On the dimension of the set of solutions for nonlocal nonlinear wave equation

G. L. Alfimov

Russian journal of nonlinear dynamics, Tome 7 (2011) no. 2, pp. 209-226.

Résumé

Nonlocal generalizations of nonlinear wave equation arise in numerous physical applications. It is known that switching from local to nonlocal description may result in new features of the problem and new types of solutions. In this paper the author analyses the dimension of the set of travelling wave solutions for а nonlocal nonlinear wave equation. The nonlocality is represented by the convolution operator which replaces the second derivative in the dispersion term. The results have been obtained for the case where the nonlinearity is bounded, and the kernel of the convolution operator is represented by a sum of exponents with weights (so-called E-type kernel). In the simplest particular case, (so-called Kac–Baker kernel) it is shown that the solutions of this equation form a 3-parametric set (assuming the equivalence of the solutions which differ by a shift with respect to the independent variable). Then it is shown that in the case of the general E-type kernel the 3-parametric set of solutions also exists, generically, under some additional restrictions. The word «generically» in this case means some transversality condition for intersection of some manifolds in a properly defined phase space.

Détail

Keywords: nonlocal nonlinear wave equation.

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