The KdV curve and Schrödinger-Airy curve

Song, Chong

doi:10.1090/S0002-9939-2011-10930-4

The KdV curve and Schrödinger-Airy curve
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by Chong Song PDF

Proc. Amer. Math. Soc. 140 (2012), 635-644 Request permission

Abstract:

Among other things, we introduce the notion of KdV curves and Schrödinger-Airy curves. These curves are stable solutions to the geometric KdV-Airy flow equation and Schrödinger-Airy flow equation respectively, which were recently proposed by Sun and Wang. We demonstrate that the KdV curves can be regarded as a 3rd-order analogue of geodesics. Other interesting properties of these curves will be addressed. Explicit examples of these curves will be provided. In addition, we will consider a perturbed KdV curve system and show the existence of multiple solutions to this system on the torus.

References

Herbert Amann, Saddle points and multiple solutions of differential equations, Math. Z. 169 (1979), no. 2, 127–166. MR 550724, DOI 10.1007/BF01215273
Kung-ching Chang, Infinite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, vol. 6, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1196690, DOI 10.1007/978-1-4612-0385-8
C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol′d, Invent. Math. 73 (1983), no. 1, 33–49. MR 707347, DOI 10.1007/BF01393824
Weiyue Ding and Youde Wang, Schrödinger flow of maps into symplectic manifolds, Sci. China Ser. A 41 (1998), no. 7, 746–755. MR 1633799, DOI 10.1007/BF02901957
X. W. Sun and Y. D. Wang; KdV geometric flows on Kähler manifolds, International Journal of Mathematics, to appear.