Asymptotic formulas of the eigenvalues for the linearization of a one-dimensional sinh-Poisson equation. (English) Zbl 1531.34077
The authors are concerned with a Neumann problem of a one-dimensional sinh-Poisson equation
\[
\begin{cases} u''+\lambda\sinh(u)&=0, \quad 0<x<1, \\
u'(0)=u'(1)&=0, \end{cases}
\]
where \(\lambda\) is a positive parameter (eigenvalue of the problem). The main result of the paper is about the set of solutions which are constructed. A complete bifurcation diagram of this problem is also obtained. The eigenvalues and eigenfunctions are obtained by using Jacobi elliptic functions and complete elliptic integrals. The asymptotic behaviour of eigenvalues as \(\lambda\to0\) is derived. The proofs only employ techniques from ODEs.
Reviewer: Smail Djebali (Riyadh)
MSC:
| 34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
References:
| [1] | Byrd, P.; Friedman, M., Handbook of Elliptic Integrals for Engineers and Scientists, Revised, Die Grundlehren der mathematischen Wissenschaften (1971), New York: Springer, New York · Zbl 0213.16602 · doi:10.1007/978-3-642-65138-0 |
| [2] | Bartsch, T.; Pistoia, A.; Weth, T., \(N\)-Vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations, Commun. Math. Phys., 297, 653-686 (2010) · Zbl 1195.35250 · doi:10.1007/s00220-010-1053-4 |
| [3] | Coddington, E.; Levinson, N., Theory of Ordinary Differential Equations (1955), New York: McGraw-Hill Book Company Inc, New York · Zbl 0064.33002 |
| [4] | Grossi, M.; Pistoia, A., Multiple blow-up phenomena for the sinh-Poisson equation, Arch. Ration. Mech. Anal., 209, 287-320 (2013) · Zbl 1303.35088 · doi:10.1007/s00205-013-0625-9 |
| [5] | Jost, J.; Wang, G.; Ye, D.; Zhou, C., The blow up analysis of solutions of the elliptic sinh-Gordon equation, Calc. Var. Partial Differ. Equ., 31, 263-276 (2008) · Zbl 1137.35061 · doi:10.1007/s00526-007-0116-7 |
| [6] | Miyamoto, Y., Takemura, H., Wakasa, T.: Asymptotic formulas of the eigenvalues for the linearization of the scalar field equation (2022) (submitted) |
| [7] | Miyamoto, Y.; Wakasa, T., Exact eigenvalues and eigenfunctions for a one-dimensional Gel’fand problem, J. Math. Phys., 60, 021506 (2019) · Zbl 1409.34034 · doi:10.1063/1.5021825 |
| [8] | Onsager, L., Statistical hydrodynamics, Nuovo Cimento Suppl., 6, 279-287 (1949) · doi:10.1007/BF02780991 |
| [9] | Pistoia, A.; Ricciardi, T., Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents, Discrete Contin. Dyn. Syst., 37, 5651-5692 (2017) · Zbl 1368.35144 · doi:10.3934/dcds.2017245 |
| [10] | Ricciardi, T.; Takahashi, R., Blow-up behavior for a degenerate elliptic sinh-Poisson equation with variable intensities, Calc. Var. Partial Differ. Equ., 55, 1-25 (2016) · Zbl 1361.35056 · doi:10.1007/s00526-016-1090-8 |
| [11] | Wakasa, T.; Yotsutani, S., Representation formulas for some 1-dimensional linearized eigenvalue problems, Commun. Pure Appl. Anal., 7, 745-763 (2008) · Zbl 1360.34050 · doi:10.3934/cpaa.2008.7.745 |
| [12] | Wakasa, T., Representation and asymptotic formulas for 1-dimensional linearized eigenvalue problems with Dirichlet boundary condition, Nonlinear Anal., 71, e2696-e2704 (2009) · Zbl 1239.34062 · doi:10.1016/j.na.2009.06.030 |
| [13] | Wakasa, T.; Yotsutani, S., Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems, Commun. Pure Appl. Anal., 9, 539-561 (2010) · Zbl 1201.34027 · doi:10.3934/cpaa.2010.9.539 |
| [14] | Wakasa, T.; Yotsutani, S., Limiting classification on linearized eigenvalue problems for 1-dimensional Allen-Cahn equation I-asymptotic formulas of eigenvalues, J. Differ. Equ., 258, 3960-4006 (2015) · Zbl 1319.34152 · doi:10.1016/j.jde.2015.01.023 |
| [15] | Wakasa, T.; Yotsutani, S., Limiting classification on linearized eigenvalue problems for 1-dimensional Allen-Cahn equation II-Asymptotic profiles of eigenfunctions, J. Differ. Equ., 261, 5465-5498 (2016) · Zbl 1355.34091 · doi:10.1016/j.jde.2016.08.016 |
| [16] | Wakasa, T., Note on parameter dependence of eigenvalues for a linearized eigenvalue problem, Bull. Kyushu Inst. Technol. Pure Appl. Math., 64, 1-12 (2017) · Zbl 1371.35283 |
| [17] | Wei, J.; Wei, L.; Zhou, F., Mixed interior and boundary nodal bubbling solutions for a sinh-Poisson equation, Pac. J. Math., 250, 225-256 (2011) · Zbl 1217.35055 · doi:10.2140/pjm.2011.250.225 |
| [18] | Wente, H., Counterexample to a conjecture of H. Hopf, Pac. J. Math., 121, 193-243 (1986) · Zbl 0586.53003 · doi:10.2140/pjm.1986.121.193 |
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.
