Diffusive-dispersive travelling waves and kinetic relations. II A hyperbolic–elliptic model of phase-transition dynamics

2002 ◽  
Vol 132 (3) ◽  
pp. 545-565 ◽  
Author(s):  
Nabil Bedjaoui ◽  
Philippe G. LeFloch
Keyword(s):  
Phase Transition ◽  
Travelling Waves ◽  
Asymptotic Properties ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Phase Boundaries ◽  
Transition Dynamics ◽  
Nonlinear Viscosity ◽  
Kinetic Relations ◽  
Elliptic Model

We deal here with a mixed (hyperbolic-elliptic) system of two conservation laws modelling phase-transition dynamics in solids undergoing phase transformations. These equations include nonlinear viscosity and capillarity terms. We establish general results concerning the existence, uniqueness and asymptotic properties of the corresponding travelling wave solutions. In particular, we determine their behaviour in the limits of dominant diffusion, dominant dispersion or asymptotically small or large shock strength. As the viscosity and capillarity parameters tend to zero, the travelling waves converge to propagating discontinuities, which are either classical shock waves or supersonic phase boundaries satisfying the Lax and Liu entropy criteria, or else are undercompressive subsonic phase boundaries. The latter are uniquely characterized by the so-called kinetic function, whose properties are investigated in detail here.

Non-classical Riemann solvers and kinetic relations. II. An hyperbolic–elliptic model of phase-transition dynamics

2002 ◽  
Vol 132 (1) ◽  
pp. 181-219 ◽  
Author(s):  
Philippe G. LeFloch ◽  
Mai Duc Thanh
Keyword(s):  
Phase Transition ◽  
Riemann Problem ◽  
Entropy Inequality ◽  
Phase Boundaries ◽  
Kinetic Relation ◽  
Two Phase ◽  
Transition Dynamics ◽  
Riemann Solvers ◽  
All Solutions ◽  
Elliptic Model

This paper deals with the Riemann problem for a partial differential equation's model arising in phase-transition dynamics and consisting of an hyperbolic–elliptic system of two conservation laws. First of all, we provide a complete description of all solutions of the Riemann problem that are consistent with the mathematical entropy inequality associated with the total energy of the system. Second, following Abeyaratne and Knowles, we impose a kinetic relation to determine the dynamics of subsonic phase boundaries. Based on the requirement that subsonic phase boundaries are preferred whenever available, we determine the corresponding wave curves associated with composite waves (shocks, rarefaction fans, phase boundaries). It turns out that even after the kinetic relation is imposed, the Riemann problem may admit up to two solutions. A nucleation criterion is necessary to select between a solution remaining in a single phase and a solution containing two phase boundaries. Alternatively, a strong assumption on the kinetic relation ensures that the Riemann solution is unique and depends continuously upon its initial data.

Nonlinear travelling internal waves with piecewise-linear shear profiles

2018 ◽  
Vol 856 ◽  
pp. 984-1013 ◽  
Author(s):  
K. L. Oliveras ◽  
C. W. Curtis
Keyword(s):  
Piecewise Linear ◽  
Travelling Waves ◽  
Tangential Velocity ◽  
Travelling Wave ◽  
Numerical Continuation ◽  
Travelling Wave Solutions ◽  
Water Wave Problem ◽  
Two Fluids ◽  
Wave Solutions ◽  
Linear Shear

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129–148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.

scholarly journals Nonuniform Continuity of the Osmosis K(2, 2) Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Aiyong Chen ◽  
Yong Ding ◽  
Wentao Huang
Keyword(s):  
Differential Equations ◽  
Small Amplitude ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Qualitative Theory ◽  
Travelling Wave Solutions ◽  
Solution Map ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Uniformly Continuous

The qualitative theory of differential equations is applied to the osmosis K(2, 2) equation. The parametric conditions of existence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

scholarly journals Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 950 ◽  
Author(s):  
María Luz Gandarias ◽  
María Rosa Durán ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Two Dimensions ◽  
Travelling Wave Solutions ◽  
Dispersion Equations ◽  
Different Dimensions ◽  
Double Dispersion ◽  
Long Nonlinear Wave ◽  
Line Soliton

In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions.

Travelling waves for a reaction–diffusion system in population dynamics and epidemiology

2005 ◽  
Vol 135 (4) ◽  
pp. 663-675 ◽  
Author(s):  
Shangbing Ai ◽  
Wenzhang Huang
Keyword(s):  
Population Dynamics ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Diffusion Constant ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Dimensional System ◽  
Travelling Wave Solutions ◽  
Wave Solutions

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

Viscous film flow coating the interior of a vertical tube. Part 1. Gravity-driven flow

2014 ◽  
Vol 745 ◽  
pp. 682-715 ◽  
Author(s):  
Roberto Camassa ◽  
H. Reed Ogrosky ◽  
Jeffrey Olander
Keyword(s):  
Time Evolution ◽  
Film Flow ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Basic Solution ◽  
Travelling Wave Solutions ◽  
Liquid Plug ◽  
Wave Solutions ◽  
Gravity Driven ◽  
Gravity Driven Flow

AbstractThe gravity-driven flow of a viscous liquid film coating the inside of a tube is studied both theoretically and experimentally. As the film moves downward, small perturbations to the free surface grow due to surface tension effects and can form liquid plugs. A first-principles strongly nonlinear model based on long-wave asymptotics is developed to provide simplified governing equations for the motion of the film flow. Linear stability analysis on the basic solution of the model predicts the speed and wavelength of the most unstable mode, and whether the film is convectively or absolutely unstable. These results are found to be in remarkable agreement with the experiments. The model is also solved numerically to follow the time evolution of instabilities. For relatively thin films, these instabilities saturate as a series of small-amplitude travelling waves, while thicker films lead to solutions whose amplitude becomes large enough for the liquid surface to approach the centre of the tube in finite time, suggesting liquid plug formation. Next, the model’s periodic travelling wave solutions are determined by a continuation algorithm using the results from the time evolution code as initial seed. It is found that bifurcation branches for these solutions exist, and the critical turning points where branches merge determine film mean thicknesses beyond which no travelling wave solutions exist. These critical thickness values are in good agreement with those for liquid plug formations determined experimentally and numerically by the time-evolution code.

Travelling waves for delayed reaction–diffusion equations with global response

2005 ◽  
Vol 462 (2065) ◽  
pp. 229-261 ◽  
Author(s):  
Teresa Faria ◽  
Wenzhang Huang ◽  
Jianhong Wu
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Index Formula ◽  
Delayed Response ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Non Local

We develop a new approach to obtain the existence of travelling wave solutions for reaction–diffusion equations with delayed non-local response. The approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain Banach space, coupled with an index formula of the associated Fredholm operator and some careful estimation of the nonlinear perturbation. The general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation, and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non-local and delayed response for the matured population.

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