Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Wavefronts for a cooperative tridiagonal system of differential equations

  • 54 Accesses

  • 40 Citations

Abstract

Consider the infinite system of nonlinear differential equations\(\dot u\) n =f(n−1, un, un+1),nεℤ, wherefεC 1,D 1 f > 0,D 3 f>0, andf(0, 0, 0) = 0 =f(1, 1, 1). Existence of wavefronts—i.e., solutions of the formu n (t) = U(n + ct), whereℝ,U(− ∞) = 0,U(+∞) = 1, andU is strictly increasing—is shown for functionsf which satisfy the condition: there existsa, 0<a<1, such thatf(x, x,x)<0 for 0<x<a andf(x, x, x) > 0 fora < x < 1.

This is a preview of subscription content, log in to check access.

References

  1. Bell, J. (1981). Some threshold results for models of myelinated nerves.Math. Biosci. 54, 181–190.

  2. Bell, J., and Cosner, C. (1984). Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons.Q. Appl. Math. 42, 1–14.

  3. Chi, H., Bell, J., and Hassard, B. (1986). Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory.J. Math. Biol. 24, 583–601.

  4. Keener, J. P. (1987). Propagation and its failure in coupled systems of discrete excitable cells.SIAM J. Appl. Math. 47, 556–572.

  5. Kolmogoroff, A., Petrovsky, I., and Piscounoff, N. (1937). étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique.Moscow Univ. Bull. Math. 1, 1–25.

  6. McKean, H. K. (1970). Nagumo's equation.Adv. Math. 4, 209–223.

  7. Rustichini, A. (1989a). Functional differential equations of mixed type: The linear autonomous case.J. Dynam. Diff. Eq. 1, 121–143.

  8. Rustichini, A. (1989b). Hopf bifurcation for functional differential equations of mixed type.J. Dynam. Diff. Eq. 1, 145–177.

  9. Smith, H. L. (1990). A discrete Lyapunov function for a class of linear differential equations.Pacific J. Math. 144, 345–360.

  10. Walter, W. (1970).Differential and Integral Inequalities, Springer, New York.

  11. Zinner, B. (1991). Stability of traveling wavefronts for the discrete Nagumo equation.SIAM J. Math. Anal. 22, 1016–1020.

  12. Zinner, B. (1992). Existence of traveling wavefront solutions for the discrete Nagumo equation.J. Diff. Eq. 96, 1–27.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hankerson, D., Zinner, B. Wavefronts for a cooperative tridiagonal system of differential equations. J Dyn Diff Equat 5, 359–373 (1993). https://doi.org/10.1007/BF01053165

Download citation

Key words

  • Traveling waves
  • Nagumo equation
  • cooperative systems
  • comparision principles