zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $$u_{tt}+ u_{xxxx}+ u^+=1\tag1$$ 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:
35L75Nonlinear hyperbolic PDE of higher $(>2)$ order
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
WorldCat.org
Full Text: DOI
References:
[1] Ammann, O. H.; Von Karman, T.; Woodruff, G. B.: The failure of the tacoma narrow Bridge. (1941)
[2] Mckenna, P. J.; Walter, W.: Traveling waves in a suspension Bridge. SIAM J. Appl. math. 50, 703-715 (1990) · Zbl 0699.73038
[3] Rabinowitz, P. H.: CBMS reg. Conf. ser. Math.. 65 (1986)
[4] Brezis, H.; Nirenberg, L.: Remarks on finding critical points. Comm. pure appl. Math. 44, 939-963 (1991) · Zbl 0751.58006
[5] Adams, R. A.: Sobolev spaces. (1975) · Zbl 0314.46030
[6] Hutson, V.; Pym, J. S.: Applications of functional analysis and operator theory. (1980) · Zbl 0426.46009
[7] Choi, Y. S.; Mckenna, P. J.; Romano, M.: A mountain pass method for the numerical solution of semilinear wave equations. Numer. math. 64, 487-509 (1993) · Zbl 0796.65109
[8] Choi, Y. S.; Mckenna, P. J.: A mountain pass method for the numerical solution of semilinear elliptic problems. (1993) · Zbl 0779.35032
[9] Strikwerda, H. C.: Finite difference schemes and partial differential equations. (1989) · Zbl 0681.65064
[10] Strauss, W. A.: CBMS reg. Conf. ser. Math.. 73 (1989)
[11] Newell, Alan C.: CBMS-NSF reg. Conf. ser. Applied math.. 48 (1985)
[12] B. Buffoni, Periodic and homoclinic orbits for Lorentz-Lagrangian systems via variational methods, Nonlinear Anal. · Zbl 0838.49004
[13] Lions, P. L.: The concentration-compactness principle in the calculus of variations, part 1. Ann. inst. H. Poincaré anal. Non linéaire 1, 109-145 (1984)
[14] P. Deift, T. Kriecherbauer, S. Venakides, Forced lattice vibrations--A videotext, Comm. Pure Appl. Math. · Zbl 0860.34005