Lower bounds for admissible values of the travelling wave speed in asymmetrically supported beam
DOI:
https://doi.org/10.12775/TMNA.2025.023Keywords
Beam equation, jumping nonlinearity, travelling wave, Mountain Pass Theorem, Fučík spectrum, Swift-Hohenberg operator, Padé approximationAbstract
We study the admissible values of the wave speed $c$ for which the beam equation with jumping nonlinearity possesses a travelling wave solution. In contrast to previously studied problems modelling suspension bridges, the presence of the term with negative part of the solution in the equation results in restrictions of $c$. In this paper, we provide the maximal wave speed range for which the existence of the travelling wave solution can be proved using the Mountain Pass Theorem. We also introduce its close connection with related Dirichlet problems and their Fučík spectra. Moreover, we present several analytical approximations of the main existence result with assumptions that are easy to verify. Finally, we formulate a conjecture that the infimum of the admissible wave speed range can be described by the Fučík spectrum of a simple periodic problem.References
G.A. Baker Jr. and P. Graves-Morris, Padé Approximants, Encyclopedia of Mathematics and its Applications, vol. 59, 2nd ed., Cambridge University Press, Cambridge, 1996, xiv+746 pp.
A.R. Champneys and P.J. McKenna, On solitary waves of a piecewise linear suspended beam model, Nonlinearity 10 (1997), no. 6, 1763.
Y. Chen, Traveling wave solutions to beam equation with fast-increasing nonlinear restoring forces, Appl. Math. J. Chinese Univ. 15 (2000), no. 2, 156–160.
Y. Chen and P.J. McKenna, Traveling waves in a nonlinearly suspended beam: Theoretical results and numerical observations, J. Differential Equations 136 (1997), 325–355.
G. Dattoli, S. Lorenzutta and C. Cesarano, Finite sums and generalized forms of Bernoulli polynomials, Rend. Mat. Appl. (7) 19 (1999), no. 3, 385–391.
P. Drábek and J. Milota, Methods of Nonlinear Analysis, Springer Basel, 2013.
L. Greenberg, An oscillation method for fourth-order, selfadjoint, two-point boundary value problems with nonlinear eigenvalues, SIAM J. Math. Anal. 22 (1991), no. 4, 1021–1042.
G. Holubová and H. Levá, Travelling wave solutions of the beam equation with jumping nonlinearity, J. Math. Anal. Appl. 527 (2023), no. 2, 127466.
P. Karageorgis and J. Stalker, A lower bound for the amplitude of traveling waves of suspension bridges, Nonlinear Anal. 75 (2012), no. 13, 5212–5214.
P. Krejčı́, On solvability of equations of the 4th order with jumping nonlinearities, Čas. Pěstovánı́ Mat. 108 (1983), 29–39.
J. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15 (1977), no. 1, 319–328.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
