ABSTRACT We discuss traveling fronts in several coupled reaction–diffusion systems that model predator–prey interactions, as well as other systems that share similar structural features. We review some cases where the traveling fronts exist and are small perturbations of fronts in certain scalar equations of Fisher–KPP or Burgers–FKPP type. As a new result consistent with the reviewed cases, we establish the existence of a previously unobserved family of traveling fronts in a mussel–algae interaction model, in a parameter regime complementary to those studied earlier. These fronts are constructed as perturbations of fronts of a Fisher–KPP‐type scalar equation using geometric singular perturbation theory.
ABSTRACT In this paper, we develop a unified framework based on generalized principal eigenvalues to study spreading speeds in time‐heterogeneous nonlocal diffusion models. First, for time‐heterogeneous environments with a positive uniform mean value, we investigate the properties and interrelationships of three types of generalized principal eigenvalues and establish sign criteria for them. Next, we use these signs to select appropriate test functions and explicitly construct upper and lower solutions on unbounded domains, thereby characterizing spreading speeds in two scenarios. One concerns a Fisher‐KPP equation with compactly supported initial data, and the other involves predator–prey systems with Holling‐type I–III functional responses, considering both coinvasion and predator‐invasion initial data. Overall, this framework offers an alternative perspective for quantifying the spatiotemporal evolution of solutions.
ABSTRACT We show that the removal of a type of reverse reactions preserves the absence of Hopf bifurcations in mass‐action reaction networks, or more precisely, the absence of single pairs of purely imaginary eigenvalues of the Jacobian of the system. The motivation is to answer the open question of whether the dual phosphorylation mechanism requiring two encounters of the enzyme and the substrate for both phosphorylation and dephosphorylation admits periodic trajectories. With the tools of model reduction, we provide evidence in favor of the absence of oscillations of this network by precluding Hopf bifurcations in any reduced network comprising three out of its four intermediate protein complexes. Our argument relies on a detailed analysis of the semi‐algebraic conditions precluding Hopf bifurcations obtained from Hurwitz determinants of the characteristic polynomial of the Jacobian of the reduced network.
ABSTRACT We consider the soliton solutions of a recently proposed coupled Sasa‐Satsuma‐modified Korteweg‐de Vries (Sasa‐Satsuma‐mKdV) equation using the Kadomtsev–Petviashvili reduction method. Under zero, nonzero and mixed boundary conditions, we derive four distinct classes of soliton solutions: bright–bright, dark–dark, bright–dark, and dark–bright. These solutions are derived from the vector Hirota equation, for which the bright, dark, and bright–dark soliton solutions are provided in the Appendix. We perform asymptotic analysis of soliton collisions for each class of solutions. Inelastic collisions are observed between bright–bright solitons. In the dark–dark case, the soliton structures are similar to those of the Sasa‐Satsuma equation which include double‐hole, Mexican hat and anti‐Mexican hat solutions. The collision between these solitons and kink solitons and the resonances between dark–dark solitons and breather–breather are new phenomena found in the coupled Sasa‐Satsuma‐mKdV equation.
ABSTRACT In this paper, we continue the study of bifurcation standing wave solutions of ‐systems started by Chen–Iooss based on Lyapunov–Schmidt method. Our study is to generalize their theories for multiple‐dimensional kernels. More precisely, we will provide wave patterns of an ‐system with both one‐dimensional kernel and two‐dimensional kernels.
ABSTRACT We consider a quasi‐one‐dimensional Poisson–Nernst–Planck (PNP) model of ionic transport through a membrane channel, incorporating two oppositely charged ion species, nonzero permanent charge distributions, and nonuniform finite ion sizes modeled via hard‐sphere potentials. First, we establish a rigorous existence and local uniqueness result for the steady‐state PNP system using geometric singular perturbation theory, treating the finite ion sizes as small parameters. Second, we conduct a detailed asymptotic analysis of zero‐current ionic flows for small permanent charge. This analysis reveals a nonlinear interplay between finite ion size effects and fixed charge, including the existence of a critical membrane potential at which the influence of permanent charge on the ionic flux reverses sign. These findings highlight novel nonlinear effects arising from finite ion size and permanent charge in ionic flows, and offer insights into biological ion channel behavior that are beyond the reach of current experimental techniques.
ABSTRACT In this paper, we develop a method for synthesis of the Weyl matrix for Schrödinger operators on carbon nano‐structures which are equivalent to the hexagonal lattice. We construct Weyl matrices for graphs that contain any number of hexagons by adding new edges and solving elementary systems of linear algebraic equations at each step. Our method can be applied in numerical simulations for studying inverse spectral problems for differential operators on hexagonal lattices.
ABSTRACT This paper systematically analyzes complex dynamics of a 3D quadratic Jerk system without equilibrium or with infinitely many equilibria, focusing on the generation mechanisms of nonchaotic behaviors and hidden chaos. First, Lyapunov stability of a line of nonhyperbolic equilibria is analyzed. Further, the concept of Jacobi stability is extended to the case where the deviation curvature tensor has a zero eigenvalue, thereby obtaining that a line of nonhyperbolic equilibria is Jacobi unstable for certain parameters. Second, we analyze three special cases: (i) a linear Jerk system for which the induced flow is proved to be weakly Li–Yorke chaos in weak topology and which has infinitely many periodic orbits; (ii) a completely integrable Jerk system, which has a homoclinic orbit and a family of periodic orbits; (iii) a simple nonlinear Jerk system, for which we prove that there exist infinitely many singularly degenerate heteroclinic cycles when invariant algebraic surfaces vanish. Third, we analyze the global dynamics on the invariant algebraic surfaces of Jerk system, covering IMSDHCs, Jacobi stability, and dynamics at infinity. Fourth, we use the averaging theory to prove that a hidden periodic orbit bifurcates from a nonisolated zero‐Hopf equilibrium at the origin and nonorigin equilibrium, and establish, based on KAM theorem, the existence of hidden nested invariant tori surrounding this periodic orbit. Meanwhile, there are island chains and hidden chaos near nested invariant tori. Finally, Poincaré map shows that hidden chaos is generated via the Feigenbaum period‐doubling route, and its existence is verified based on the topological horseshoe theory with a computer‐assisted proof.
ABSTRACT Exact solutions for the steady normal shock in the Navier–Stokes–Fourier framework for calorically perfect Newtonian gases are presented. The governing equations are reduced to an Abel differential equation of the second kind for general local transport laws of the longitudinal viscosity and thermal conductivity that maintain a constant longitudinal Prandtl number, while shock profiles are obtained by subsequent quadrature. For certain combinations of the specific heat ratio , longitudinal Prandtl number and upstream Mach number , exact solutions are obtained on one‐parameter curves in ‐space. The resulting longitudinal Prandtl numbers range from values close to zero up to 36. Exact shock profiles are developed for the constant‐coefficient case. Most solutions determine the fluid velocity implicitly, while one solution yields an explicit representation of the velocity as a function of the spatial coordinate.
ABSTRACT In this paper we consider the fourth‐order (or biharmonic) nonlinear Schrödinger (NLS) equation in dimensions 1, 2, and 3, where the potential term is expressed as a power nonlinearity (for any positive power) and the dispersion operator has the fourth ‐order combined with the lower second ‐order. The fourth order NLS equation has recently attracted the attention of researchers since quartic solitons have been experimentally obtained in optics, and thus, the mathematical theory of solutions to the fourth‐order NLS equation in physical dimensions is timely to develop. In this work we show the local well‐possesses of the fourth‐order or biharmonic NLS equation on a weighted subset of a Sobolev space as well as in spaces.
ABSTRACT We consider the Cauchy problem for the nonlinear Dirac equation on a noncompact ‐star metric graph , where , , and denotes the self‐adjoint Dirac–Kirchhoff operator on . Using Bourgain‐type spaces defined through the spectral resolution of , together with elementary bounds for the Dirac flow and fractional Nemytskii estimates below the trace threshold, we prove local well‐posedness for initial data The corresponding solution belongs to Moreover, is conserved along the solution on the existence interval. We also establish a blow‐up alternative in the combined and space–time control norm.
In this article, we examine a system of hyperbolic balance laws governing macroscopic production model which describes high-volume product flows. Our primary focus is on the nonlinear wave interactions involving shock and rarefaction. Employing the theory of differential constraints, we first demonstrate that the governing system is compatible with a set of differential constraints, which allows us to handle the nonhomogeneous term by making the governing system diagonalized. We show that the governing system admits a strictly convex entropy-entropy flux pair which leads to the well-posedness of the solution. Furthermore, we obtain the solution, that consists of shocks and rarefaction waves, of the Riemann problem as well as the generalized Riemann problem. Consequently, we discuss all possible interactions of elementary waves of the Riemann problem. Here, the primary challenge is to identify the solution structure of the generalized Riemann problem which is formed at the time of collision of shocks and rarefaction waves as solution of the Riemann problem. In addition, we implement an upwind-based splitting scheme to validate the analytical results through various test cases.
We study the long-term dynamics of followers that selectively follow one of multiple leaders on Riemannian manifolds, where the leaders interact through repulsive forces while remaining cohesively bounded. We propose a multileader-follower multiagent system defined on Riemannian manifolds. In our model, each follower chooses exactly one leader among several leaders and follows it, while the leaders interact with each other through repulsive forces that prevent collisions but do not allow excessive dispersion. Through follower-leader interactions, each group of followers converges to its corresponding leader. For the theoretical analysis, we review the Rauch Comparison Theorem and the main geometric concepts related to it. We also introduce the classical Barbalat Lemma. Furthermore, we extend the Barbalat Lemma to a manifold setting so that it can be applied to vector fields on Riemannian manifolds. We then present several sufficient conditions on the initial data, system parameters, and kernel functions. Using a suitable energy function, we obtain several energy estimates, which in particular guarantee collision avoidance among the leaders and leader cohesion in the sense that the interleader distances remain uniformly bounded. By combining these energy estimates with the Rauch Comparison Theorem, the classical Barbalat Lemma, and its manifold extension developed in this paper, we rigorously prove that each selectively assigned group of followers asymptotically converges to its corresponding leader on Riemannian manifolds. Finally, we provide numerical simulations to validate and illustrate our theoretical results.
We consider the initial-boundary value (IBV) problem for the modified Camassa-Holm (mCH) equation on the half-line . We provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann-Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated with the initial and boundary values of the solution, whose compatibility is characterized in spectral terms.
This paper investigates the global dynamics of planar piecewise linear hysteretic systems. First, we analyze the Hopf bifurcation in a general piecewise linear hysteretic systems, corresponding to the classification HLB17 in Simpsons taxonomy (Simpson, 2022, Physics Reports), and then deduced the stability criteria via the derivative of Poincar & eacute; map. Next, we examine a symmetric subclass of such systems, known as planar feedback systems, and prove that they admit at most one limit cycle. Furthermore, we provide a complete characterization of global phase portraits in the Poincar & eacute; disk, elucidating their topological structure.As an application, we construct a physical model consisting of a metal ball confined within a pipe and subjected to magnetic forces from symmetrically placed contacts. This setup yields a planar feedback system, and our analysis reveals two distinct regimes of periodic motion, depending on the relative strength of the magnetic interactions. A comprehensive bifurcation diagram is derived, delineating the transition between these dynamical behaviors.
We study the orbital stability and asymptotic stability problems for KdV solitons on the right half-line for nonhomogeneous boundary conditions in the energy space . This paper improves the results of Cavalcante and Mu & ntilde;oz [Revista Matem & aacute;tica Iberoamericana 35, no. 6 (2019); and SIAM Journal on Mathematical Analysis 55, no. 5 (2023): 4193-5992], which treat the homogeneous case. One of the key components of the stability argument in this work is the refinement of the estimate for the Lyapunov functional, given that the mass and energy, unlike in the context of homogeneous boundary conditions, do not exhibit a dissipative mechanism. As a consequence of this analysis, we obtain global control of the trace of the first and second derivatives of the solution to the model. This is, as far as we understand, the first orbital and asymptotic stability result for solitons posed on a half-line in the context of nonhomogeneous boundary conditions.
The paper studies a singularly perturbed Korteweg-de Vries (KdV) equation for , where is a small parameter and are real constants. The equation arises in the study of surface waves in finite-depth water with small surface tension on the free surface. It is known that the equation admits homoclinic solutions with small oscillatory tails at infinity, called generalized solitary-wave solutions (or generalized one-hump solutions). Here, the multi-hump solutions are formally constructed. First, two-hump solutions are derived by drawing on ideas and methods from rigorous existence studies of two-hump solutions for various equations, including the singularly perturbed KdV equation. Three-hump solutions are then formally obtained via a matching procedure, although a rigorous mathematical justification for the existence of three-hump or other multi-hump solutions remains open. Finally, the ideas and methods for the formal construction of multi-hump solutions with arbitrary number of humps are discussed.
Phenotypic plasticity significantly influences species interactions, especially via inducible defensive mechanisms in predator-prey dynamics. This work proposes and explores a predator-prey scenario whereby the prey species demonstrates inducible defence mechanisms against predators. The model demonstrates a diverse array of complex dynamical characteristics, notably highlighting the significant stabilizing influence of defence on population dynamics. We further investigate the system under conditions of spatio-temporal diffusion inside a confined region, demonstrating that the Turing instability domain diminishes as the defence level escalates, thereby decreasing the probability of spatial pattern emergence. To achieve more realistic ecological interactions, the model is augmented by including a nonlocal factor into the intraspecific competition of the prey population. The study shows that inducible defence suppresses pattern creation in the local model, but adding nonlocal interactions changes this behavior considerably by making the Turing domain wider as the range of interactions gets expanded. These results show how important inducible defence and spatial linkages are for keeping ecosystems stable and diverse in space. In general, the results give new insight into how behavioral adaptations and nonlocal impacts work together to affect species coexistence, pattern development, and the robustness of ecological systems.
This paper presents a unified and systematic approach to constructing the modified Kadomtsev-Petviashvili (mKP), BKP, and modified BKP (mBKP) hierarchies through the representation theory of polynomial Lie algebras. Based on the polynomial Lie algebras gl(infinity)((n)) and so(infinity)((n)), we naturally derive the complete structure of these integrable systems, including bilinear identities, tau functions, wave matrices, dressing operators, and Lax pairs, emerge naturally without external constraints. A key result is the automatic emergence of the BKP constraint L-BKP* = -partial derivative L-BKP(partial derivative-1) from so(infinity)((n)) symmetry, contrasting with traditional approaches where such constraints are artificially imposed. The polynomial algebra framework provides a unifying principle where matrix structures arise intrinsically through fermionic Fock space decompositions, offering a powerful methodology for constructing and analyzing integrable hierarchies in higher dimensions.
This paper analyzes delta-shock wave interactions in a one-dimensional model of magnetohydrodynamics (MHD) described by a nonlinear hyperbolic system of conservation laws. The study is conducted within a distributional framework that generalizes the classical weak solution concept, allowing for Dirac delta-type initial data. A Cauchy problem formulated as a Riemann problem with singularities in both state variables is solved exactly. The resulting solutions display coherent, soliton-like dynamics, and a subclass of distributional solitons is identified as a special case of delta-shock waves. These findings advance the analytical understanding of nonlinear conservation laws with singular data and shed light on novel wave structures in MHD.
