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The aim of this paper is to explore a Schrödinger equation that incorporates a higher‐order operator. Traditional models for electron dynamics have utilized a second‐order diffusion Schrödinger equation, where oscillatory behavior is achieved through complex domain formulations. Incorporating a higher‐order operator enables the induction of oscillatory spatial patterns in solutions. Our analysis initiates with a variational formulation within generalized spaces, facilitating the examination of solution boundedness. Subsequently, we delve into the oscillatory characteristics of solutions, drawing upon a series of lemmas originally applied to the Kuramoto–Sivashinsky equation, the Cahn–Hilliard equation, and other equations that employ higher‐order operators. Specific solution types, such as standing waves, are numerically investigated to illustrate the oscillatory spatial patterns. The discussion then extends to the theory of traveling waves to establish general conditions for positive solutions. A contribution of this work is the precise evaluation of a critical traveling wave speed, denoted as c∗$$ {c}^{\ast } $$, above which the first minimum remains positive. For values of the traveling wave speed c$$ c $$ significantly greater than c∗$$ {c}^{\ast } $$, the solutions can be entirely positive. [ABSTRACT FROM AUTHOR]

Invasive‐invaded species problems are of relevance in mathematics applied to population dynamics. In this paper, the mentioned dynamics is introduced based on a fourth‐order parabolic operator, together with coupled non‐linear reaction terms. The fourth‐order operator allows us to model a heterogeneous diffusion, as introduced by the Landau–Ginzburg free energy approach. The reaction terms are given by a coupled non‐linear effect in the invasive species, to account for the action of the invaded species and limited resources, and by a non‐Lipschitz term in the invaded species, to account for possible sprouts, once the invasion occurs. The analysis starts by the proof of existence and uniqueness of solutions, making use of the semi‐group theory and a fixed point argument. Asymptotic solutions to the invasive species are explored with an exponential scaling. Afterward, the problem is analyzed with traveling wave profiles, for which a region of positive solutions is explored. [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonlocal dispersal susceptible–infected–recovered (SIR) epidemic model adopted with the mass action infection mechanism. We mainly study the existence and non‐existence of traveling waves connecting the infection‐free equilibrium state and the endemic equilibrium state. The main difficulty lies in the fact that the semiflow generated here does not admit the order‐preserving property. Meanwhile, this new model brings some new challenges due to the unboundedness of the nonlinear term. We overcome these difficulties to obtain the boundedness of traveling waves with the speed c>cmin$$ c>{c}_{\mathrm{min}} $$ by some analysis techniques firstly and then prove the existence of traveling waves by employing Lyapunov–LaSalle theorem and Lebesgue dominated convergence theorem. By utilizing an approximating method, we study the existence of traveling waves with the critical wave speed cmin$$ {c}_{\mathrm{min}} $$. Our results on this new model may provide some implications on disease modeling and controls. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the time fractional Keller‐Segel system with a small parameter. After the fractional order traveling wave transformation, the heteroclinic orbit of the degenerate time fractional Keller‐Segel system is demonstrated through the method of constructing a suitable invariant region. Moreover, the persistence of traveling waves in the system with a small parameter can be further illustrated. The results are mainly reliance on the application of geometric singular perturbation theory and Fredholm theorem, which are fundamental theoretical frameworks for dealing with problems of complexity and high dimensionality. Eventually, the asymptotic behavior is depicted by the asymptotic theory to illustrate the rate of decay for traveling waves. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and nonexistence of traveling waves of a virus infection model with humoral immunity, cell‐to‐cell transmission, and general nonlinear incidence. Our results show that the existence of traveling wave solutions is determined not only by the basic reproduction number R0$$ {\mathcal{R}}_0 $$ of virus infection and the antibody response reproduction number R1$$ {\mathcal{R}}_1 $$ but also by the critical wave speed c∗$$ {c}^{\ast } $$. More precisely, we obtain the existence of traveling wave solution connecting infection‐free equilibrium and antibody‐free infection equilibrium for R0>1,R1<1$$ {\mathcal{R}}_0>1,{\mathcal{R}}_1<1 $$, and c>c∗$$ c>{c}^{\ast } $$ and connecting infection‐free equilibrium and antibody‐present infection equilibrium for R0>1,R1>1$$ {\mathcal{R}}_0>1,{\mathcal{R}}_1>1 $$, and c>c∗$$ c>{c}^{\ast } $$. The existence of traveling wave solution connecting infection‐free equilibrium and antibody‐present infection equilibrium is discussed. Some numerical simulations are carried out to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

Following a straightforward proof for symmetric solutions to be traveling waves by Pei (Exponential decay and symmetry of solitary waves to Degasperis‐Procesi equation. Journal of Differential Equations. 2020;269(10):7730‐7749), we prove that classical symmetric solutions of the highly nonlinear shallow water equation recently derived by Quirchmayr (A new highly nonlinear shallow water wave equation. Journal of Evolution Equations. 2016;16(3):539‐556) are indeed traveling waves, with further information on their steady structures. We also provide a simple proof that symmetric waves are traveling waves to the free surface evolution equation of moderate amplitude waves in shallow water. [ABSTRACT FROM AUTHOR]

In this paper, we show how the chemotactic model ∂tρ=d1Δxρ−∇x·(ρ∇xc)∂tc=d2Δxc+F(ρ,c,∇xρ,∇xc,Δxρ)$$ \left\{\begin{array}{l}{\partial}_t\rho &#x0003D;{d}_1{\Delta}_x\rho -{\nabla}_x\cdotp \left(\rho {\nabla}_xc\right)\\ {}{\partial}_tc&#x0003D;{d}_2{\Delta}_xc&#x0002B;F\left(\rho, c,{\nabla}_x\rho, {\nabla}_xc,{\Delta}_x\rho \right)\end{array}\right. $$introduced in Alejo and López (2021), which accounts for a chemical production–degradation operator of Hamilton–Jacobi type involving first‐ and second‐order derivatives of the logarithm of the cell concentration, namely, F=μ+τc−σρ+AΔxρρ+B|∇xρ|2ρ2+C|∇xc|2,$$ F&#x0003D;\mu &#x0002B;\tau c-\sigma \rho &#x0002B;A\frac{\Delta_x\rho }{\rho }&#x0002B;B\frac{{\left&#x0007C;{\nabla}_x\rho \right&#x0007C;}&#x0005E;2}{\rho&#x0005E;2}&#x0002B;C{\left&#x0007C;{\nabla}_xc\right&#x0007C;}&#x0005E;2, $$with μ,τ,σ,A,B,C∈ℝ$$ \mu, \tau, \sigma, A,B,C\in \mathbb{R} $$, can be formally reduced to a repulsive Keller–Segel model with logarithmic sensitivity ∂tρ=D1Δxρ+χ∇x·(ρ∇xlog(c))∂tc=D2Δxc+λρc−βc,χ,λ,β>0,$$ \left\{\begin{array}{l}{\partial}_t\rho &#x0003D;{D}_1{\Delta}_x\rho &#x0002B;\chi {\nabla}_x\cdotp \left(\rho {\nabla}_x\log (c)\right)\\ {}{\partial}_tc&#x0003D;{D}_2{\Delta}_xc&#x0002B;\lambda \rho c-\beta c\end{array}\right.,\kern0.30em \chi, \lambda, \beta >0, $$whenever the chemotactic parameters are appropriately chosen and the cell concentration keeps strictly positive. In this way, some explicit solutions (namely, traveling waves and stationary cell density profiles) of the former system can be transferred to a number of variants of the the latter by means of an adequate change of variables. [ABSTRACT FROM AUTHOR]

This paper is considering the problem of traveling wave solutions (TWS) for a susceptible‐exposed‐infectious‐recovered (SEIR) epidemic model with discrete diffusion. The threshold condition for the existence and nonexistence of TWS is obtained. More specifically, such kind of solutions are governed by the threshold number ℜ0. We can find a critical wave speed c∗ if ℜ0 > 1, by employing the Schauder's fixed point theorem, limiting argument and two‐sided Laplace transform, we confirm that there exists TWS for c > c∗, while there exists no TWS for c < c∗. We also obtain the nonexistence of TWS for ℜ0 ≤ 1. At last, we give some biological explanations from the epidemiological perspective. [ABSTRACT FROM AUTHOR]

We study the existence and nonexistence of traveling waves of the reaction‐diffusion equations that describes a diffusive SEIR model with nonlocal reaction between the infected subpopulation and the susceptible subpopulation, where the total population is not constant. The existence of traveling waves depends on the basic reproduction number R0 of the corresponding ordinary differential equations and the minimal wave speed c∗. The main difficulty is the lack of order‐preserving property of our general system. Its proof is showed by constructing a closed and convex set and applying Schauder fixed point theorem. The proof of nonexistence result is obtained by two‐side Laplace transform. Finally, we present some numerical results of the minimal wave speed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper looking for traveling wave solutions, we find that when the polynomial of velocity is quintic the generalized Ostrovsky equation (GOE) has abundant exact solutions that can be expressed in terms of the Jacobi elliptic functions. Hence, the GOE has a plenty of periodic waves, solitary waves, compactons, etc. These solutions are derived from the solutions of a simple non‐linear ordinary differential equation. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction-diffusion system with delay arising from the spread of an epidemic by oral-faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c⩾ c∗. Then we show that the traveling wave fronts with speed c> c∗ are exponentially asymptotically stable. [ABSTRACT FROM AUTHOR]

In this work, we study the approximation of traveling wave solutions propagated at minumum speeds c0( h) of the delayed Nicholson's blowflies equation: [ABSTRACT FROM AUTHOR]

D. Zeidan In this paper, we consider the existence of traveling waves in a generalized nonlinear dispersive-dissipative equation, which is found in many areas of application including waves in a thermoconvective liquid layer and nonlinear electromagnetic waves. By using the theory of dynamical systems, specifically based on geometric singular perturbation theory and invariant manifold theory, Fredholm theory, and the linear chain trick, we construct a locally invariant manifold for the associated traveling wave equation and use this invariant manifold to obtain the traveling waves for the nonlinear dispersive-dissipative equation. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In the present paper, we construct exact solutions to a system of partial differential equations iu x + v + u | v | 2 = 0, iv t + u + v | u | 2 = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinear wave in terms of the amplitudes of both waves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of u or v with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of traveling waves to a predator-prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper-lower solutions and by using the cross-iteration method as well as the Schauder's fixed-point theorem, the existence of a traveling wave solution connecting the semi-trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents an asymptotic analysis of non-discrete radiating modes with applications in waveguide theory. As a main application, the radiating modes of an open waveguide structure with circular geometry is considered. A generalized Jordan's lemma is used to justify that field components can be calculated as the sum of discrete and non-discrete modes, that is, as the sum of residues of poles and an integral along the branch-cut defined by the transversal wavenumber of the exterior domain. An asymptotic expression is derived for field components at large distance along the waveguide and supplemented with rigorous upper and lower error bounds. A numerical example regarding the axial symmetric 0th order transverse magnetic modes of a thin copper wire in water is included to demonstrate that there may be a non-trivial balance between the contributions from discrete and non-discrete modes. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic stability of a composite wave consisting of two traveling waves to a hyperbolic-parabolic system modeling repulsive chemotaxis. On the basis of elementary energy estimates, we show that the composite wave is asymptotically stable under general initial perturbations, which are not necessarily zero integral. As an application, we obtain a similar result for this system in the presence of a boundary. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]