Traveling Wave Problem Research Articles

Well-posedness of the stationary and slowly traveling wave problems for the free boundary incompressible Navier-Stokes equations

We establish that solitary stationary waves in three dimensional viscous incompressible fluids are a general phenomenon and that every such solution is a vanishing wave-speed limit along a one parameter family of traveling waves. The setting of our result is a horizontally-infinite fluid of finite depth with a flat, rigid bottom and a free boundary top. A constant gravitational field acts normal to bottom, and the free boundary experiences surface tension. In addition to these gravity-capillary effects, we allow for applied stress tensors to act on the free surface region and applied forces to act in the bulk. These are posited to be in either stationary or traveling form.In the absence of any applied stress or force, the system reverts to a quiescent equilibrium; in contrast, when such sources of stress or force are present, stationary or traveling waves are generated. We develop a small data well-posedness theory for this problem by proving that there exists a neighborhood of the origin in stress, force, and wave speed data-space in which we obtain the existence and uniqueness of stationary and traveling wave solutions that depend continuously on the stress-force data, wave speed, and other physical parameters. To the best of our knowledge, this is the first proof of well-posedness of the solitary stationary wave problem and the first continuous embedding of the stationary wave problem into the traveling wave problem. Our techniques are based on vector-valued harmonic analysis, a novel method of indirect symbol calculus, and the implicit function theorem.

Traveling/non-traveling phase transition and non-ergodic properties in the random transverse-field Ising model on the Cayley tree

We study the random transverse field Ising model on a finite Cayley tree. This enables us to probe key questions arising in other important disordered quantum systems, in particular the Anderson transition and the problem of dirty bosons on the Cayley tree, or the emergence of non-ergodic properties in such systems. We numerically investigate this problem building on the cavity mean-field method complemented by state-of-the art finite-size scaling analysis. Our numerics agree very well with analytical results based on an analogy with the traveling wave problem of a branching random walk in the presence of an absorbing wall. Critical properties and finite-size corrections for the zero-temperature paramagnetic-ferromagnetic transition are studied both for constant (independent of the system volume) and algebraically vanishing (scaling as an inverse power law with the system volume) boundary conditions. In the later case, we reveal a regime which is reminiscent of the non-ergodic delocalized phase observed in other systems, thus shedding some light on critical issues in the context of disordered quantum systems, such as Anderson transitions, the many-body localization or disordered bosons in infinite dimensions.

Traveling wave solutions for a two-species competitive Keller–Segel chemotaxis system

In this paper, the traveling wave problem for a two-species competition reaction–diffusion–chemotaxis Lotka–Volterra system is investigated. Upper and lower solutions method and fixed point theory are employed to show the existence of traveling wave solutions connecting the coexistence constant steady state with zero state for all large enough wave speed c, and conversely, when c is small, we prove there is no traveling wave solution.

Mass and spring dimer Fermi–Pasta–Ulam–Tsingou nanopterons with exponentially small, nonvanishing ripples

AbstractWe study traveling waves in mass and spring dimer Fermi–Pasta–Ulam–Tsingou (FPUT) lattices in the long wave limit. Such lattices are known to possess nanopteron traveling waves in relative displacement coordinates. These nanopteron profiles consist of the superposition of an exponentially localized “core,” which is close to a Korteweg–de Vries solitary wave, and a periodic “ripple,” whose amplitude is small beyond all algebraic orders of the long wave parameter, although a zero amplitude is not precluded. Here we deploy techniques of spatial dynamics, inspired by results of Iooss and Kirchgässner, Iooss and James, and Venney and Zimmer, to construct mass and spring dimer nanopterons whose ripples are both exponentially small and also nonvanishing. We first obtain “growing front” traveling waves in the original position coordinates and then pass to relative displacement. To study position, we recast its traveling wave problem as a first‐order equation on an infinite‐dimensional Banach space; then we develop hypotheses that, when met, allow us to reduce such a first‐order problem to one solved by Lombardi. A key part of our analysis is then the passage back from the reduced problem to the original one. Our hypotheses free us from working strictly with lattices but are easily checked for FPUT mass and spring dimers. We also give a detailed exposition and reinterpretation of Lombardi's methods, to illustrate how our hypotheses work in concert with his techniques, and we provide a dialog with prior methods of constructing FPUT nanopterons, to expose similarities and differences with the present approach.

CRITICAL TRAVELLING WAVES IN ONE MODEL OF THE ”REACTION-DIFFUSION” TYPE

The paper is devoted to the order reduction for critical traveling wave problems for a reaction-diffusion type systems. The mathematical apparatus is based on the geometric theory of singular perturbations and the canards technique. The use of the method of invariant manifolds of singularly perturbed systems allows us to replace the study of traveling waves of the original PDE system by analyzing their profiles in a ODE system of a lower order.

Minimal-speed selection of traveling fronts to a three components lattice competition system

In this paper, we study the traveling wave problem for a three species lattice competition system with monostable nonlinearity. We are interested in the linear or nonlinear determinacy of the minimal wave speed. Some conditions on the linear determinacy has been obtained in [J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math. 19 (2015) 1805–1829]. In this paper, we first provided some general conditions for the linear determinacy. Then, we derive some novel results on the nonlinear determinacy. Finally, some numerical simulations are carried out to illustrate our analytic results. To our knowledge, this may be the first investigation on the nonlinear determinacy of the three species competition system.

Spatial segregation limit of traveling wave solutions for a fully nonlinear strongly coupled competitive system

<abstract><p>The paper is concerned with a singular limit for the bistable traveling wave problem in a very large class of two-species fully nonlinear parabolic systems with competitive reaction terms. Assuming existence of traveling waves and enough compactness, we derive and characterize the limiting problem. The assumptions and results are discussed in detail. The free boundary problem obtained at the limit is specified for important applications.</p></abstract>

Unstable dynamics of solitary traveling waves in a lattice with long-range interactions

In this work we revisit the existence, stability and dynamics of traveling solitary waves in the context of lattice dynamical systems. We consider a nonlinear lattice of an α-Fermi-Pasta–Ulam type with the additional feature of all-to-all harmonic long-range interactions whose strength decays exponentially with distance. The competition between the nonlinear nearest-neighbor terms and the longer-range linear terms yields two parameter regimes where the dependence of the energy H of the traveling waves on their velocity c is non-monotonic and multivalued, respectively. We examine both cases, and identify the exact (up to a prescribed numerical tolerance) traveling waves. To investigate the stability of the obtained solutions, we compute their Floquet multipliers, thinking of the traveling wave problem as a periodic one modulo shifts. We show that in the general case when the relationship between H and c is not necessarily single-valued, a stability threshold corresponds to H′(s)=0, where s is the parameter along the energy–velocity curve. Perturbing the unstable solutions along the corresponding eigenvectors, we identify two different scenarios of the dynamics of their transition to stable branches. In the first case, the perturbed wave slows down after expelling a dispersive wave. The second scenario involves an increase in the velocity of the perturbed wave accompanied by the formation of a slower small-amplitude traveling solitary wave.

Stability analysis of traveling wave solutions of a generalized Korteweg–de Vries–Burgers equation with variable dissipation parameter

We consider traveling wave solutions of a generalized Korteweg–de Vries–Burgers equation. The dissipation coefficient depends on coordinate and time and has a smoothed step-like form at every instant of time. Small-scale processes of dissipation and dispersion determine the solution of the traveling wave problem in the high-gradient region. The flux function is non-convex and has two inflection points. It is shown that traveling wave solutions with the same wave speed can converge to the three different limiting values behind the wave. The Evans function technique is used to conduct linear stability analysis. We demonstrate that traveling wave solutions can be linearly stable for each of the three possible cases. Thus, we found linearly stable solutions in the form of a traveling wave, which correspond to admissible discontinuities in hyperbolic model. These discontinuities have the same speed but different states behind them.

Solitary wave solutions of a Whitham–Boussinesq system

The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in Dinvay et al. (2019), where it was numerically shown to be stable and a good approximation to the incompressible Euler equations. In subsequent papers (Dinvay, 2019; Dinvay et al., 2019) the initial-value problem was studied and well-posedness in classical Sobolev spaces was proved. Here we prove existence of solitary wave solutions and provide their asymptotic description. Our proof relies on a variational approach and a concentration-compactness argument. The main difficulties stem from the fact that in the considered Euler–Lagrange equation we have a non-local operator of positive order appearing both in the linear and non-linear parts. Our approach allows us to obtain solitary waves for a particular Boussinesq system as well.

Order reduction for critical traveling wave problems

The paper deals with the order reduction for critical traveling wave problems. The specificity of such traveling waves is that they separate waves with qualitatively different behaviors. W e show how the application of the geometric theory of singular perturbations allows us to reduce the traveling wave problem for the original PDE system to the analysis the projection of the system onto its slow invariant manifold. W e illustrate this approach to the problem of finding the point-to-periodic traveling wave for the reaction-diffusion model.

Order reduction for problems with traveling wave solutions to reaction–diffusion systems

The paper is devoted to the order reduction for traveling wave problems for reaction-diffusion systems. The use of the method of invariant manifolds of singularly perturbed systems allows us to replace the study of the traveling wave of the original PDE system with the analysis of their profiles in the ODE system with a lower order.

TRAVELING WAVES AND THEIR EVOLUTION FOR THE ZK(N, 2N, -N) EQUATION

<abstract> In this paper, using the approach of dynamical systems we investigate the traveling waves for the ZK(n, 2n, -n) equation including the types and evolution of traveling waves. The traveling wave problem is converted into the analysis of phase portraits of the corresponding traveling wave system, which is 5-parametric and has a singular line in its phase space. The orbits passing through this singular line in phase portraits are determined by a time rescaling. After converting the orbits in these phase portraits into traveling waves, we state all types of traveling waves and give at least one exact traveling wave solution for each type of bounded traveling waves in our main results. Finally, we discuss the evolution of these traveling waves among themselves when parameters vary by the bifurcations happening in the phase portraits of the traveling wave system. </abstract>

Existence of pulses for a reaction-diffusion system of blood coagulation in flow.

A reaction-diffusion system describing blood coagulation in flow is studied. We prove the existence of stationary solutions provided that the speed of the travelling wave problem for the limiting value of the velocity is positive. The implications to the problem of clot growth are discussed.

Periodic travelling interfacial hydroelastic waves with or without mass II: Multiple bifurcations and ripples

In a prior work, the authors proved a global bifurcation theorem for spatially periodic interfacial hydroelastic travelling waves on infinite depth, and computed such travelling waves. The formulation of the travelling wave problem used both analytically and numerically allows for waves with multi-valued height. The global bifurcation theorem required a one-dimensional kernel in the linearization of the relevant mapping, but for some parameter values, the kernel is instead two-dimensional. In the present work, we study these cases with two-dimensional kernels, which occur in resonant and non-resonant variants. We apply an implicit function theorem argument to prove existence of travelling waves in both of these situations. We compute the waves numerically as well, in both the resonant and non-resonant cases.

Traveling fronts for lattice neural field equations

We show existence and uniqueness of traveling front solutions to a class of neural field equations set on a lattice with infinite range interactions in the regime where the kinetics of each individual neuron is of bistable type. The existence proof relies on a regularization of the traveling wave problem allowing us to use well-known existence results for traveling front solutions of continuous neural field equations. We then show that the traveling front solutions which have nonzero wave speed are unique (up to translation) by constructing appropriate sub and super solutions. The spectral properties of the traveling fronts are also investigated via a careful study of the linear operator around a traveling front in co-moving frame where we crucially use Fredholm properties of nonlocal differential operators previously obtained by the author in an earlier work. For the spectral analysis, we need to impose an extra exponential localization condition on the interactions.

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

We study the effect of domain growth on the orientation of striped phases in a Swift-Hohenberg equation. Domain growth is encoded in a step-like parameter dependence that allows stripe formation in a half plane, and suppresses patterns in the complement, while the boundary of the pattern-forming region is propagating with fixed normal velocity. We construct front solutions that leave behind stripes in the pattern-forming region that are parallel to or at a small oblique angle to the boundary. Technically, the construction of stripe formation parallel to the boundary relies on ill-posed, infinite-dimensional spatial dynamics. Stripes forming at a small oblique angle are constructed using a functional-analytic, perturbative approach. Here, the main difficulties are the presence of continuous spectrum and the fact that small oblique angles appear as a singular perturbation in a traveling-wave problem. We resolve the former difficulty using a farfield-core decomposition and Fredholm theory in weighted spaces. The singular perturbation problem is resolved using preconditioners and boot-strapping.

The Tip Region of a Near-Surface Hydraulic Fracture

This paper investigates the tip region of a hydraulic fracture propagating near a free-surface via the related problem of the steady fluid-driven peeling of a thin elastic layer from a rigid substrate. The solution of this problem requires accounting for the existence of a fluid lag, as the pressure singularity that would otherwise exist at the crack tip is incompatible with the underlying linear beam theory governing the deflection of the thin layer. These considerations lead to the formulation of a nonlinear traveling wave problem with a free boundary, which is solved numerically. The scaled solution depends only on one number K, which has the meaning of a dimensionless toughness. The asymptotic viscosity- and toughness-dominated regimes, respectively, corresponding to small and large K, represent the end members of a family of solutions. It is shown that the far-field curvature can be interpreted as an apparent toughness, which is a universal function of K. In the viscosity regime, the apparent toughness does not depend on K, while in the toughness regime, it is equal to K. By noting that the apparent toughness represents an intermediate asymptote for the layer curvature under certain conditions, the obtention of time-dependent solutions for propagating near-surface hydraulic fractures can be greatly simplified. Indeed, any such solutions can be constructed by a matched asymptotics approach, with the outer solution corresponding to a uniformly pressurized fracture and the inner solution to the tip solution derived in this paper.

Traveling waves for a model of individual clustering with logistic growth rate

We consider a traveling wave problem for a one-dimensional model of individual clustering or aggregation. This model, originally formulated by Grindrod [J. Math. Biol. 26(6), 651–660 (1988)], describes the mechanism of individual dispersion when individuals are able to utilize information about their local environment. Specifically, it assumes that each individual disperses randomly with probability δ and disperses deterministically so as to improve his reproductive rate with probability 1−δ. In this paper, we prove the existence of a traveling wave solution when the probability of random dispersion exceeds a critical value δ* uniquely determined by the biased individual velocity.

The existence of minimum speed of traveling wave solutions to a non-KPP isothermal diffusion system

The reaction–diffusion system at=axx−abn,bt=Dbxx+abn, where n≥1 and D>0, arises from many real-world chemical reactions. Whereas n=1 is the KPP type nonlinearity, which is much studied and very important results obtained in literature not only in one dimensional spatial domains, but also multi-dimensional spaces, but n>1 proves to be much harder. One of the interesting features of the system is the existence of traveling wave solutions. In particular, for the traveling wave solution a(x,t)=a(x−vt), b(x,t)=b(x−vt), where v>0, if we fix limx→−∞⁡(a,b)=(0,1) it was proved by many authors with different bounds v⁎(n,D)>0 such that a traveling wave solution exists for any v≥v⁎ when n>1. For the latest progress, see [7]. That is, the traveling wave problem exhibits the mono-stable phenomenon for traveling wave of scalar equation ut=uxx+f(u) with f(0)=f(1)=0, f(u)>0 in (0,1) and, u=0 is unstable and u=1 is stable. A natural and significant question is whether, like the scalar case, there exists a minimum speed. That is, whether there exists a minimum speed vmin>0 such that traveling wave solution of speed v exists iff v≥vmin? This is an open question, in spite of many works on traveling wave of the system in last thirty years. This is duo to the reason, unlike the KPP case, the minimum speed cannot be obtained through linear analysis at equilibrium points (a,b)=(0,1) and (a,b)=(1,0). In this work, we give an affirmative answer to this question.