Geodesic
Quasitravelling waves
Sbornik. Mathematics, Tome 201 (2010) no. 12, pp. 1731-1775 Cet article a éte moissonné depuis la source Math-Net.Ru

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A finite difference analogue of the wave equation with potential perturbation is investigated, which simulates the behaviour of an infinite rod under the action of an external longitudinal force field. For a homogeneous rod, describing solutions of travelling wave type is equivalent to describing the full space of classical solutions to an induced one-parameter family of functional differential equations of point type, with the characteristic of the travelling wave as parameter. For an inhomogeneous rod, the space of solutions of travelling wave type is trivial, and their ‘proper’ extension is defined as solutions of ‘quasitravelling’ wave type. By contrast to the case of a homogeneous rod, describing the solutions of quasitravelling wave type is equivalent to describing the quotient of the full space of impulsive solutions to an induced one-parameter family of point-type functional differential equations by an equivalence relation connected with the definition of solutions of quasitravelling wave type. Stability of stationary solutions is analyzed. Bibliography: 9 titles.
Keywords: functional differential equations, scale of function spaces, wave equation, travelling waves.
Mots-clés : impulsive solutions 
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L. A. Beklaryan. Quasitravelling waves. Sbornik. Mathematics, Tome 201 (2010) no. 12, pp. 1731-1775. http://geodesic.mathdoc.fr/item/SM_2010_201_12_a1/

[1] L. A. Beklaryan, Vvedenie v teoriyu funktsionalno-differentsialnykh uravnenii. Gruppovoi podkhod, Faktorial Press, M., 2007

[2] L. A. Beklaryan, M. B. Kruchenov, “On the solvability of a linear homogeneous functional-differential equation of point type”, Differ. Equ., 44:4 (2008), 453–463 | DOI | MR | Zbl

[3] J. Frenkel, T. Kontorova, “On the theory of plastic deformation and twinning”, Acad. Sci. U.S.S.R. J. Phys., 1 (1939), 137–149 | MR | Zbl

[4] L. D. Pustyl'nikov, “Infinite-dimensional non-linear ordinary differential equations and the KAM theory”, Russian Math. Surveys, 52:3 (1997), 551–604 | DOI | MR | Zbl

[5] L. A. Beklaryan, “Equations of advanced-retarded type and solutions of traveling-wave type for infinite-dimensional dynamic systems”, J. Math. Sci. (N. Y.), 124:4 (2004), 5098–5109 | DOI | MR | Zbl

[6] J. K. Hale, S. M. Verduyn Lunel, Introduction to functional differential equations, Appl. Math. Sci., 99, Springer-Verlag, New York, NY, 1993 | MR | Zbl

[7] Yu. D. Latushkin, A. M. Stëpin, “Weighted translation operators and linear extensions of dynamical systems”, Russian Math. Surveys, 46:2 (1991), 95–165 | DOI | MR | Zbl

[8] A. Antonevich, A. Lebedev, Functional-differential equations. I. $C^*$-theory, Pitman Monogr. Surveys Pure Appl. Math., 70, Longman, Harlow, 1994 | MR | Zbl

[9] Ju. L. Dalec'kiǐ, M. G. Kreǐn, Stability of solutions of differential equations in Banach space, Amer. Math. Soc., Providence, RI, 1974 | MR | MR | Zbl | Zbl