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Stability of Semi-Discrete Shock Profiles by Means of an Evans Function in Infinite Dimensions

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Abstract

Semi-discrete shock profiles are traveling wave solutions of hyperbolic systems of conservation laws under discretization in space. The existence of semi-discrete shocks has been investigated in earlier papers. Here the spectral stability of those nonlinear waves is addressed, and formulated in terms of a variational delay differential operator. Constructing a generalized Evans function, in infinite dimensions, it is shown how to derive stability criteria. Some examples are given when the criterion is fully explicit, e.g., for extreme Lax shocks. Additionally, connection is made with the alternative approach proposed by Chow, Mallet-Paret, and Shen (Journal of Differential Equations 1998), regarding the stability of traveling waves in general Lattice Dynamical Systems.

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REFERENCES

  1. Alexander, J. C., Gardner, R., and Jones, C. K. R. T. (1990). A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212.

    Google Scholar 

  2. Battelli, F. (1990). Melnikov functions and heteroclinic orbits in delay differential equations. J. Math. Anal. Appl. 150(2), 319–334.

    Google Scholar 

  3. Benzoni-Gavage, S. (1998). Semi-discrete shock profiles for hyperbolic systems of conservation laws. Phys. D 115(1/2), 109–123.

    Google Scholar 

  4. Benzoni-Gavage, S. (1999). On the stability of semidiscrete shock profiles by means of an Evans function in infinite dimension. C. R. Acad. Sci. Paris, Série I 329, 377–382.

    Google Scholar 

  5. Benzoni-Gavage, S., Serre, D., and Zumbrun, K. (2001). Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32(5), 929–962.

    Google Scholar 

  6. Bridges, T. J., and Derks, G. (1999). Hodge duality and the Evans function. Phys. Lett. A 251(6), 363–372.

    Google Scholar 

  7. Chow, S.-N., Mallet-Paret, J., and Shen, W. (1998). Traveling waves in lattice dynamical systems. J. Differential Equations 149(2), 248–291.

    Google Scholar 

  8. Coppel, W. A. (1978). Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin.

    Google Scholar 

  9. Delfour, M. C., and Manitius, A. (1980). The structural operator F and its role in the theory of retarded systems. I. J. Math. Anal. Appl. 73(2), 466–490.

    Google Scholar 

  10. Diekmann, O., van Gils, S. A., Verduyn Lunel, S. M., and Walther, H.-O. (1995). Delay Equations, Functional, complex, and nonlinear analysis, Springer-Verlag, New York.

    Google Scholar 

  11. Evans, J. W. (1971/72). Nerve axon equations. I. Linear approximations. Indiana Univ. Math. J. 21, 877–885.

    Google Scholar 

  12. Evans, J. W. (1972/73). Nerve axon equations. II. Stability at rest. Indiana Univ. Math. J. 22, 75–90.

    Google Scholar 

  13. Evans, J. W. (1972/73). Nerve axon equations. III. Stability of the nerve impulse. Indiana Univ. Math. J. 22, 577–593.

    Google Scholar 

  14. Evans, J. W. (1974/75). Nerve axon equations. IV. The stable and the unstable impulse. Indiana Univ. Math. J. 24(12), 1169–1190.

    Google Scholar 

  15. Fries, C. (1998). Nonlinear asymptotic stability of general small-amplitude viscous Laxian shock waves. J. Differential Equations 146(1), 185–202.

    Google Scholar 

  16. Gardner, R. A., and Jones, C. K. R. T. (1990). Traveling waves of a perturbed diffusion equation arising in a phase field model. Indiana Univ. Math. J. 39(4), 1197–1222.

    Google Scholar 

  17. Gardner, R. A., and Zumbrun, K. (1998). The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51(7), 797–855.

    Google Scholar 

  18. Godillon, P. (2001). Linear stability of shock profiles for systems of conservation laws with semi-linear relaxation. Phys. D 148, 289–316.

    Google Scholar 

  19. Goodman, J. (1986). Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95(4), 325–344.

    Google Scholar 

  20. Hale, J. K., and Lunel, S. M. V. (1993). Introduction to Functional Differential Equations Springer-Verlag.

  21. Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin.

    Google Scholar 

  22. Hille, E., and Phillips, R. S. (1957). Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., rev. ed.

    Google Scholar 

  23. Huot, P. Existence and Stability of Semi-Discrete Shock Profiles, Ph.D. thesis, E´ cole Normale Supérieure de Lyon, in preparation.

  24. Kapitula, T. (1999). The Evans function and generalized Melnikov integrals. SIAM J. Math. Anal. 30(2), 273–297 (electronic).

    Google Scholar 

  25. Kapitula, T., and Sandstede, B. (1998). Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations. Phys. D 124(1-3), 58–103.

    Google Scholar 

  26. Kappel, F., and Schappacher, W. (1979). Nonlinear functional-differential equations and abstract integral equations. Proc. Roy. Soc. Edinburgh Sect. A 84(1/2), 71–91.

    Google Scholar 

  27. Kato, T. (1966). Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag, New York, New York.

    Google Scholar 

  28. Lin, X.-B. (1986). Exponential dichotomies and homoclinic orbits in functional-differential equations. J. Differential Equations 63(2), 227–254.

    Google Scholar 

  29. Liu, T.-P. (1985). Nonlinear stability of shock waves for viscous conservation laws. Mem. Amer. Math. Soc. 56(328), v+108.

    Google Scholar 

  30. Liu, T.-P., and Yu, S.-H. (1999). Continuum shock profiles for discrete conservation laws, I. construction. Comm. Pure Appl. Math. 52(1), 85–127.

    Google Scholar 

  31. Liu, T.-P., and Yu, S.-H. (1999). Continuum shock profiles for discrete conservation laws. II. Stability. Comm. Pure Appl. Math. 52(9), 1047–1073.

    Google Scholar 

  32. Majda, A., and Ralston, J. (1979). Discrete shock profiles for systems of conservation laws. Comm. Pure Appl. Math. 32(4), 445–482.

    Google Scholar 

  33. Mallet-Paret, J. (1999). The Fredholm alternative for functional differential equations of mixed type. J. Dynam. Differential Equations 11(1), 1–47.

    Google Scholar 

  34. Matsumura, A., and Nishihara, K. (1985). On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2(1), 17–25.

    Google Scholar 

  35. Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York.

    Google Scholar 

  36. Pego, R. L., and Weinstein, M. I. (1992). Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340(1656), 47–94.

    Google Scholar 

  37. Rudin, W. (1987). Real and Complex Analysis, 3rd ed., McGraw-Hill, New York.

    Google Scholar 

  38. Schechter, M. (1971). Spectra of Partial Differential Operators, North-Holland, Amsterdam.

    Google Scholar 

  39. Swinton, J. (1992). The stability of homoclinic pulses: a generalisation of Evans's method. Phys. Lett. A 163(1/2), 57–62.

    Google Scholar 

  40. Szepessy, A., and Xin, Z. P. (1993). Nonlinear stability of viscous shock waves. Arch. Rational Mech. Anal. 122(1), 53–103.

    Google Scholar 

  41. Vanderbauwhede, A., and Iooss, G. (1992). Center manifold theory in infinite dimensions. Dyn. Reported 1, 125–163.

    Google Scholar 

  42. Zumbrun, K., and Howard, P. (1998). Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47(3), 741–871.

    Google Scholar 

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Benzoni-Gavage, S. Stability of Semi-Discrete Shock Profiles by Means of an Evans Function in Infinite Dimensions. Journal of Dynamics and Differential Equations 14, 613–674 (2002). https://doi.org/10.1023/A:1016391200280

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