Narrow Results

A representation formula for second-order nonhomogeneous nonlinear ordinary differential equations (ODEs) has been recently constructed by M. Frasca using its Green’s function, i.e. the solution of the corresponding nonlinear differential equation with a Dirac delta function instead of its nonhomogeneity. It has been shown that the first-order term–the convolution of the nonlinear Green’s function and the right-hand side, analogous to the Green’s representation formula for linear equations — provides a numerically efficient solution of the original equation, while the higher order terms add complementary corrections. The cases of square and sine nonlinearities have been studied by Frasca. Some new cases of explicit determination of nonlinear Green’s function have been studied previously by us. Here, we gather nonlinear equations and their explicitly determined Green’s functions from existing references, as well as investigate new nonlinearities leading to implicit determination of nonlinear Green’s function. Some transformations allowing to reduce second-order nonlinear partial differential equations (PDEs) to nonlinear ODEs are considered, meaning that Frasca’s method can be applied to second-order PDEs as well. We perform a numerical error analysis for a generalized Burgers’ equation and a nonlinear wave equation with a damping term in comparison with the method of lines.

Approximation and analysis are used for investigating accurate soliton solutions of the ill-posed Boussinesq (IPB) equation. The investigated model explains shallow-water gravitational waves. It examines one-dimensional nonlinear strings and lattices. IPB explains small-amplitude surface waves on nonlinear strings and lattices. We provide unique analytical solutions to analyze numerical beginning and boundary conditions. A solution’s quality is judged by its divergence from analytical predictions. Physical wave properties are illustrated.

The Korteweg–de Vries (KdV)-type bilinear equations always allow 2-soliton solutions. In this paper, for a general KdV-type bilinear equation, we interpret how the so-called extended homoclinic orbit solutions arise from a special case of its 2-soliton solution. Two properties of bilinear derivatives are developed to deal with bilinear equation deformations. A non-integrable (3+1)-dimensional bilinear equation is employed as an example.

A nonlinear micro-polar non-Newtonian fluid model is investigated. By the complete discrimination system for polynomial method, we give the classification of the traveling wave patterns and analysis of topological stability and dynamical behaviors. We also discuss the physical realizations and the conditions of existence of these patterns. As a result, when some of parameters are not intrinsic parameters we show that in general only two stable patterns will be observed.

In this paper, the investigation is conducted on a (2 + 1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation for an incompressible fluid. Via the Riemann theta function, periodic-wave solutions are derived, and breather-wave solutions are constructed with the aid of the extended homoclinic test approach. Based on the polynomial expansion method, several traveling-wave solutions are derived. Besides, we observe that the amplitude of the breather keeps unchanged during the propagation and the traveling wave which is kink shaped propagates stably. Furthermore, we analyze the transition between the periodic-wave and soliton solutions, which implies that the periodic-wave solutions tend to the soliton solutions via a limiting procedure.

In this paper, the generalized Kudryashov (GK) approach and the sine-Gordon expansion approach are used for constructing new specific analytical solutions of the deoxyribonucleic acid model, which include the well-known bell-shaped soliton, kink, singular kink, periodic soliton, contracted bell-shaped soliton and anti-bell-shaped soliton. The efficacy of these strategies demonstrates their utility and efficiency in addressing a wide range of integer and fractional-order nonlinear evolution problems. The physical relevance of the demonstrated results has been proven using three-dimensional forms. It is interesting to mention that the solutions achieved here using the provided methods are extra-extensive and may be used to explain the internal interaction of the deoxyribonucleic acid model originating in mathematical biology. The suggested approach was utilized to get exact traveling wave solutions for fractional nonlinear partial differential equations appearing in nonlinear science.

In this paper, we establish the existence and describe the global structure of traveling waves for a class of lattice delay differential equations describing cellular neural networks with distributed delayed signal transmission. We describe the transition of wave profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. We also describe an interval of the wave speed where the structure of the wave solution is unknown since the corresponding profile equation involves distributed argument of both advanced and retarded types, and we present some preliminary numerical simulation to illustrate the complexity.

In this paper we investigate the onset of instabilities in a model describing the propagation of the steady planar premixed combustion wave. In particular, we are interested in determining the Bogdanov–Takens bifurcation condition, which is investigated semi-analytically. We derive an analytic condition for the existence of this type of bifurcation and based on this criterion we numerically determine the parameter values for which the Bogdanov–Takens bifurcation occurs. This numerical method is found to be more efficient than the previous methods.

We study the stability of steady states and establish the existence of traveling waves for a diffusive host-vector epidemic with a nonlocal spatiotemporal interaction. We develop the techniques of contracting-convex-sets, limit argument, singular perturbation and fixed point theorems.

In this paper, we are concerned with the wave propagation for a system of 2-D lattice differential equations with delay. Under the monostable assumption, the asymptotic behavior, the monotonicity and uniqueness of traveling wave are established when the wave speed is greater than or equal to the minimal wave speed c_*(θ) > 0. In addition, the directional dependence of the minimal wave speed is analyzed numerically.

A wide variety of spatio-temporal models are available in literature which are unable to generate stationary patterns through Turing bifurcation. Introduction of nonlocal terms to the same model can produce Turing patterns and this is true even for a single species population model. In this paper, we consider a prey–predator model of Holling–Tanner type with a generalist predator and a nonlocal interaction in the intra-specific competition term of the prey population. Nonmonotonic functional response is assumed to describe consumption rate of the prey by the predator. The Turing instability condition has been studied for the model without the nonlocal term around coexisting steady states. We also determine the Turing domain in the presence of nonlocal interaction term. The spatial-Hopf bifurcation has been studied and it plays an important role to find the pure Turing domain for the nonlocal model. Furthermore, in the presence of nonlocal interaction, the nonlocal model produces traveling wave solution. Using linear stability analysis, we have obtained the wave speed for the traveling wave front analytically. With the help of numerical simulation, we have verified that the speed of the traveling wave front for the complete nonlinear nonlocal model matches with the analytical approximation. The emergence of wave trains has also been established for higher range of nonlocal interaction.

In this paper, the method of dynamical systems developed in [Li & Chen, 2007] is applied to the rotation-two-component Camassa–Holm system. Through qualitative analysis, under given parameter conditions, exact explicit solitary wave solution, pseudo-peakon solution, peakon and periodic peakon, as well as compacton solution, are obtained. Some parameter conditions constraints are derived for ensuring the existence of these solutions.

We consider the dynamical effects of electromagnetic flux on the discrete Chialvo neuron model. It is shown that the model can exhibit rich dynamical behaviors such as multistability, firing patterns, antimonotonicity, closed invariant curves, various routes to chaos, and fingered chaotic attractors. The system enters a chaos regime via period-doubling cascades, reverse period-doubling route, antimonotonicity, and via a closed invariant curve to chaos. The results were confirmed using the techniques of bifurcation diagrams, Lyapunov exponent diagram, phase portraits, basins of attraction, and numerical continuation of bifurcations. Different global bifurcations are also shown to exist via numerical continuation. After understanding a single neuron model, a network of Chialvo neurons is explored. A ring-star network of Chialvo neurons is considered and different dynamical regimes such as synchronous, asynchronous, and chimera states are revealed. Different continuous and piecewise continuous wavy patterns were also found during the simulations for negative coupling strengths.

In this paper, we study the traveling wave solutions of a generalized reaction–diffusion system based on the classical Fisher-type system. Through qualitative analysis and blow-up techniques, we prove the existence of traveling fronts of the system. Moreover, we detect the limit points (i.e. $\omega$-limit points or $\alpha$-limit points) of all traveling wave solutions by analyzing the global topological phase portraits of the equivalent system.

We study the existence of traveling wave solutions and spreading properties for single-layer delayed neural field equations. We focus on the case where the kinetic dynamics are of monostable type and characterize the invasion speeds as a function of the asymptotic decay of the connectivity kernel. More precisely, we show that for exponentially bounded kernels the minimal speed of traveling waves exists and coincides with the spreading speed, which further can be explicitly characterized under a KPP type condition. We also investigate the case of algebraically decaying kernels where we prove the non-existence of traveling wave solutions and show the level sets of the solutions eventually locate in-between two exponential functions of time. The uniqueness of traveling waves modulo translation is also obtained.

Stop-and-go waves, also called phantom jams, are often observed in real traffic flows but can be produced neither by the classical Lighthill–Whitham–Richards (LWR) model nor by its known variants. To capture stop-and-go waves, we add hysteresis to the LWR model. For the model we propose, all possible viscous waves are found, and necessary and sufficient conditions for their existence are provided. In particular, deceleration and acceleration shocks appear; the latter were never rigorously defined before, in spite of the fact that they were observed in real traffic flows. Stop-and-go waves can be constructed by a pair of deceleration and acceleration shocks that completes a hysteresis cycle, illustrating how hysteresis loops lead to stop-and-go waves. In contrast, in the phase region where anticipation (i.e. negative hysteresis) loops exist, stop-and-go waves are not present, and speed variations decay. Riemann solutions are then found for all possible Riemann data. We explicitly show that, in the phase region where hysteresis loops exist, a sufficient deviation in speed of a few vehicles in an otherwise uniform car platoon can generate stop-and-go waves, confirming observations of real traffic experiments.

We consider a hyperbolic–parabolic system arising from a chemotaxis model in tumor angiogenesis, which is described by a Keller–Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost $L^2$-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.

In this work, we describe a hyperbolic model with cell–cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call “pressure”) which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posedness property of the associated Cauchy problem on the real line. We start from bounded initial conditions and we consider some invariant properties of the initial conditions such as the continuity, smoothness and monotony. We also describe in detail the behavior of the level sets near the propagating boundary of the solution and we find that an asymptotic jump is formed on the solution for a natural class of initial conditions. Finally, we prove the existence of sharp traveling waves for this model, which are particular solutions traveling at a constant speed, and argue that sharp traveling waves are necessarily discontinuous. This analysis is confirmed by numerical simulations of the PDE problem.

We consider a controlled reaction–diffusion equation, motivated by a pest eradication problem. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. In this direction, the first part of the paper studies the optimal control of 1-dimensional traveling wave profiles. Using Stokes’ formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. In the last section, we introduce a family of optimization problems for a moving set. We show how these can be derived from the original parabolic problems, by taking a sharp interface limit.

Analysis of the speed of propagation in parabolic operators is frequently carried out considering the minimal speed at which its traveling waves (TWs) move. This value depends on the solution concept being considered. We analyze an extensive class of Fisher-type reaction–diffusion equations with flows in divergence form. We work with regular flows, which may not meet the standard elliptical conditions, but without other types of singularities. We show that the range of speeds at which classic TWs move is an interval unbounded to the right. Contrary to classic examples, the infimum may not be reached. When the flow is elliptic or over-elliptic, the minimum speed of propagation is achieved. The classic TW speed threshold is complemented by another value by analyzing an extension of the first-order boundary value problem to which the classic case is reduced. This singular minimum speed can be justified as a viscous limit of classic minimal speeds in elliptic or over-elliptic flows. We construct a singular profile for each speed between the minimum singular speed and the speeds at which classic TWs move. Under additional assumptions, the constructed profile can be justified as that of a TW of the starting equation in the framework of bounded variation functions. We also show that saturated fronts verifying the Rankine–Hugoniot condition can appear for strictly lower speeds even in the framework of bounded variation functions.