We consider the extended Korteweg-de Vries (eKdV) equation as a model for long moderately nonlinear surface water waves. In the slow time formulation this equation generates fast propagating resonant radiation due to the non-convexity of its linear dispersion curve, which is not present in the strongly nonlinear Serre-Su-Gardner-Green-Naghdi (SSGGN) parent system. We show that the extended KdV-Whitham approximation and the slow space formulation of the eKdV equation are suitable regularisations of the eKdV equation in several cases of interest, and even for moderate amplitudes. Numerical comparisons are made between the SSGGN system and the respective reduced models, where simulations are initiated with an approximate soliton solution of the eKdV equation, constructed by use of Kodama-Fokas-Liu near-identity transformation to the KdV equation.

We use a difference Lax form to construct simultaneous integrals of motion of the fourth Painlevé equation and the difference second Painlevé equation over fields with finite characteristic $p>0$. For $p\neq 3$, we show that the integrals can be normalised to be completely invariant under the corresponding extended affine Weyl group action. We show that components of reducible fibres of integrals correspond to reductions to Riccatti equations. We further describe a method to construct non-rational algebraic solutions in a given positive characteristic. We also discuss a projective reduction of the integrals.

We study large but finite neural networks that, in the thermodynamic limit, admit an exact low-dimensional mean-field description. We assume that the governing mean-field equations describing macroscopic quantities such as the mean firing rate or mean membrane potential are known, while their parameters are not. Moreover, only a single scalar macroscopic observable from the finite network is assumed to be measurable. Using time-series data of this observable, we infer the unknown parameters of the mean-field equations and reconstruct the dynamics of unobserved (hidden) macroscopic variables. Parameter estimation is carried out using the differential evolution algorithm. To remove the dependence of the loss function on the unknown initial conditions of the hidden variables, we synchronize the mean-field model with the finite network throughout the optimization process. We demonstrate the methodology on two networks of quadratic integrate-and-fire neurons: one exhibiting periodic collective oscillations and another displaying chaotic collective dynamics. In both cases, the parameters are recovered with relative errors below $1\%$ for network sizes exceeding 1000 neurons.

In this paper discrete equations are derived from Bäcklund transformations of the fifth Painlevé equation, including a new discrete equation which has ternary symmetry. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalised Laguerre polynomials and the other in terms of the generalised Umemura polynomials, both of which can be expressed as Wronskians of Laguerre polynomials. Hierarchies of rational solutions of the discrete equations are derived in terms of the generalised Laguerre and generalised Umemura polynomials. It is known that there is nonuniqueness of some rational solutions of the fifth Painlevé equation. Pairs of nonunique rational solutions are used to derive distinct hierarchies of rational solutions which satisfy the same discrete equation.

Higher-order networks encode the many-body interactions of complex systems ranging from the brain to biological transportation networks. Simplicial and cell complexes are ideal higher-order network representations for investigating higher-order topological dynamics where dynamical variables are not only associated with nodes, but also with edges, triangles, and higher-order simplices and cells. Global Topological Synchronization (GTS) refers to the dynamical state in which identical oscillators associated with higher-dimensional simplices and cells oscillate in unison. On standard unweighted and undirected complexes this dynamical state can be achieved only under strict topological and combinatorial conditions on the underlying discrete support. In this work we consider generalized higher-order network representations including directed and hollow complexes. Based on an in depth investigation of their topology defined by their associated algebraic topology operators and Betti numbers, we determine under which conditions GTS can be observed. We show that directed complexes always admit a global topological synchronization state independently of their topology and structure. However, we demonstrate that for directed complexes this dynamical state cannot be asymptotically stable. While hollow complexes require more stringent topological conditions to sustain global topological synchronization, these topologies can favor both the existence and the stability of global topological synchronization with respect to undirected and unweighted complexes.

In the present work we study the nucleation of Dispersive shock waves (DSW) in the {defocusing}, discrete nonlinear Schr{ö}dinger equation (DNLS), a model of wide relevance to nonlinear optics and atomic condensates. Here, we study the dynamics of so-called dam break problems with step-initial data characterized by two-parameters, one of which corresponds to the lattice spacing, while the other being the right hydrodynamic background. Our analysis bridges the anti-continuum limit of vanishing coupling strength with the well-established continuum integrable one. To shed light on the transition between the extreme limits, we theoretically deploy Whitham modulation theory, various quasi-continuum asymptotic reductions of the DNLS and existence and stability analysis and connect our findings with systematic numerical computations. Our work unveils a sharp threshold in the discretization across which qualitatively continuum dynamics from the dam breaks are observed. Furthermore, we observe a rich multitude of wave patterns in the small coupling limit including unsteady (and stationary) Whitham shocks, traveling DSWs, discrete NLS kinks and dark solitary waves, among others. Besides, we uncover the phenomena of DSW breakdown and the subsequent formation of multi-phase wavetrains, due to generalized modulational instability of \textit{two-phase} wavetrains. We envision this work as a starting point towards a deeper dive into the apparently rich DSW phenomenology in a wide class of DNLS models across different dimensions and for different nonlinearities.

We study the bifurcation scenario of a three-degree-of-freedom Hamiltonian system, a model based on the Lagrange restricted 3-body problem: a test particle moving in the gravitational field of a pair of interacting dwarf galaxies. The phase space of this system has 3 fundamental normally hyperbolic invariant manifolds (NHIMs) and their invariant stable and unstable manifolds form homoclinic/heteroclinic tangles. As the perturbation parameter increases, the NHIMs begin to lose normal hyperbolicity and their constituent KAM tori break, creating transient chaotic dynamics around them. We also observe a certain kind of coordination between the bifurcation scenarios of these NHIMs. We analyse this phenomenon using Poincaré maps and the delay time function.

Understanding the emergence of cooperation in social networks has advanced through pairwise interactions, but the corresponding theory for group-based public goods games (PGGs) remains less explored. Here, we provide theoretical conditions under which cooperation thrives in PGGs on arbitrary population structures, which are accurate under weak selection. We find that a class of networks that would otherwise fail to produce cooperation, such as star graphs, are particularly conducive to cooperation in PGGs. More generally, PGGs can support cooperation on almost all networks, which is robust across all kinds of model details. This fundamental advantage of PGGs derives from self-reciprocity realized by group separations and from clustering through second-order interactions. We also apply PGGs to empirical networks, which shows that PGGs could be a promising interaction mode for the emergence of cooperation in real-world systems.