| 1. |
Shunsaku Nii, Genericity of interactions with potentials being Morse functions, Journal of Geometry and Physics, 10.1016/j.geomphys.2018.03.013, 129, 233-237, 2018.07, Potential functions defined on a d-dimensional lattice by nearest neighbor coupling are considered. The question is whether the potential functions are Morse functions. It is shown that for generic nearest neighbor couplings the potential functions are Morse functions for any system size.. |
| 2. |
Ayuki Sekisaka, Shunsaku Nii, Computer assisted verification of the eigenvalue problem for one-dimensional Schro ̈dinger operator, Mathematical Challenges to a New Phase of Materials Science, 145-157, 2016.01, 周期的ポテンシャル項を持つシュレーディンガー作用素に指数減衰する摂動を施したものの固有値問題について、Maslov Index のアイディアを用いて計算機援用証明を行った。. |
| 3. |
Ayuki Sekisaka, Shunsaku Nii, Computer assisted verification of the eigenvalue problem for one-dimensional Schrödinger operator, International Conference on Mathematical Challenges in a New Phase of Materials Science, 2014
Mathematical Challenges in a New Phase of Materials Science, 10.1007/978-4-431-56104-0_8, 145-157, 2016.01, We propose a rigorous computational method for verifying the isolated eigenvalues of one-dimensional Schrödinger operator containing a periodic potential and a perturbation which decays exponentially at ±∞. We show how the original eigenvalue problem can be reformulated as the problem of finding a connecting orbit in a Lagrangian-Grassmanian. Based on the idea of the Maslov theory for Hamiltonian systems, we set up an integer-valued topological measurement, the rotation number of the orbit in the resulting one-dimensional projective space. Combining the interval arithmetic method for dynamical systems, we demonstrate a computer-assisted proof for the existence of isolated eigenvalues within the first spectral gap.. |
| 4. |
Shunsaku Nii, Bifurcations of stationary solutions with triple junctions in phase separation problems --A new view of bifurcation analysis--, Physica D, 238 (2009), pp. 1050-1055, 2009.06. |
| 5. |
Shunsaku Nii, Bifurcations of stationary solutions with triple junctions in phase separation problems - A new view of bifurcation analysis, Physica D: Nonlinear Phenomena, 10.1016/j.physd.2009.02.016, 238, 13, 1050-1055, 2009.06, Investigated here are bifurcations of stationary solutions with triple junctions to the curvature-driven motion of curves, which arises in phase separation problems. We demonstrate that, when seen from a new perspective, typical bifurcations under perturbations of domain shape can be analyzed using only elementary geometry.. |
| 6. |
Shunsaku Nii, $\pi_1$- and $\pi_2$-theories of operators (Problems in the Calculus of Variations and Related Topics), RIMS Kokyuroku, 1628, 58-62, 2009.02. |
| 7. |
Jian Deng, Shunsaku Nii, An Infinite-dimensional Evans function theory for Elliptic Boundary Value Problems, Journal of Differential Equations , vol. 244 pp. 753--765, 2008.02. |
| 8. |
Jian Deng, Shunsau Nii, An infinite-dimensional Evans function theory for elliptic boundary value problems, Journal of Differential Equations, 10.1016/j.jde.2007.10.037, 244, 4, 753-765, 2008.02, An infinite-dimensional Evans function E (λ) and a stability index theorem are developed for the elliptic eigenvalue problem in a bounded domain Ω ⊂ Rm. The number of zero points of the Evans function in a bounded, simply connected complex domain D is shown to be equal to the number of eigenvalues of the corresponding elliptic operator in D. When the domain Ω is star-shaped, an associated unstable bundle E (D) based on D is constructed, and the first Chern number of E (D) also gives the number of eigenvalues of the elliptic operator inside D.. |
| 9. |
Jian Deng, Shunsaku Nii, Infinite-dimensional Evans function theory for elliptic eigenvalue problems in a channel, Journal of Differential Equations, 225 pp.57-89, 2006.06. |
| 10. |
Jian Deng, Shunsaku Nii, Infinite-dimensional Evans function theory for elliptic eigenvalue problems in a channel, Journal of Differential Equations, 10.1016/j.jde.2005.09.007, 225, 1, 57-89, 2006.06, An infinite-dimensional Evans function theory is developed for the elliptic eigenvalue problem associated with the stability of travelling solitary waves in a channel. Also, a bundle is constructed over the complex domain, so that its first Chern number gives the number of eigenvalues inside the domain.. |
| 11. |
BIFURCATIONS AND STABILITY OF TRAVELING WAVES (Nonlinear Diffusive Systems and Related Topics). |
| 12. |
S.Nii, Topological methods in stability analysis of travelling waves, Amer. Math. Soc. Transl, 204, pp45--62, 2001.01. |
| 13. |
Shunsaku Nii, The accumulation of eigenvalues in a stability problem, Physica D: Nonlinear Phenomena, 10.1016/S0167-2789(00)00061-0, 142, 1-2, 70-86, 2000.08, When two waves propagating in a one-dimensional medium are locked together as a composite wave, a natural question arises as to whether the new wave is stable. An interesting and novel instability mechanism is exposed here in which a cascade of eigenvalues accumulates at a distinguished point in the unstable half plane. The underlying assumption is that the transition between the two waves occurs at an unstable, homogeneous steady state of the partial differential equations. This causes the individual waves to have an unstable continuous spectrum, but the instability of the full wave cannot be predicted from the configuration of these spectra alone.. |
| 14. |
Shunsaku Nii, Pitchfork and Hopf bifurcations of traveling pulses generated by coexisting front and back waves, Methods and Applications of Analysis, 7 pp. 615-640, 2000.01. |
| 15. |
Shunsaku Nii, Accumulation of eigenvalues in a stability problem, Physica D, 142 pp. 70-86, 2000.01. |
| 16. |
Shunsaku Nii, Pitchfork and Hopf bifurcations of traveling pulses generated by coexisting front and back waves, Methods and Applications of Analysis, 7, 615-640, 2000. |
| 17. |
Shunsaku Nii, A topological approach to stability of pulses bifurcating from an inclination-flip homoclinic orbit, Methods and Applications of Analysis, 7, 205-232, 2000. |
| 18. |
Shunsaku Nii, A topological proof of stability of N-front solutions of the FitzHugh-Nagumo equations, Jounal of Dynamics and Differential Equations, 11 pp. 515-555, 1999.01. |
| 19. |
Shunsaku Nii, A topological proof of stability of N-front solutions of the FitzHugh-Nagumo equations, Journal of Dynamics and Differential Equations, 10.1023/A:1021965920761, 11, 3, 515-555, 1999.01, Consideration is devoted to traveling N-front wave solutions of the FitzHugh-Nagumo equations of the bistable type. Especially, stability of the N-front wave is proven. In the proof, the eigenvalue problem for the N-front wave bifurcating from coexisting simple front and back waves is regarded as a bifurcation problem for projectivised eigenvalue equations, and a topological index is employed to detect eigenvalues.. |
| 20. |
Shunsaku Nii, An extension of the stability index for traveling-wave solutions and its application to bifurcations, SIAM Journal on Mathematical Analysis, 10.1137/S003614109427878X, 28, 2, 402-433, 1997.03, We treat the stability index for traveling-wave solutions of one-dimensional reaction-diffusion equations due to Alexander, Gardner, and Jones [J. Reine Angew. Math., 410(1990), pp. 167-212]. An extension of the stability index which makes the index robust to perturbation is given and, using the extension, an additive formula for a gluing bifurcation of traveling waves is proven. We also consider certain heteroclinic bifurcations as an application, some specific examples of which are discussed.. |
| 21. |
Shunsaku Nii, Stability of travelling multiple-front (multiple-back) wave solutions of the FitzHugh-Nagumo equations, SIAM Journal on Mathematical Analysis, 28, 1094-1112, 1997. |
| 22. |
Shunsaku Nii, 進行波に沿った線型化固有値問題へのTOPOLOGICALなアプローチ(函数解析を用いた偏微分方程式の研究), RIMS Kokyuroku, 969, 18-28, 1996.10. |
| 23. |
Shunsaku Nii, N-homoclinic bifurcations for homoclinic orbits changing their twisting, Journal of Dynamics and Differential Equations, 10.1007/BF02218844, 8, 4, 549-572, 1996.01, We study bifurcations, called N-homoclinic bifurcations, which produce homoclinic orbits rounding N times (N≥2) in some tubular neighborhood of original homoclinic orbit A family of vector fields undergoes such a bifurcation when it is a perturbation of a vector field with a homoclinic orbit. N-Homoclinic bifurcations are divided into two cases; one is that the linearization at the equilibrium has only real principal eigenvalues, and the other is that it has complex principal eigenvalues. We treat the former case, espcially that linearization has only one unstable eigenvalue. As main tools we use a topological method, namely, Conley index theory, which enables us to treat more degenerate cases than those studied by analytical methods.. |
