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Traveling waves in a nonlinearly suspended beam: Theoretical results and numerical observations. (English) Zbl 0879.35113

The existence of traveling wave solutions of the equation \[ u_{tt}+ u_{xxxx}+ u^+=1\tag{1} \] is proved by a variational method. It is shown by using the concentrated compactness ideas of P. L. Lions, that the mountain pass lemma can be applied to get the nontrival critical point of an associated functional, which is a nontrivial solution of (1). To find the traveling wave solution the mountain pass algorithm is used. This algorithm is similar to that for finding the mountain pass type critical points in the direct variational formulation of a semilinear elliptic equation, the main difference is that the steepest descent direction in being sought in the \(H^2\) norm in the present case. Results of computations show that the piecewise linear nonlinearity is a source of some numerical errors, and the smooth nonlinearity gives mountain pass type solutions which have extremely interesting properties.

MSC:

35L75 Higher-order nonlinear hyperbolic equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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