##
**Numerical mountain pass solutions of a suspension bridge equation.**
*(English)*
Zbl 0877.35126

The author develops a new global numerical method to find the large amplitude oscillations of a nonlinear model for suspension bridges. The central idea of the method is to find the solutions in a straightforward global way, without requiring good initial guesses as to where solutions might be. The framework of the algorithm is based on a work by Choi and McKenna. The mountain pass algorithm is a constructive implementation of the mountain pass theorems of Ambrosetti, Rabinowitz, and Ekeland.

The basic idea of the method is quite intuitive. On a finite-dimensional approximation subspace, take a piecewise linear path joining the local minimum and a point whose image is lower. After calculating the maximum of the functional along the path, the path is deformed by pushing the point at which the maximum is located in the direction of the steepest descent. One repeats this step, stopping only when the critical point is reached. The algorithm is extremely robust and global in nature and it was tested in the cases where the mathematical proof guarantees the existence of large amplitude periodic solutions and when the solutions can be reduced to solutions of ordinary differential equations. Some solutions are obtained where there are no proofs of existence.

The basic idea of the method is quite intuitive. On a finite-dimensional approximation subspace, take a piecewise linear path joining the local minimum and a point whose image is lower. After calculating the maximum of the functional along the path, the path is deformed by pushing the point at which the maximum is located in the direction of the steepest descent. One repeats this step, stopping only when the critical point is reached. The algorithm is extremely robust and global in nature and it was tested in the cases where the mathematical proof guarantees the existence of large amplitude periodic solutions and when the solutions can be reduced to solutions of ordinary differential equations. Some solutions are obtained where there are no proofs of existence.

Reviewer: L.Vazquez (Madrid)

### MSC:

35Q72 | Other PDE from mechanics (MSC2000) |

65N99 | Numerical methods for partial differential equations, boundary value problems |

35G20 | Nonlinear higher-order PDEs |

35A15 | Variational methods applied to PDEs |

PDFBibTeX
XMLCite

\textit{L. D. Humphreys}, Nonlinear Anal., Theory Methods Appl. 28, No. 11, 1811--1826 (1997; Zbl 0877.35126)

Full Text:
DOI

### References:

[1] | DOOLE S. H. & HOGAN S. J., Piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation. (To appear.); DOOLE S. H. & HOGAN S. J., Piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation. (To appear.) · Zbl 0855.34041 |

[2] | Choi, Y. S.; Jen, K. C.; McKenna, P. J., The structures of the solution set for periodic oscillations in a suspension bridge model, IMA J. Appl. Math., 47, 283-306 (1991) · Zbl 0756.73041 |

[3] | Lazer, A. C.; McKenna, P. J., Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32, 537-578 (1990) · Zbl 0725.73057 |

[4] | McKenna, P. J.; Walter, W., Nonlinear oscillations in a suspension bridge, Arch. ration. Mech. Analysis, 98, 167-177 (1987) · Zbl 0676.35003 |

[5] | Choi, Q. H.; Jung, T., The study of a nonlinear suspension bridge equation by a variational reduction method, Appl. Analysis, 50, 73-92 (1993) · Zbl 0739.35044 |

[6] | Humphreys, L., Numerical and theoretical results on large amplitude periodic solutions of a suspension bridge equation, (Ph.D. thesis (1994), University of Connecticut: University of Connecticut New York) |

[7] | Choi, Y. S.; McKenna, P. J., A mountain pass method for the numerical solutions of semilinear elliptic problems, Nonlinear Analysis, 20, 417-437 (1993) · Zbl 0779.35032 |

[8] | Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. funct. Analysis, 14, 349-381 (1973) · Zbl 0273.49063 |

[9] | Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, (Conf. Board Math. Sci. Reg. Conf. Ser. Math., 65 (1986)), 1-44 · Zbl 0609.58002 |

[10] | Choi, Y. S.; McKenna, P. J.; Romano, M., A mountain pass method for the numerical solution of semilinear wave equations, Num. Math., 64, 487-509 (1993) · Zbl 0796.65109 |

[11] | CHEN Y., Travelling wave solutions for nonlinear beam equations. Ph.D. thesis, University of Connecticut. (In preparation.); CHEN Y., Travelling wave solutions for nonlinear beam equations. Ph.D. thesis, University of Connecticut. (In preparation.) |

[12] | Coron, J. M., Periodic solutions of a nonlinear wave equation without assumptions of monotonicity, Math. Annln., 262, 273-285 (1983) · Zbl 0489.35061 |

[13] | Brezis, H., Periodic solutions of nonlinear vibrating strings, (Proceedings AMS Symposium on the Mathematical Heritage of H. Poincaré (April 1980)) · Zbl 0543.35065 |

[14] | Brezis, H.; Nirenberg, L., Forced vibrations for a nonlinear wave equation, Communs pure appl. Math., 31, 1-30 (1978) · Zbl 0378.35040 |

[15] | Strang, G., Introduction to Applied Mathematics (1986), Wellesley Press · Zbl 0618.00015 |

[16] | Burden, R. L.; Faires, J. D., Numerical Analysis (1989), PWS-Kent: PWS-Kent Cambridge · Zbl 0671.65001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.