Approximate travelling waves for generalized KPP equations and classical mechanics

1994 ◽  
Vol 446 (1928) ◽  
pp. 529-554 ◽  
Keyword(s):  
Travelling Waves ◽  
Classical Mechanics ◽  
Initial Distribution ◽  
Travelling Wave ◽  
Step Function ◽  
Initial Position ◽  
Classical Path ◽  
Huygens Principle ◽  
Kpp Equations ◽  
Complete Riemannian Manifolds

We consider the existence of approximate travelling waves of generalized KPP equations in which the initial distribution can depend on a small parameter μ which in the limit μ → 0 is the sum of some δ -functions or a step function. Using the method of Elworthy & Truman (1982) we construct a classical path which is the backward flow of a classical newtonian mechanics with given initial position and velocity before the time at which the caustic appears. By the Feynman–Kac formula and the Maruyama–Girsanov–Cameron–Martin transformation we obtain an identity from which, with a late caustic assumption, we see the propagation of the global wave front and the shape of the trough. Our theory shows clearly how the initial distribution contributes to the propagation of the travelling wave. Finally, we prove a Huygens principle for KPP equations on complete riemannian manifolds without cut locus, with some bounds on their volume element, in particular Cartan–Hadamard manifolds.

Numerical solution of Fisher's equation

1974 ◽  
Vol 11 (03) ◽  
pp. 445-457 ◽  
Author(s):  
Jenö Gazdag ◽  
José Canosa
Keyword(s):  
Water Waves ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Step Function ◽  
Local Perturbation ◽  
Fisher’S Equation ◽  
Minimal Speed ◽  
Fisher's Equation

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration.

An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

2011 ◽  
Vol 141 (6) ◽  
pp. 1141-1173 ◽  
Author(s):  
Jared C. Bronski ◽  
Mathew A. Johnson ◽  
Todd Kapitula
Keyword(s):  
Blow Up ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Classical Mechanics ◽  
Index Theorem ◽  
Travelling Wave ◽  
Kdv Equations ◽  
The Kdv Equation ◽  
De Vries ◽  
The Stability

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

On gradients of approximate travelling waves for generalised KPP equations

1997 ◽  
Vol 127 (2) ◽  
pp. 423-439 ◽  
Author(s):  
H. Z. Zhao
Keyword(s):  
Travelling Waves ◽  
Probabilistic Approach ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Difficult Problem ◽  
Logarithmic Transformation ◽  
Reaction Diffusion Equations ◽  
Kpp Equation ◽  
Dirac Delta ◽  
Kpp Equations

In this paper we use stochastic semiclassical analysis and the logarithmic transformation to study the gradients of the approximate travelling wave solutions for the generalised KPP equations with Gaussian and Dirac delta initial distributions. We apply the logarithmic transformation to the nonlinear reaction diffusion equations and obtain a Maruyama–Girsanov–Cameron–Martin formula for the driftμ2∇loguμ,uμbeing a solution of a generalised KPP equation. We obtain thatμ2|∇ loguμ(t,x)| is bounded and the trough is flat. The difficult problem in this paper is to prove that the corresponding crest is flat. A probabilistic approach is used in this paper to treat this problem successfully.

Numerical solution of Fisher's equation

1974 ◽  
Vol 11 (3) ◽  
pp. 445-457 ◽  
Author(s):  
Jenö Gazdag ◽  
José Canosa
Keyword(s):  
Water Waves ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Step Function ◽  
Local Perturbation ◽  
Fisher’S Equation ◽  
Minimal Speed ◽  
Fisher's Equation

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration.

Stochastic generalised KPP equations

1996 ◽  
Vol 126 (5) ◽  
pp. 957-983 ◽  
Author(s):  
I. M. Davies ◽  
A. Truman ◽  
H. Z. Zhao
Keyword(s):  
White Noise ◽  
Numerical Solutions ◽  
Travelling Waves ◽  
Wave Structure ◽  
Travelling Wave ◽  
Wave Form ◽  
Kpp Equations ◽  
Image Position ◽  
Theoretical Results ◽  
Good Agreement

We classify multiplicative white noise perturbationsk(·)dw, of generalised KPP equations and their effects on deterministic approximate travelling wave solutions by the behaviour of, the solutions of the stochastic generalised KPP equations converge to deterministic approximate travelling waves and ifbeing an associated potential energy, Фsa solution of the corresponding classical mechanical equations of Newton,Dbeing a certain domain inR1×Rrthen the white noise perturbations essentially destroy the wave structure and force the solutions to die down.For the case(suppose the existence of the limit) we show that there is a residual wave form but propagating at a different speed from that of the unperturbed equations. Numerical solutions are included and give good agreement with theoretical results.

Travelling Wave Control of Rotating Discs and Analysis of its Spillover Effect

1988 ◽  
Vol 202 (2) ◽  
pp. 119-127 ◽  
Author(s):  
C-S Kim ◽  
C-W Lee
Keyword(s):  
Travelling Waves ◽  
Spillover Effect ◽  
Polynomial Equation ◽  
Travelling Wave ◽  
Rotating Disc ◽  
System A ◽  
Finite Dimensional ◽  
Control Scheme ◽  
Actuator System ◽  
Wave Control

A modal control scheme for rotating disc systems is developed based upon the finite-dimensional sub-system model including a few lower backward travelling waves important to the disc response. For the single discrete sensor and actuator system, a polynomial equation, which determines the closed-loop system poles, is derived and the spillover effect is analysed, providing a sufficient condition for stability. Finally, simulation studies are performed to show the effectiveness of the travelling wave control scheme proposed.

TRAVELLING WAVES AND OSCILLATIONS IN SAL’NIKOV’S COMBUSTION REACTION IN A COMPRESSIBLE GAS

2015 ◽  
Vol 56 (3) ◽  
pp. 233-247 ◽  
Author(s):  
RHYS A. PAUL ◽  
LAWRENCE K. FORBES
Keyword(s):  
Asymptotic Solution ◽  
Multiple Scales ◽  
Travelling Waves ◽  
Method Of Lines ◽  
Travelling Wave ◽  
Reaction Scheme ◽  
Compressible Viscous Gas ◽  
The Method Of Lines ◽  
Temperature Waves ◽  
Parameter Values

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

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 Refined long-time asymptotics for Fisher–KPP fronts

2019 ◽  
Vol 21 (07) ◽  
pp. 1850072 ◽  
Author(s):  
James Nolen ◽  
Jean-Michel Roquejoffre ◽  
Lenya Ryzhik
Keyword(s):  
Travelling Waves ◽  
American Mathematical Society ◽  
Front Propagation ◽  
Step Function ◽  
Kolmogorov Equation ◽  
Compact Set ◽  
Kpp Equation ◽  
Branching Brownian Motion ◽  
Long Time ◽  
Unstable States

We study the one-dimensional Fisher–KPP equation, with an initial condition [Formula: see text] that coincides with the step function except on a compact set. A well-known result of Bramson in [Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978) 531–581; Convergence of Solutions of the Kolmogorov Equation to Travelling Waves (American Mathematical Society, Providence, RI, 1983)] states that, as [Formula: see text], the solution converges to a traveling wave located at the position [Formula: see text], with the shift [Formula: see text] that depends on [Formula: see text]. Ebert and Van Saarloos have formally derived in [Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Phys. D 146 (2000) 1–99; Front propagation into unstable states, Phys. Rep. 386 (2003) 29–222] a correction to the Bramson shift, arguing that [Formula: see text]. Here, we prove that this result does hold, with an error term of the size [Formula: see text], for any [Formula: see text]. The interesting aspect of this asymptotics is that the coefficient in front of the [Formula: see text]-term does not depend on [Formula: see text].

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.

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