Travelling Waves for Fourth Order Parabolic Equations

We study travelling wave solutions for a class of fourth order parabolic equations. Travelling wave fronts of the form u(x, t )= U (x + ct), connecting homogeneous states, are proven to exist in various cases: connections between two stable states, as well as connections between an unstable and a stable state, are considered.

[1]  William C. Troy,et al.  Pulse-like Spatial Patterns Described by Higher-Order Model Equations☆ , 1998 .

[2]  Dee Gt,et al.  Bistable systems with propagating fronts leading to pattern formation. , 1988 .

[3]  R. Gardner,et al.  TRAVELING WAVES OF A PERTURBED DIFFUSION EQUATION ARISING IN A PHASE FIELD MODEL , 1990 .

[4]  J. Kwapisz,et al.  Homotopy Classes for Stable Periodic and Chaotic¶Patterns in Fourth-Order Hamiltonian Systems , 2000 .

[5]  J. B. Berg.  Uniqueness of solutions for the extended Fisher-Kolmogorov equation , 1998 .

[6]  Linearly induced vector fields and R2-actions on spheres , 1975 .

[7]  J. McLeod,et al.  The approach of solutions of nonlinear diffusion equations to travelling front solutions , 1977 .

[8]  J. Kwapisz.  Uniqueness of the Stationary Wave for the Extended Fisher-Kolmogorov Equation , 2000 .

[9]  F. Dumortier,et al.  Local Study of Planar Vector Fields: Singularities and Their Unfoldings , 1991 .

[10]  D. Aronson,et al.  Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation , 1975 .

[11]  M. Karlsson,et al.  Radiationless optical solitons with oscillating tails , 1994 .

[12]  Freddy Dumortier,et al.  Structures in Dynamics: Finite Dimensional Deterministic Studies , 1991 .

[13]  Alan R. Champneys,et al.  Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system , 1996 .

[14]  C. Conley.  Isolated Invariant Sets and the Morse Index , 1978 .

[15]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[16]  B. Buuoni.  Shooting Methods and Topological Transversality , 1996 .

[17]  Mark A. Peletier,et al.  NON-EXISTENCE AND UNIQUENESS RESULTS FOR FOURTH-ORDER HAMILTONIAN SYSTEMS , 1999 .

[18]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[19]  Freddy Dumortier,et al.  Singularities of vector fields on the plane , 1977 .

[20]  William C. Troy,et al.  A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation , 1995 .

[21]  J. Kwapisz,et al.  Homotopy Classes for Stable Connections between Hamiltonian Saddle-Focus Equilibria , 1998 .

[22]  J. Hulshof,et al.  TRAVELLING WAVE SOLUTIONS OF A FOURTH-ORDER SEMILINEAR DIFFUSION EQUATION , 1998 .

[23]  L. Perko.  Differential Equations and Dynamical Systems , 1991 .

[24]  J. F. Toland,et al.  Homoclinic orbits in the dynamic phase-space analogy of an elastic strut , 1992, European Journal of Applied Mathematics.

[25]  G. E. Bredon.  Topology and geometry , 1993 .

[26]  William D. Kalies,et al.  MULTITRANSITION HOMOCLINIC AND HETEROCLINIC SOLUTIONS OF THE EXTENDED FISHER-KOLMOGOROV EQUATION , 1996 .

[27]  W. Saarloos,et al.  UNIVERSAL ALGEBRAIC RELAXATION OF FRONTS PROPAGATING INTO AN UNSTABLE STATE AND IMPLICATIONS FOR MOVING BOUNDARY APPROXIMATIONS , 1997, patt-sol/9707004.

[28]  V. Rottschäfer,et al.  Existence and Stability of Traveling Fronts in the Extended Fisher-Kolmogorov Equation , 2001 .

[29]  William C. Troy,et al.  Spatial patterns described by the extended Fisher-Kolmogorov (EFK) equation: kinks , 1995, Differential and Integral Equations.