Abstract
This paper begins with a review of earlier work on extending the notion of an eigenvalue problem for a linear differential equation to an eigenvalue problem for a nonlinear differential equation. In previous work it was argued that in the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that spawn the separatrix play the role of an eigenvalue. Previously discussed nonlinear differential equations that have discrete eigenvalue structure include the first-order equation y′ = cos(πxy) and the Painlevé transcendents P-I and P-II. In new work it is shown here that the concept of a nonlinear eigenvalue problem extends to huge classes of nonlinear differential equations. Numerical and analytical results on the eigenvalue behavior for some of these new differential equations are presented.
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References
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Bender, C.M., Komijani, J., Wang, Qh. (2017). Nonlinear eigenvalue problems. In: Fauvet, F., Manchon, D., Marmi, S., Sauzin, D. (eds) Resurgence, Physics and Numbers. CRM Series, vol 20. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-613-1_2
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DOI: https://doi.org/10.1007/978-88-7642-613-1_2
Publisher Name: Edizioni della Normale, Pisa
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