Exact travelling wave solutions of a class of nonlinear diffusion equations by reduction to a quadrature

Otwinowski, M.; Paul, R.; Laidlaw, W. G.

doi:10.1016/0375-9601(88)90880-8

Cited by 102 publications

(61 citation statements)

References 10 publications

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“…We here discuss, how to find analytical solutions for uniformly translating fronts φ(ξ ) in the equation • We rephrase and straighten the method from [65] (see also [108,109]) how to find analytical front solutions.…”

Section: Appendix C Analytical Solutions For Pushed Nonlinear Diffusmentioning

confidence: 99%

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Ebert

Saarloos

2000

Physica D: Nonlinear Phenomena

360

596

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Fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes: pulled fronts and pushed fronts. The term "pulled front" expresses that these fronts are "pulled along" by the spreading of linear perturbations about the unstable state. Accordingly, their asymptotic speed v * equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state. The central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towards v * . We show that when such fronts evolve from "sufficiently steep" initial conditions, which initially decay faster than e −λ * x for x → ∞, they have a universal relaxation behavior as time t → ∞: the velocity of a pulled front always relaxes algebraically likeThe parameters v * , λ * , and D are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state. This front velocity is independent of the precise value of the front amplitude, which one tracks to measure the front position. The interior of the front is essentially slaved to the leading edge, and develops universally asis a uniformly translating front solution with velocity v < v * .Our result, which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well-known nonlinear diffusion equation of type ∂ t φ = ∂ 2 x φ + φ − φ 3 , where the invaded unstable state is φ = 0. Even for this simple case, the subdominant t −3/2 term extends an earlier result of Bramson. Our analysis is then generalized to more general (sets of) partial differential equations with higher spatial or temporal derivatives, to PDEs with memory kernels, and also to difference equations such as those that occur in numerical finite difference codes. Our universal result for pulled fronts thus implies independence (i) of the level curve which is used to track the front position, (ii) of the precise nonlinearities, (iii) of the precise form of the linear operators in the dynamical equation, and (iv) of the precise initial conditions, as long as they are sufficiently steep. The only remainders of the explicit form of the dynamical equation are the nonlinear solutions v and the three saddle point parameters v * , λ * , and D. As our simulations confirm all our analytical predictions in every detail, it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts, if they are uniformly translating for t → ∞. An immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions. In addition to our main result for pulled fronts, we also discuss the propagation and convergence of fronts emerging from * Corresponding author. Present address: CWI, Postbus 94079, 1090 GB Amsterdam, Netherlands 0167-2789/00/$ -see front matter © 2000 Else...

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Section: Appendix C Analytical Solutions For Pushed Nonlinear Diffusmentioning

confidence: 99%

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Ebert

Saarloos

2000

Physica D: Nonlinear Phenomena

360

596

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“…In the mathematical literature [31] there has been a classification of solutions of equations of this form. For certain forms of the potential [1] the solutions include kink solitons that may be responsible for dissipation-free energy transfer in biological cells [9]:…”

Section: Tubulin Microtubules and Coherent Statesmentioning

confidence: 99%

Qed-Cavity Model of Microtubules Implies Dissipationless Energy Transfer and Biological Quantum Teleportation

Mavromatos

Mershin²,

Nanopoulos

2002

Int. J. Mod. Phys. B

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We refine a QED-cavity model of microtubules (MTs), proposed earlier by two of the authors (N.E.M. and D.V.N.), and suggest mechanisms for the formation of biomolecular mesoscopic coherent and/or entangled quantum states, which may avoid decoherence for times comparable to biological characteristic times. This refined model predicts dissipationless energy transfer along such "shielded" macromolecules at near room temperatures as well as quantum teleportation of states across MTs and perhaps neurons.

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“…Moreover, we seeking the exact solutions of the NLPDEs play an important role in the nonlinear problems, a number of methods have been developed, such as inverse scattering theory (Gardner et al, 1967), Hirota's bilinear methods (Hirota, 1973), the truncated Painlevé expansion (Weiss et al, 1983;Cariello and Tabor, 1991), homogeneous balance method (Wang, 1995(Wang, , 1996, reduction of the NLPDE to a quadrature problem (Otwinowski et al, 1988), etc.…”

Section: Introductionmentioning

confidence: 99%

Conservation Laws and Exact Solutions for some Nonlinear Partial Differential Equations

2006

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is used for finding the local conservation laws for some nonlinear partial differential equations. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that of finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. Different methods to construct new exact solution classes for the same nonlinear partial differential equations are also presented, which are named hyperbolic function method and the Bäcklund transformations. On the other hand, other methods and transformations are developed to obtain exact solutions for some nonlinear partial differential equations.

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Exact travelling wave solutions of a class of nonlinear diffusion equations by reduction to a quadrature

Cited by 102 publications

References 10 publications

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Qed-Cavity Model of Microtubules Implies Dissipationless Energy Transfer and Biological Quantum Teleportation

Conservation Laws and Exact Solutions for some Nonlinear Partial Differential Equations

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