The adiabatic limit of wave map flow on a two-torus

Speight, J.

doi:10.1090/tran/6538

The adiabatic limit of wave map flow on a two-torus
HTML articles powered by AMS MathViewer

by J. M. Speight PDF

Trans. Amer. Math. Soc. 367 (2015), 8997-9026 Request permission

Abstract:

The $S^2$ valued wave map flow on a Lorentzian domain $\mathbb {R}\times \Sigma$, where $\Sigma$ is any flat two-torus, is studied. The Cauchy problem with initial data tangent to the moduli space of holomorphic maps $\Sigma \rightarrow S^2$ is considered, in the limit of small initial velocity. It is proved that wave maps, in this limit, converge in a precise sense to geodesics in the moduli space of holomorphic maps, with respect to the $L^2$ metric. This establishes, in a rigorous setting, a long-standing informal conjecture of Ward.

References

Piotr Bizoń, Tadeusz Chmaj, and Zbisław Tabor, Formation of singularities for equivariant $(2+1)$-dimensional wave maps into the 2-sphere, Nonlinearity 14 (2001), no. 5, 1041–1053. MR 1862811, DOI 10.1088/0951-7715/14/5/308
S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
J. Eells and J. C. Wood, Restrictions on harmonic maps of surfaces, Topology 15 (1976), no. 3, 263–266. MR 420708, DOI 10.1016/0040-9383(76)90042-2
M. Haskins and J. M. Speight, The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps, J. Math. Phys. 44 (2003), no. 8, 3470–3494. Integrability, topological solitons and beyond. MR 2006760, DOI 10.1063/1.1586480
N. J. Hitchin, G. B. Segal, and R. S. Ward, Integrable systems, Oxford Graduate Texts in Mathematics, vol. 4, The Clarendon Press, Oxford University Press, New York, 1999. Twistors, loop groups, and Riemann surfaces; Lectures from the Instructional Conference held at the University of Oxford, Oxford, September 1997. MR 1723384
Fritz John, Partial differential equations, 3rd ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York-Berlin, 1978. MR 514404, DOI 10.1007/978-1-4684-0059-5
J. Krieger, W. Schlag, and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), no. 3, 543–615. MR 2372807, DOI 10.1007/s00222-007-0089-3
Derek F. Lawden, Elliptic functions and applications, Applied Mathematical Sciences, vol. 80, Springer-Verlag, New York, 1989. MR 1007595, DOI 10.1007/978-1-4757-3980-0
Robert Leese, Low-energy scattering of solitons in the $\textbf {C}\textrm {P}^1$ model, Nuclear Phys. B 344 (1990), no. 1, 33–72. MR 1071448, DOI 10.1016/0550-3213(90)90684-6
Robert A. Leese, Michel Peyrard, and Wojciech J. Zakrzewski, Soliton stability in the $\textrm {O}(3)\ \sigma$-model in $(2+1)$ dimensions, Nonlinearity 3 (1990), no. 2, 387–412. MR 1054581, DOI 10.1088/0951-7715/3/2/007
André Lichnerowicz, Applications harmoniques et variétés kähleriennes, Symposia Mathematica, Vol. III (INDAM, Rome, 1968/69) Academic Press, London, 1968/1969, pp. 341–402 (French). MR 0262993
Jean Marie Linhart and Lorenzo A. Sadun, Fast and slow blowup in the $S^2$ $\sigma$-model and the $(4+1)$-dimensional Yang-Mills model, Nonlinearity 15 (2002), no. 2, 219–238. MR 1888849, DOI 10.1088/0951-7715/15/2/301
J. A. McGlade and J. M. Speight, Slow equivariant lump dynamics on the two sphere, Nonlinearity 19 (2006), no. 2, 441–452. MR 2199397, DOI 10.1088/0951-7715/19/2/011
N. S. Manton, A remark on the scattering of BPS monopoles, Phys. Lett. B 110 (1982), no. 1, 54–56. MR 647883, DOI 10.1016/0370-2693(82)90950-9
Nicholas Manton and Paul Sutcliffe, Topological solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. MR 2068924, DOI 10.1017/CBO9780511617034
Yu. N. Ovchinnikov and I. M. Sigal, On collapse of wave maps, Phys. D 240 (2011), no. 17, 1311–1324. MR 2831768, DOI 10.1016/j.physd.2011.04.014
Pierre Raphaël and Igor Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 1–122. MR 2929728, DOI 10.1007/s10240-011-0037-z
Igor Rodnianski and Jacob Sterbenz, On the formation of singularities in the critical $\textrm {O}(3)$ $\sigma$-model, Ann. of Math. (2) 172 (2010), no. 1, 187–242. MR 2680419, DOI 10.4007/annals.2010.172.187
L. A. Sadun and J. M. Speight, Geodesic incompleteness in the $\mathbf C\textrm {P}^1$ model on a compact Riemann surface, Lett. Math. Phys. 43 (1998), no. 4, 329–334. MR 1620737, DOI 10.1023/A:1007433724535
Jalal Shatah and Michael Struwe, Geometric wave equations, Courant Lecture Notes in Mathematics, vol. 2, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998. MR 1674843
J. M. Speight, Low-energy dynamics of a $\textbf {C}\textrm {P}^1$ lump on the sphere, J. Math. Phys. 36 (1995), no. 2, 796–813. MR 1312081, DOI 10.1063/1.531157
J. M. Speight, Lump dynamics in the $\textbf {C}\textrm {P}^1$ model on the torus, Comm. Math. Phys. 194 (1998), no. 3, 513–539. MR 1631469, DOI 10.1007/s002200050367
J. M. Speight, The $L^2$ geometry of spaces of harmonic maps $S^2\to S^2$ and ${\Bbb R}\textrm {P}^2\to {\Bbb R}\rm P^2$, J. Geom. Phys. 47 (2003), no. 2-3, 343–368. MR 1991480, DOI 10.1016/S0393-0440(02)00227-9
D. Stuart, Dynamics of abelian Higgs vortices in the near Bogomolny regime, Comm. Math. Phys. 159 (1994), no. 1, 51–91. MR 1257242, DOI 10.1007/BF02100485
D. Stuart, The geodesic approximation for the Yang-Mills-Higgs equations, Comm. Math. Phys. 166 (1994), no. 1, 149–190. MR 1309545, DOI 10.1007/BF02099305
David M. A. Stuart, Analysis of the adiabatic limit for solitons in classical field theory, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2087, 2753–2781. MR 2360179, DOI 10.1098/rspa.2007.0130
Hajime Urakawa, Calculus of variations and harmonic maps, Translations of Mathematical Monographs, vol. 132, American Mathematical Society, Providence, RI, 1993. Translated from the 1990 Japanese original by the author. MR 1252178, DOI 10.1090/mmono/132
R. S. Ward, Slowly-moving lumps in the $\textbf {C}\textrm {P}^1$ model in $(2+1)$ dimensions, Phys. Lett. B 158 (1985), no. 5, 424–428. MR 802039, DOI 10.1016/0370-2693(85)90445-9