# American Institute of Mathematical Sciences

eISSN:
2163-2480

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## Evolution Equations & Control Theory

March 2019 , Volume 8 , Issue 1

Special issue on nonlinear wave phenomena in continuum physics: Some recent findings

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2019, 8(1): ⅰ-ⅲ doi: 10.3934/eect.20191i +[Abstract](2438) +[HTML](394) +[PDF](152.61KB)
Abstract:
2019, 8(1): 1-29 doi: 10.3934/eect.2019001 +[Abstract](3014) +[HTML](551) +[PDF](1353.34KB)
Abstract:

Strongly nonlinear perturbation theory would seem to be an oxymoron, that is, a contradiction of terms. Nonetheless, we here describe perturbation methods for wave categories that are intrinsically nonlinear including solitons (solitary waves), bound states of solitons (bions) and spatially periodic traveling waves (cnoidal waves). Examples include the Kortweg-deVries and Benjamin-Ono equations with general power law nonlinearity and the Fifth Order KdV equation. The perturbation strategies include (ⅰ) the Gorshkov-Ostrovsky-Papko near-equal-amplitude soliton interaction theory (ⅱ) perturbation series in the Newton-homotopy parameter and (ⅲ) approximations for large values of the nonlinearity exponent. A long section places strongly nonlinear perturbation theory for waves in a larger context as a subset of unconventional perturbation expansions including phase transition theory in \begin{document}$4 - \epsilon$\end{document} dimensions, the \begin{document}$\epsilon = 1/D$\end{document} expansion where \begin{document}$D$\end{document} is the dimension in quantum chemistry, the renormalized quantum anharmonic oscillator, the Yakhot-Orszag expansion in the exponent of the energy spectrum in hydrodynamic turbulence, and the Newton homotopy expansion.

2019, 8(1): 31-42 doi: 10.3934/eect.2019002 +[Abstract](2371) +[HTML](446) +[PDF](360.14KB)
Abstract:

The model of rigid linear heat conductor with memory is reconsidered focussing the interest on the heat relaxation function. Thus, the definitions of heat flux and thermal work are revised to understand where changes are required when the heat flux relaxation function \begin{document}$k$\end{document} is assumed to be unbounded at the initial time \begin{document}$t = 0$\end{document}. That is, it is represented by a regular integrable function, namely \begin{document}$k\in L^1( \mathbb{R}^+)$\end{document}, but its time derivative is not integrable, that is \begin{document}$\dot k\notin L^1( \mathbb{R}^+)$\end{document}. The study takes its origin in [2]: the heat conductor model described therein is modified in such a way to adapt it to the case of a heat flux relaxation function \begin{document}$k$\end{document} which is unbounded at \begin{document}$t = 0$\end{document}. Notably, also when these relaxed assumptions on \begin{document}$k$\end{document} are introduced, whenever two different thermal states which correspond to the same heat flux are considered, then both states correspond also to the same thermal work. Accordingly, the notion of equivalence can be introduced, together with its physical relevance, both in the regular kernel case in [2] as well as in the singular kernel case analysed in the present investigation.

2019, 8(1): 43-55 doi: 10.3934/eect.2019003 +[Abstract](2652) +[HTML](438) +[PDF](705.11KB)
Abstract:

Connections via Bäcklund transformations among different nonlinear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Bäcklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Bäcklund charts in the case of the Burgers -heat equation, on one side, and KdV-type equations, on the other, are considered. The Bäcklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.

2019, 8(1): 57-72 doi: 10.3934/eect.2019004 +[Abstract](2223) +[HTML](433) +[PDF](639.7KB)
Abstract:

An exactly solvable model in heat conduction is considered. The \begin{document}$C$\end{document}-integrable (i.e., change-of-variables-integrable) equation for second sound (i.e., heat wave) propagation in a thin, rigid dielectric heat conductor uniformly heated on its lateral side by a surrounding medium under the Stefan-Boltzmann law is derived. A simple change-of-variables transformation is shown to exactly map the nonlinear governing partial differential equation to the classical linear telegrapher's equation. In a one-dimensional context, known integral-transform solutions of the latter are adapted to construct exact solutions relevant to heat transfer applications: (ⅰ) the initial-value problem on an infinite domain (the real line), and (ⅱ) the initial-boundary-value problem on a semi-infinite domain (the half-line). Possible "second law violations" and restrictions on the \begin{document}$C$\end{document}-transformation are noted for some sets of parameters.

2019, 8(1): 73-100 doi: 10.3934/eect.2019005 +[Abstract](2880) +[HTML](466) +[PDF](2889.93KB)
Abstract:

The influence of asymmetry in the coupling between repulsive particles is studied. A prominent example is the social force model for pedestrian dynamics in a long corridor where the asymmetry leads to anisotropy in the repulsion such that pedestrians in front, i.e., in walking direction, have a bigger influence on the pedestrian behavior than those behind. In addition to one- and two-lane free flow situations, a new traveling regime is found that is reminiscent of peristaltic motion. We study the regimes and their respective stabilityboth analytically and numerically. First, we introduce a modified social forcemodel and compute the boundaries between different regimes analytically bya perturbation analysis of the one-lane and two-lane flow. Afterwards, theresults are verified by direct numerical simulations in the parameter plane ofpedestrian density and repulsion strength from the walls.

2019, 8(1): 101-116 doi: 10.3934/eect.2019006 +[Abstract](2530) +[HTML](463) +[PDF](213.62KB)
Abstract:

We consider acoustic propagation in a particle-laden fluid, specifically, a perfect gas, under a model system based on the theories of Marble (1970) and Thompson (1972). Our primary aim is to understand, via analytical methods, the impact of the particle phase on the acoustic velocity field. Working under the finite-amplitude approximation, we investigate singular surface and traveling wave phenomena, as admitted by both phases of the flow. We show, among other things, that the particle velocity field admits a singular surface one order higher than that of the gas phase, that the particle-to-gas density ratio plays a number of critical roles, and that traveling wave solutions are only possible for sufficiently small values of the Mach number.

2019, 8(1): 117-137 doi: 10.3934/eect.2019007 +[Abstract](2722) +[HTML](451) +[PDF](393.31KB)
Abstract:

In this paper we discuss discrete regularization, more specifically about how to add finite dissipation to the discretized Euler equations so as to ensure the stability and convergence of numerical solutions of high Reynolds number flows. We will briefly review regularization strategies widely used in Lagrangian shockwave simulations (artificial viscosity), in Eulerian nonoscillatory finite volume simulations, and in Eulerian simulations of turbulent flow (explicit and implicit large eddy simulations). We will describe an alternate strategy for regularization in which we introduce a finite length scale into the discrete model by volume averaging the equations over a computational cell. The new equations, which we term Finite Scale Navier-Stokes, contain explicit (inviscid) dissipation in a uniquely specified form and obey an entropy principle. We will describe features of the new equations including control of the small scales of motion by the larger resolved scales, a principle concerning the partition of total flux of conserved quantities into advective and diffusive components, and a physical basis for the inviscid dissipation.

2019, 8(1): 139-147 doi: 10.3934/eect.2019008 +[Abstract](2492) +[HTML](415) +[PDF](306.86KB)
Abstract:

We investigate traveling wave (TW) solutions to modified versionsof the Burgers and Fisher PDE’s. Both equations are nonlinear parabolicPDE’s having square-root dynamics in their advection and reaction terms.Under certain assumptions, exact forms are constructed for the TW solutions.

2019, 8(1): 149-162 doi: 10.3934/eect.2019009 +[Abstract](2225) +[HTML](427) +[PDF](363.92KB)
Abstract:

This paper is addressed to the analysis of wave propagation in electroelastic materials. First the balance equations are reviewed and the entropy inequality is established. Next the constitutive equations are considered for a deformable and heat-conducting dielectric. To allow for discontinuity wave propagation, an appropriate objective rate equation of the heat flux is considered. The thermodynamic consistency of the whole set of constitutive equations is established. Next the nonlinear evolution equations so determined are tested in relation to wave propagation properties. Waves are investigated in the form of weak discontinuities and the whole system of equations for the jumps is obtained. As a particular simple case the propagation into an unperturbed region is examined. Both the classical electromagnetic waves and the thermal waves are found to occur. In both cases the mechanical term is found to be induced by the electrical or the thermal wave discontinuity.

2019, 8(1): 163-202 doi: 10.3934/eect.2019010 +[Abstract](3464) +[HTML](500) +[PDF](2274.47KB)
Abstract:

The goal of this work is to improve focusing of high-intensity ultrasound by modifying the geometry of acoustic lenses through shape optimization. We formulate the shape optimization problem by introducing a tracking-type cost functional to match a desired pressure distribution in the focal region. Westervelt's equation, a nonlinear acoustic wave equation, is used to model the pressure field. We apply the optimize first, then discretize approach, where we first rigorously compute the shape derivative of our cost functional. A gradient-based optimization algorithm is then developed within the concept of isogeometric analysis, where the geometry is exactly represented by splines at every gradient step and the same basis is used to approximate the equations. Numerical experiments in a \begin{document}$2$\end{document}D setting illustrate our findings.

2019, 8(1): 203-220 doi: 10.3934/eect.2019011 +[Abstract](3023) +[HTML](414) +[PDF](506.35KB)
Abstract:

We study the third order in time linear dissipative wave equation known as the Moore-Gibson-Thompson equation, that appears as the linearization of a the Jordan-Moore-Gibson-Thompson equation, an important model in nonlinear acoustics. The same equation also arises in viscoelasticity theory, as a model which is considered more realistic than the usual Kelvin-Voigt one for the linear deformations of a viscoelastic solid. In this context, it is known as the Standard Linear Viscoelastic model. We complete the description in [13] of the spectrum of the generator of the corresponding group of operators and show that, apart from some exceptional values of the parameters, this generator can be made to be a normal operator with a new scalar product, with a complete set of orthogonal eigenfunctions. Using this property we also obtain optimal exponential decay estimates for the solutions as \begin{document}$t\to\infty$\end{document}, whether the operator is normal or not.

2019, 8(1): 221-230 doi: 10.3934/eect.2019012 +[Abstract](2915) +[HTML](450) +[PDF](1194.51KB)
Abstract:

We focus on the problem of shock wave formation in a model of blood flow along an elastic artery. We analyze the conditions under which this phenomenon can appear and we provide an estimation of the instant of shock formation. Numerical simulations of the model have been conducted using the Discontinuous Galerkin Finite Element Method. The results are consistent with certain phenomena observed by practitioners in patients with arteriopathies, and they could predict the possible formation of a shock wave in the aorta.

2019, 8(1): 231-246 doi: 10.3934/eect.2019013 +[Abstract](2353) +[HTML](447) +[PDF](8395.58KB)
Abstract:

This paper reports a study of transient dynamic responses of the anti-plane shear Lamb's problem on random mass density field with fractal and Hurst effects. Cellular automata (CA) is used to simulate the shear wave propagation. Both Cauchy and Dagum random field models are used to capture fractal dimension and Hurst effects in the mass density field. First, the dynamic responses of random mass density are evaluated through a comparison with the homogenerous computational results and the classical theoretical solution. Then, a comprehensive study is carried out for different combinations of fractal and Hurst coefficients. Overall, this investigation determines to what extent fractal and Hurst effects are significant enough to change the dynamic responses by comparing the signal-to-noise ratio of the response versus the signal-to-noise ratio of the random field.

2019  Impact Factor: 0.953