variables facilitates the computation of the stability diagram through numerical solutions of the eigenvalue problem as a specific parameter $\varepsilon$ varies within a bounded interval. For $\kappa<2$, an oscillatory instability is predicted analytically and confirmed numerically. Our principal result establishes that for $\kappa \geq 2$ , there exists a critical value of $\varepsilon$ beyond which the unstable soliton becomes stable, exhibiting oscillatory stability.

The transport and accumulation of inertial particles driven by surface gravity waves play a vital role in shaping coastal processes such as sediment dynamics, pollutant dispersion, and the fate of floating impurities. While significant progress has been made in deep-water settings under monochromatic wave conditions, particularly in studies such as those by DiBenedetto, et al. (2022), and Santamaria et al. (2013), this study advances the field by addressing more complex and realistic environments. Specifically, we extend the analysis to shallow-water regimes characterized by nonlinear wave phenomena, including cnoidal and solitary waves, which produce intricate particle trajectories due to their strong nonlinearity. By incorporating the dynamics of finite-size particles with varying shapes and densities into a refined theoretical framework tailored to shallow waters—building on and extending the model developed by DiBenedetto and coauthors—we offer new insights into how wave motion governs particle behavior in nearshore waters. This work represents a significant step toward understanding the complex interplay between wave dynamics and particle transport in coastal systems. This is a work in collaboration with H. Kalisch and F. Desquines.

Non-Hermitian systems with nonreciprocal couplings have gathered significant attention due to their ability to exhibit exotic wave phenomena absent in their Hermitian counterparts. In the linear regime, such systems display distinctive spectral features, including the accumulation of eigenstates at the system boundaries; known as the non-Hermitian skin effect (NHSE); as well as direction-dependent amplification of traveling waves. I will investigate how nonlinearity alters the wave dynamics of nonreciprocal systems. I will first extend the concept of the NHSE into the nonlinear regime. Then, considering more realistic conditions where dissipative effects are present, I will unveil a novel control mechanism in which nonlinearity mediates the interplay between nonreciprocity and dissipation. This interplay could give rise to unidirectional solitons, even within a wave amplification dominated phase. These unidirectional solitons generalize the conventional bidirectional Boussinesq ones through a balance between dispersion, nonlinearity, dissipation, and nonreciprocity. This work is in collaboration with Vassos Achilleos, Ricardo Carretero-González, Panayotis Kevrekidis, Dimitrios Frantzeskakis, Sayan Jana and Lea Sirota.

We consider optical Kerr frequency combs generated in a nonlinear two-mode forced microresonator. In physical set-ups it has been observed that two-mode forcing increases several performance metrics and leads to better synchronization of the radio-frequency oscillator with the comb. Kerr frequency combs are modeled by stationary spatially localized solutions of the Lugiato-Lefever equation $$ iu_{t} = −du_{xx} + icu_{x} + (\zeta − i)u − |u|^2 u + i f (x), $$ which is a damped, detuned, and driven nonlinear Schrödinger equation. We show by using the Lyapunov-Schmidt reduction method that two-mode forcing leads to localized 1-pulse solutions with oscillatory tails upon bifurcating from the bright NLS-soliton. Adopting a spatial dynamics approach we then construct multi-pulses resembling well-separated multiple copies of the 1-pulses and we study their stability. We observe that the periodic background state traps the 1- and multi-pulses yielding better stability properties as opposed to 1-mode forcing. Numerical simulations with pde2path complement our analytical findings. This is based on a joint project with Björn de Rijk (KIT).

Exploiting integrablity in dynamical systems is often a great starting point for analyzing more complex, nonintegrable equations. However, it is difficult to even recognize if a given system is integrable before investing effort into studying it from this perspective. Therefore, we formulate the automated discovery of integrability in dynamical systems, specifically as a symbolic regression problem. Loosely speaking, we seek to maximize the compatibility between the known Hamiltonian and a conjectured Lax pair of the system. Our approach is tested on a variety of systems ranging from nonlinear oscillators to canonical Hamiltonian PDEs. We test robustness of the framework against nonintegrable perturbations, and, in all examples, reliably confirm or deny integrability. Moreover, using a thresholded regularization to promote sparsity, we recover expected and discover new Lax pairs despite wide hypotheses on the operators. We will discuss future directions for adapting our framework toward further automated discoveries in mathematical physics.

We consider the $L^2$ gradient flow of the Helfrich model for lipid bilayers. The model incorporates constraints on membrane inextensibility (area constraint) and the absence of osmotic exchange (volume constraint). These constraints give rise to Lagrange multipliers, which appear as non-local terms. The lipid bilayer serves as a simplified model for the shape of red blood cells as well as other self-organizing cellular structures in biology. We studied bifurcation of closed vesicles using {\tt pde2path}. We extend this to cylindrical topology. Well-known phenomena are the pearling instability of the cylindrical shape and transitions to coiled structures solution. We find them and other bifurcations using center manifold analysis and numerical continuation. Due to the presence of Lagrange multipliers, we take a non-standard approach to derive amplitude equations for each bifurcation scenario. Within this framework, we analyze the stability of the bifurcation branches. To provide a broader perspective on the shape transitions of cylindrical structures, we employ numerical continuation, similar to our approach for closed vesicles. However the analysis shows some discrepancies with experiments.

Atomic Bose-Einstein condensates are routinely produced in many laboratories, and their control and detection has been developed to the point where we can consider these systems as quantum/classical field simulators. The terminology of simulator implies that the readout of the field configurations as well as the control of the initial condition are under perfect control, allowing to dial in questions of interest. I will report on two of our recent results. The first is an extension of our result on the study of quantum field dynamics of expanding spacetime [1], namely the observation of square lattice structures in oscillating spacetime [2][3]. Using our control of the local densities of our quasi-two-dimensional quantum gas of potassium atoms, we also characterise the emergent pattern via its excitations. In doing so, we go beyond the mere demonstration of pattern formation by classifying the emergent pattern as a new class of materials - smectic-A superliquid crystal phase [4][5]. As a second example, I will present our very recent findings on sine-Gordon type solitons in spinor condensates. Here, we use the quasi-one-dimensional spinor gas of rubidium. With the spatially resolved readout of the nematic observables and the ability to imprint specific phase patterns, we show that a long-lived solitonic structures can be prepared in the spinor phase. Their spatial shape as well as their collisional properties suggest that these solitons are to a good approximation sine-Gordon solitons. References [1] C. Viermann, et al. , Nature, 611, 260 (2022). [2] N. Liebster, PRX 15, 011026 (2025). [3] K. Fuji, et al. Phys. Rev. A, 109, L051301 (2024). [4] J. Hoffmann, and W. Zwerger, J. Stat. Mech., 2021, 033104 (2021). [5] N. Liebster, arXiv:2503.10519 (2025).

We numerically examine the effects of superfluid shear in two-dimensional harmonically trapped rotating Bose-Einstein condensates. Our approach uses a modified Gross-Pitaevskii equation that allows for spatially dependent damping of the condensate, such as would be provided by a spatially inhomogeneous thermal atomic cloud. With an initial condensate in equilibrium in a rotating frame of reference, the subsequent application of spatially dependent damping enables us to study the effects of continuous superfluid shear and differential rotation. We observe the emergence of coherent structures in the non-equilibrium condensate within certain regimes of damping and rotation, with possible connections to structures and flow patterns observed in the dynamics of classical fluids.

Exciton-polariton condensates, formed by the strong coupling between excitons and photons in semiconductor microcavities, present a rich platform for exploring nonlinear phenomena in driven-dissipative quantum fluids. The inherent nonlinearity arising from exciton-exciton interactions enables the formation of solitonic structures, such as dark and bright solitons, and vortex states, characterized by phase singularities and quantized circulation, allowing the study of topological excitations in non-equilibrium condensates. At higher densities, polariton condensates enter a turbulent regime, where coherent vortices and disordered flows coexist, leading to cascades reminiscent of classical fluid turbulence but with quantum signatures. Additionally, exciton-polaritons can be engineered to interact with photonic bound states in the continuum (BICs), creating localized states that remain confined despite residing in the spectral continuum. These bound states, supported by destructive interference and symmetry protection, exhibit enhanced nonlinear response and provide a novel pathway for manipulating polariton interactions. The combination of solitons, vortices, turbulence, and BICs in polariton condensates underscores the versatility of these systems as platforms for investigating emergent nonlinear dynamics and exploring analogies between fluid mechanics, quantum optics, and condensed matter physics.

We present several numerical tools using classical finite elements with mesh adaptivity to solve the Bogoliubov-de Gennes (BdG) eigenvalue problem, representing a linearization of the Gross-Pitaevskii (GP) equation. The BdG model is commonly used to assess on the stability of stationary solutions of the GP equation. Revealing the existence of such solutions (wave configurations, exotic vortices or vortex structures) in Bose-Einstein condensates is an active and timely research topic in experimental and theoretical physics. For the computation of the GP stationary solutions we use a Newton algorithm coupled with a continuation method exploring the parameter space (the chemical potential or the interaction constant). Bogoliubov-de Gennes equations are then solved using dedicated libraries (ARPACK or SLEPc) for the associated eigenvalue problem. For 3D configurations, we use domain decomposition techniques enabling parallel computing. The programs are written as toolboxes for FreeFem++ (www.freefem.org), a free finite-element software, allowing to easily implement various numerical algorithms. Validations and illustrations are presented for computing difficult configurations with vortices observed in physical experiments: dark solitons, single-line vortex, dark/anti-dark solitons in one or two-component Bose-Einstein condensates. I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels and P. G. Kevrekidis, Vector dark-antidark solitary waves in multicomponent Bose-Einstein condensates, Phys. Rev. A 94, 053617, 2016. G. Sadaka, V. Kalt, I. Danaila and F. Hecht, A finite element toolbox for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates, Computer Physics Communications, 294, 108948(1-17), 2024. G. Sadaka, P. Jolivet, E. G. Charalampidis, I. Danaila, Parallel finite-element codes for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates, 306, p. 109378(1-17), 2025.

Scale-by-scale scalar statistics are theoretically investigated from first principles, with particular emphasis on higher-order moments, which reflect rare or extreme events. The explicit dependence of scalar statistics on large-scale gradients, advection, waves, and coherent structures is highlighted. The investigated scalar quantities include kinetic energy, temperature, humidity, and salinity. Examples illustrating these phenomena are provided for various flow conditions, including: i) Direct Numerical Simulation (DNS) of homogeneous isotropic turbulence subjected to mean scalar gradients. ii) Quantum turbulence as described by the Hall–Vinen–Bekharevich–Khalatnikov (HVBK) model. iii) Atmospheric flows simulated numerically using the Weather Research and Forecasting (WRF) model with Large Eddy Simulation (LES). This example specifically addresses heat waves over France. Collectively, these diverse examples underscore the strong dependence of small-scale scalar statistics on large-scale flow dynamics.

I will discuss one-dimensional Bose-Einstein condensates in quasi-periodic potentials formed by the superposition of two lattices with incommensurate periods and comparable amplitudes. In this setting, all atomic states with energies below the mobility edge become localized, giving rise to a variety of unconventional linear and nonlinear dynamical regimes. The system supports the realization of bosonic Josephson junctions involving pairs, quartets, or even larger groups of coupled modes localized at distinct spatial positions; exhibits oscillatory dynamics under a weak additional lattice tilt, corresponding to Bloch-Landau-Zener oscillations in a quasiperiodic potential; and enables lattice dynamics governed solaly by nonlinear coupling.

A quantum graph is a metric graph with a coordinate system defined along each edge. Functions are defined on the edges subject to consistency conditions at the vertices. Any mathematical question on a quantum graph should be amenable to numerical computations. The QGLAB project set out to make setting up and solving such problems simple, allowing users to easily construct and discretize quantum graphs and their Laplacian operators, solve linear and nonlinear problems, and visualize the results while working at a high level of abstraction. The principal idea underlying this package is the rectangular differentiation matrix, which allows the vertex conditions to be handled straightforwardly for a wide variety of problems.

We simulate the dynamics of about 600 quantum vortices in a spinning-down cylindrical container using a Gross–Pitaevskii model. For the first time, we find convincing spatial-temporal evidence of avalanching behaviour resulting from vortex depinning and collective motion. During a typical avalanche, around 10 to 20 vortices exit the container in a short period, producing a glitch in the superfluid angular momentum and a localised void in the vorticity. After the glitch, vortices continue to depin and circulate around the vorticity void in a similar manner to that seen in previous point vortex simulations. We also show that the effective Magnus force can be used to predict when and where avalanches will occur.

In quantum matter, vortices are topological excitations characterized by quantized circulation of the velocity field. They are often modeled as funnel-like holes around which the quantum fluid exhibits a swirling flow. In this perspective, vortex cores are nothing more than empty regions where the superfluid density goes to zero. In the last few years, this simple view has been challenged and it is now increasingly clear that, in many real systems, vortex cores are not that empty: thermal atoms, quasi-particle excitations, tracer atoms, just to name a few examples, can be commonly found in the cores of quantum vortices. These particles provide the vortex with an effective inertial mass. In this talk, I will discuss the dynamics of two-dimensional point-like vortices whose cores are filled by massive particles [1]. I will show that the introduction of core mass in the standard point-vortex model constitutes a singular perturbation, as it alters the order of the equations of motion [2]. I will also discuss the new dynamical regimes that are unlocked by the presence of core mass. The simplest example is a single vortex within a rigid circular boundary, where a massless vortex can only precess uniformly. In contrast, the presence of a massive core can lead to small-amplitude radial oscillations, which are, in turn, clear signatures of the associated inertial effect. I will also show that, as opposed to their massless counterpart [3], massive vortices can collide, resulting into vortex/antivortex annihilation processes or into the stabilization of doubly charged vortices [4]. Eventually, I will demonstrate that, in Fermi superfluids, vortices have an intrinsic inertial mass, originating from the normal component localized at their cores [5] and that this mass modifies the instability growth rates of many-vortex systems [6]. References: [1] A. Richaud, V. Penna, R. Mayol, and M. Guilleumas, Phys. Rev. A 101, 013630 (2020) [2] A. Richaud, V. Penna, and A. L. Fetter, Phys. Rev. A 103, 023311 (2021) [3] M. Caldara, A. Richaud, P. Massignan, and A. L. Fetter, SciPost Phys. 17, 039 (2024) [4] A. Richaud, G. Lamporesi, M. Capone, and A. Recati, Phys. Rev. A 107, 053317 (2023) [5] A. Richaud, M. Caldara, P. Massignan, M. Capone, and G. Wlazłowski, arXiv:2410.12417 [6] M. Caldara, A. Richaud, M. Capone, and P. Massignan, SciPost Phys. 17, 076 (2024)

We present recent results on the dynamics of solitons in the sine-Gordon equation modified with a Caputo-type fractional temporal derivative and a Riesz-type fractional spatial derivative. Our findings show that the fractional time derivative induces a deceleration in kink motion, while the fractional spatial derivative enables the formation of kink–antikink bound states resembling breathers. We also briefly discuss the effects of both types of fractional derivatives on breather dynamics, in both continuous and discrete frameworks.

Each algebraic soliton of the massive Thirring model corresponds to a simple embedded eigenvalue in the Kaup-Newell spectral problem and attains the maximal mass among the family of solitary waves traveling with the same speed. I will review a hierarchy of rational solutions describing dynamics of nearly identical algebraic solitons with a quantized mass. Each solution of the family corresponds to an embedded eigenvalue of a higher algebraic multiplicity. Dynamics suggest that the algebraic solitons are stable coherent structures.

This talk is divided in two parts. In the first part I will brefly review the Whitham modulation equations for the Kadomtsev-Petviashvili equation and their application to study the dynamics of bent solitons as well as oblique interactions between soliton stems and dispersive shock waves. The second part of the talk will be devoted to the oblique collisions and dynamical interference patterns of two-dimensional dispersive shock waves, which are studied numerically and analytically via the temporal dynamics induced by wedge-shaped initial conditions. Various asymptotic wave patterns are identified, classified and characterized in terms of the incidence angle and the amplitude of the initial step, which can give rise to either subcritical or supercritical configurations, including the generalization to dispersive shock waves of the Mach reflection and expansion of viscous shocks and line solitons, and an eightfold amplification of the amplitude of an obliquely incident flow upon a wall at the critical angle.

The integrable discrete Manakov system is a vector generalization of the Ablowitz-Ladik model, and it admits a variety of discrete vector soliton solutions, referred to as fundamental solitons, fundamental breathers, and composite breathers. In this talk, a full characterization of the interactions of these solitons and breathers, including the explicit forms of their polarization vectors before and after the interaction, will be given. Additionally, the results will be interpreted in terms of a Yang-Baxter refactorization property for the transmission coefficients associated with the interacting solitons.

We consider Maxwell equations for Kerr nonlinear dispersive (i.e. frequency dependent) media under the presence of material interfaces; a possible application being to surface plasmons. Due to the inherent losses, monochromatic solutions with real frequencies do not exist. In the linear setting the problem amounts to a non-self-adjoint Maxwell operator pencil. We determine its spectrum and in the spatially one dimensional case use a simple eigenvalue as the main frequency in the construction of a polychromatic solution in the nonlinear setting. This polychromatic solution has the form of an infinite series of odd harmonics. Detailed resolvent estimates are needed to prove the convergence of the series. In the time domain dispersive media correspond to a memory effect in the polarization. We show that polychromatic solutions require that the memory be only over a finite interval in the past. This work is in collaboration with Max Hanisch (Halle) and Runan He (Madrid).

In this talk, we aim to highlight the crucial role that internal vibrational modes play in the dynamics of topological defects. Using paradigmatic examples such as the kink solution in the φ⁴ model and vortices in the Abelian Higgs model, we illustrate how their dynamics can be significantly modified when these defects are excited. In particular, we discuss how such excitations can lead to unexpected behaviors in scattering processes, both in kink-antikink collisions and in vortex interactions.

We consider the problem of transferring an electron along a lattice of phonons obeying periodic on-site and linear longitudinal interactions in Holstein’s approach. Then, we find that the long wave limit produces the coupling between the linear Schrödinger (LS) and sine-Gordon (sG) equations. Taking advantage of the existence of trapped Kink-Anti Kink solutions in the sG equation, we are able to variationally describe traveling localized coupled solutions in the LS-sG system. We show the significance of permitting longitudinal interactions in the Holstein’s approach in order to hold trapped localized solutions. It is also shown a critical ratio between longitudinal and on-site interactions, as depending on the velocity of propagation, from where coupled localized solutions exist.

Social dinner @ Carmen de la Victoria

We investigate steady propagation of subsonic phase transition fronts in a viscoelastic chain with up-down-up elastic interaction forces. Employing a combination of semi-analytical computations, numerical iterations of a nonlinear map and direct numerical simulations, we obtain traveling wave solutions, which are periodic modulo a shift by one lattice spacing, as well as bifurcating solution branches that feature period doubling. For a case of piecewise linear elastic interactions, we systematically investigate linear stability of traveling wave solutions using the Floquet analysis. This analysis is complemented by numerical simulations that explore the fate of unstable solutions perturbed along the corresponding eigenmodes and identify additional bifurcating branches. We show that smooth approximations of the piecewise linear interactions may have a significant effect on stability of low-velocity motion and the form of bifurcating branches.

This talk will focus on the formation of rogue waves, in the form of time-periodic and spatially localized solutions known as Kuznetsov-Ma (KM) breathers in the discrete, focusing nonlinear Schr\""odinger (DNLS) equation and Ablowitz-Ladik (AL) systems. In doing so, we will investigate the existence, stability and dynamics of KM breathers in the Salerno model which itself interpolates between the DNLS and AL systems. We will explore the configuration space of KM breathers by varying the homotopy parameter associated with the Salerno model (connecting the AL and DNLS models) as well as the period of the solution. We will show that on one hand, the KM breather in the AL model is not the only one solution since more KM solutions bearing oscillatory tails are shown to be present therein. On the other hand, and as per the DNLS model, novel KM breathers will be presented in this case. Then, upon using a proximity argument, KM breathers on a flat background will be shown to exist for the DNLS case. The results will be complemented by discussing the stability of the solutions using Floquet theory and direct dynamical simulations. More recent results on the defocusing AL system will be presented too (if time permits) and open problems and questions will be discussed.

Supratransmission [1] is a fascinating nonlinear wave phenomenon, which enables plane wave transmission through frequency band gaps. This phenomenon reveals the type of waves that can exist in discrete systems [2-3] apart from its numerous applications in physics and engineering. In this work, we generate the discrete rogue waves [4] by the supratransmission way with a chain of pendula submitted to a harmonic-driving source with constant amplitude and parametrical excitation. This opens the door for the application of discrete rogue waves within a simple device. $$ $$ This is join work with Masayuki Kimura, Yusuke Doi and Juan FR Archilla. $$ $$ Funding: Alain B Togueu Motcheyo is supported by grant VII PPITUS-2025 from the University of Seville., MK acknowledges support from JSPS Kakenhi (C) No. 24K07393, YD acknowledges JSPS Kakenhi No. (C) No. 24K14978. JFRA thanks project MICIU PID2022-138321NB-C22. $$ $$ [1] F. Geniet and J. Leon, Energy transmission in the forbidden band gap of a nonlinear chain, Phys. Rev. Lett. 89, (2002) 134102. [2] A. B. Togueu Motcheyo, M. Kimura, Y. Doi and C. Tchawoua, Supratransmission in discrete one- dimensional lattices with the cubic-quintic nonlinearity, Nonlinear Dyn 95, (2019) 2461. [3] A. B. Togueu Motcheyo, J. E. Macias-Diaz, Nonlinear bang gap transmission with zero frequency in cross-stitch lattice, Chaos, Solitons and Fractals 170 (2023) 113349. [4] A. B. Togueu Motcheyo, M. Kimura, Y. Doi and, Juan F.R. Archilla, Nonlinear bandgap transmission by discrete rogue waves induced in a pendulum chain, Physics Letters A 497 (2024) 129334.

Many-body Hamiltonian systems describe a rich variety of physical systems and natural phenomena that range from atomic scale particles to celestial bodies. In recent years there has been an increasing interest in non-equilibrium phenomena which appear in such systems. This talk will focus on the Fermi-Pasta-Ulam-Tsingou (FPUT) model and its relevance to the integrable Toda lattice. In particular, we discuss the role of Toda integrals in the FPUT model for identifying energy diffusion and estimating equilibrium times. Finally, we examine the presence of KAM tori regimes at low energies and how their structure evolves as the system size increases.

We present a new method for monitoring snow and lakes in mountainous regions using synthetic aperture radar (SAR) satellite imagery. SAR data are beneficial for Earth observation due to their comprehensive information content, allowing for detecting changes in snow cover and water bodies among other environmental applications. SAR data are sensitive to various physical properties, including water in the snow and the radar backscattering mechanisms in the observed areas. One key advantage of SAR data is their ability to monitor mountain ranges regardless of weather conditions. However, the analysis and interpretation of SAR data are pretty complex as they do not provide direct information about snow or open water; instead, they provide indirect information through backscatter measurements that require careful interpretation. Our method is inspired by the spatially discrete Nagumo equation and utilizes bistable dynamics to effectively segment SAR composite images. We enhance this approach by substituting the discrete Laplacian with a long-range coupling operator that incorporates terrain information (altitude, slope, and orientation) to adjust the coupling strength between the image's pixels. We validated our snow-detection algorithm during the 2017–2018 season on a steep mountain massif in the French Alps, obtaining better results than the Copernicus European reference wet snow products (James et al. 2024). More recently, we have upgraded our algorithm by incorporating additional data, such as cloud-free optical satellite images, to accurately detect and monitor lakes in the French Alps, including challenging situations like glacial lakes.

Building on a recent energy-based theory of autoresonance in chains of coupled oscillators, we investigate this phenomenon in both isolated oscillators and networks of coupled nonlinear oscillators. In the linear limit, we derive analytical expressions for the forces that sustain autoresonance. These results are subsequently extended to nonlinear networks using a perturbative approach.

Breathers are considered a rare phenomenon in the context of nonlinear PDEs. Going beyond the case of completely integrable PDEs, one can prove the existence of breathers for PDEs with spatially varying coefficients whose periodicity is carefully tailored. We will discuss the stability of such breathers and will give some preliminary results obtained using numerical stability and bifurcation analysis. We will further discuss possibilities for a less stringent notion of stability of families of solutions that are in some sense close enough to the original breather. This will be motivated through the use of this result in nonlinear photonics. This is join work with Panos Kevrekidis and Jesús Cuevas-Maraver.

The existence of a local curve of corotating vortex pairs was proven by Hmidi and Mateu via a desingularization of a pair of point vortices. In this talk, we will present a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. This is a collaboration with Susanna V. Haziot.

Systems whose physical properties change in time have gained significant recent attention in many scientific fields. In this talk, a time-varying system in the realm of pressure waves will be examined. It will be shown that “wavenumber gap breathers exists”, namely ones that are localized in time, periodic in space and have wavenumber in a gap. This solution is a direct counterpart of the famous “breather” solution, which is localized in space and periodic in time. Numerical simulations, experiments and rigorous analysis will be brought together to study this novel solution within the emerging area of time varying media.

A wide variety of stationary or moving spatially localized structures (LSs) is present in evolution problems governed by high-order spatially reversible nonlinear partial differential equations. LSs arise due to a double balance between nonlinearity and spatial coupling on the one hand, and energy dissipation and gain on the other hand. In general, the key ingredients for LS formation are {\it bistability} and {\it front pinning} [Int. J. Bifurc. Chaos {\bf 12}, 2445 (2002)]. Bistability means that two different, but potentially stable states of the system, coexist over a range of parameter values. When bistability is provided, forward and backward fronts connecting these coexisting states may form and, under some conditions, they may lock, leading to LSs of different extensions.\\ There are two bistable scenarios that generally lead to spatial localization. In one of them, the two coexisting states are spatially uniform ({\it uniform-bistability}), and LSs consist of one uniform plateau embedded in the other state. These states organize in a bifurcation structure known as {\it collapsed homoclinic snaking} [Phys. D, {\bf 206}, 82 (2005)]. Another typical example of bistability appears when a uniform state undergoes a subcritical Turing bifurcation, creating a nonuniform spatially periodic state that coexists with the former one. We refer two this scenario as {\it Turing bistability}. The LSs emerging here consist in a slug of the pattern surrounded by the uniform state, and undergo a bifurcation structure known as {\it standard homolinic snaking} [Phys. D, {\bf 129}, 147 (1999)].\\ These two bistable scenarios may coexist in the same system, and even in the same parameter regime, leading to {\it tristability}, where two uniform states coexist with a subcritical Turing pattern. This last scenario leads to an extraordinarily great variety of new LS configurations and complex bifurcation schemes which involve the transition between standard and collapsed homoclinic snaking via the formation of new types of hybrid states [SIAM J. Ap. Dyn. Sys. 22 (4), 2693-2731 (2023)].\\ In this work, we characterize such transition in different pattern forming models including the prototypical Swift-Hohenberg equation [SIAM J. Ap. Dyn. Sys. 22 (4), 2693-2731 (2023)], the FitzHugh-Nagumo equation[arXiv:2501.10271v1], and the Lugiato-Lefever equation [Chaos, Solitons & Fractals, 186, 115201 (2024)]. We find that the transformation between the previous bifurcation structures is related to a sequence of {\it necking bifurcations} where the aforementioned structures merge with a number of coexisting isolas, forming new hybrid bifurcation organizations.\\

We investigate the linear stability of two exact stationary solutions of the parametrically driven, damped nonlinear Dirac equation. By linearizing the system around these solutions and analyzing the resulting eigenvalue problem, we establish that one solution is always unstable, confirming earlier predictions from a variational approach. In contrast, the second solution can be stabilized by sufficiently strong dissipation. An oscillatory instability, previously reported in the context of the nonlinear Schrödinger equation, is also observed here. We derive the stability curve that delineates stable and unstable regions in parameter space and examine how this diagram depends on the driving frequency.

This tutorial introduces machine learning tools for discovering, modeling, and predicting coherent structures in nonlinear dynamical systems. Emphasis will be on structure-preserving and data-driven methods, with examples from soliton-bearing PDEs and lattice systems.