Physica D: Nonlinear Phenomena最新文献

The aim of this work is multifold. Firstly, it intends to present a complete, quantitative and self-contained description of the periodic traveling wave solutions and Whitham modulation equations for the Toda lattice, combining results from different previous works in the literature. Specifically, we connect the Whitham modulation equations and a detailed expression for the periodic traveling wave solutions of the Toda lattice. Along the way, some key details are filled in, such as the explicit expression of the characteristic speeds of the genus-one Toda–Whitham system. Secondly, we use these tools to obtain a detailed quantitative characterization of the dispersive shocks of the Toda system. Lastly, we validate the relevant analysis by performing a detailed comparison with direct numerical simulations.

This paper investigates a continuous time model for the baleen whale population, which is a diverse and widely distributed parvorder of carnivorous marine mammals. We use theoretical and schematic designs to explore stability charts, rightmost characteristic roots, and supercritical Hopf bifurcation of the positive equilibrium. Our research on the Hopf bifurcation and stability of the bifurcating periodic solutions is based on the center manifold reduction and Poincaré normal form theory. Interestingly, the characteristic equation has pure imaginary roots at the second, third, and subsequent critical values. However, Hopf bifurcation theorem is not satisfied because all other characteristic roots of the characteristic equation at these critical values do not have strictly negative real parts, except the pure imaginary roots. We also use the parameter values reported in the previous studies to simulate the unstable periodic solutions at the second and third critical values through bifurcation diagrams. The numerical results obtained under specific parameter values align closely with our theoretical derivations.

Rogue waves in a reverse space nonlocal nonlinear Schrödinger (NLS) equation with real and parity-symmetric nonlinearity-induced potential are considered. This equation has clear physical meanings since it can be derived from the Manakov system with a special reduction. The $N$-fold Darboux transformation and its generalized form for the nonlocal NLS equation are constructed. As an application, the multiparametric $N$th-order rogue wave solution in terms of Schur polynomials for the nonlocal NLS equation with focusing case is derived by the limit technique. The significant differences of rogue wave dynamics between the nonlocal NLS equation and its usual (local) counterpart are illustrated through two types of specific rogue wave solutions. Unlike the eye-shaped (Peregrine type) rogue waves, the rogue wave doublets which involve an eye-shaped rogue wave and a dark/four-petaled rogue wave merging or separating with each other, and the rogue wave sextets that are characterized by the superpositions of three eye-shaped rogue waves and three dark/four-petaled rogue waves with fundamental, triangular and quadrilateral patterns are shown. Moreover, some wave characteristics including the difference between the light intensity and the plane-wave background, and the pulse energy of the rogue wave doublets are discussed.

Traditional discrete learning methods involve discretizing continuous equations using difference schemes, necessitating considerations of stability and convergence. Integrable nonlinear lattice equations possess a profound mathematical structure that enables them to revert to continuous integrable equations in the continuous limit, particularly retaining integrable properties such as conservation laws, Hamiltonian structure, and multiple soliton solutions. The pseudo grid-based physics-informed convolutional-recurrent network (PG-PhyCRNet) is proposed to investigate the localized wave solutions of integrable lattice equations, which significantly enhances the model’s extrapolation capability to lattice points beyond the temporal domain. We conduct a comparative analysis of PG-PhyCRNet with and without pseudo grid by investigating the multi-soliton solutions and rational solitons of the Toda lattice and self-dual network equation. The results indicate that the PG-PhyCRNet excels in capturing long-term evolution and enhances the model’s extrapolation capability for solitons, particularly those with steep waveforms and high wave speeds. Finally, the robustness of the PG-PhyCRNet method and its effect on the prediction of solutions in different scenarios are confirmed through repeated experiments involving pseudo grid partitioning.

We consider a generic higher order mass aggregation term for interactions between particles exhibiting oscillatory clumping and disaggregation behavior in the F ring of Saturn, using a novel predator–prey model that relates the mean mass aggregate (prey) and the square of the relative dispersion velocity (predator) of the interacting particles. The resulting cyclic dynamic behavior is demonstrated through time series plots, phase portraits and their stroboscopic phase maps.

Employing an eigenvalue stability analysis of the Jacobian of the system, we find out that there are two distinct regimes depending on the exponent and the amplitude of the higher order interactions of the nonlinear mass term. In particular, the system exhibits a limit cycle oscillatory stable behavior for a range of values of these parameters and a non-cyclic behavior for another range, separated by a curve across which phase transitions would occur between the two regimes. This shows that the observed clumping dynamics in Saturn’s F ring, corresponding to a limit cycle stability regime, can be systematically maintained in presence of physical higher order mass aggregation terms in the introduced model.

In this paper, we study the vanishing dissipation limit of the 2D anisotropic Boussinesq equations with the Navier-slip boundary condition for velocity field and the fixed flux boundary condition for temperature in the upper half plane. By constructing boundary layer correctors to compensate for the discrepancies between dissipative equations and non-dissipative equations at the boundary, we prove that the solutions of the anisotropic Boussinesq equations converge to the solutions of the non-dissipative Boussinesq equations in $L^2$-norm. Particularly, we find that the anisotropic dissipation coefficients only affect the rate of convergence, which is different from the phenomenon of the Dirichlet problem of the anisotropic Boussinesq equations in Wang & Xu (2021).

Using nonlinear projections and preserving structure in model order reduction (MOR) are currently active research fields. In this paper, we provide a novel differential geometric framework for model reduction on smooth manifolds, which emphasizes the geometric nature of the objects involved. The crucial ingredient is the construction of an embedding for the low-dimensional submanifold and a compatible reduction map, for which we discuss several options. Our general framework allows capturing and generalizing several existing MOR techniques, such as structure preservation for Lagrangian- or Hamiltonian dynamics, and using nonlinear projections that are, for instance, relevant in transport-dominated problems. The joint abstraction can be used to derive shared theoretical properties for different methods, such as an exact reproduction result. To connect our framework to existing work in the field, we demonstrate that various techniques for data-driven construction of nonlinear projections can be included in our framework.

Cahn–Hilliard–Navier–Stokes (CHNS) systems describe flows with two-phases, e.g., a liquid with bubbles. Obtaining constitutive relations for general dissipative processes for such systems, which are thermodynamically consistent, can be a challenge. We show how the metriplectic 4-bracket formalism (Morrison and Updike, 2024) achieves this in a straightforward, in fact algorithmic, manner. First, from the noncanonical Hamiltonian formulation for the ideal part of a CHNS system we obtain an appropriate Casimir to serve as the entropy in the metriplectic formalism that describes the dissipation (e.g. viscosity, heat conductivity and diffusion effects). General thermodynamics with the concentration variable and its thermodynamics conjugate, the chemical potential, are included. Having expressions for the Hamiltonian (energy), entropy, and Poisson bracket, we describe a procedure for obtaining a metriplectic 4-bracket that describes thermodynamically consistent dissipative effects. The 4-bracket formalism leads naturally to a general CHNS system that allows for anisotropic surface energy effects. This general CHNS system reduces to cases in the literature, to which we can compare.

We study the dynamics of the collinear points in the planar, restricted three-body problem, assuming that the primaries move on an elliptic orbit around a common barycenter. The equations of motion can be conveniently written in a rotating–pulsating barycentric frame, taking the true anomaly as independent variable. We consider the Hamiltonian modeling this problem in the extended phase space and we implement a normal form to make a center manifold reduction. The normal form provides an approximate solution for the Cartesian coordinates, which allows us to construct several kinds of orbits, most notably planar and vertical Lyapunov orbits, and halo orbits. We compare the analytical results with a numerical simulation, which requires special care in the selection of the initial conditions.