Abstract
A geometric approach to immersion formulas for soliton surfaces is provided through new cohomologies on spaces of special types of \({\mathfrak {g}}\)-valued differential forms. We introduce Poincaré-type lemmas for these cohomologies, which appropriately describe the integrability conditions of Lax pairs associated with systems of PDEs. Our methods clarify the structure and properties of the deformations and soliton surfaces for the aforesaid Lax pairs. Our findings allow for the generalization of the theory of soliton surfaces in Lie algebras to general soliton submanifolds. Techniques from the theory of infinite-dimensional jet manifolds and diffieties enable us to justify certain common assumptions of the theory of soliton surfaces. Theoretical results are illustrated through \({\mathbb {C}}P^{N-1}\) sigma models.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ablowitz, M.J.: Nonlinear Phenomena. Springer, Berlin (1982)
Babelon, O., Bernard, D., Talon, M.: Introduction to Classical Integrable Systems (Cambridge Monographs on Mathematical Physics). Cambridge University Press, Cambridge (2006)
Biernacki, W., Cieslinski, J.L.: A compact form of the Darboux-Backlund transformation for some spectral problems in Clifford algebras. Phys. Lett. A 288, 167–172 (2001)
Bobenko, A.I.: Surfaces in terms of 2 by 2 matrices. Old and new integrable cases. In: Fordy, A.P., Wood, J.C. (eds.) Harmonic Maps and Integrable Systems. Aspects of Mathematics, vol. E 23. Vieweg+Teubner Verlag, Wiesbaden (1994)
Bobenko, A., Eitner, U.: Painlevé Equations in the Differential Geometry of Surfaces. Lecture Notes in Mathematics, vol. 1753. Springer, Berlin (2000)
Bolton, J., Jensen, G.R., Rigoli, M., Woodward, L.M.: On conformal minimal immersions of \(S^2\) into \({\mathbb{C}}P^n\). Math. Ann. 279, 599–620 (1988)
Cartan, E.: Sur la structure des groupes infinis de transformation in Les systèmes différentiels en Involution. Gauthier-Villars, Paris (1953)
Chavolin, J., Joanny, J.F., Zinn-Justin, J.: Liquids at Interfaces. Elsevier, Amsterdam (1989)
Chen, F.F.: Introduction to Plasma Physics and Controlled Fusion. Plasma Physics, vol. 1. Plenum Press, New York (1983)
Cieśliński, J.: A generalized formula for integrable classes of surfaces in Lie algebras. J. Math. Phys. 38, 4255–4272 (1997)
Cieśliński, J.L.: Geometry of submanifolds derived from spin-valued spectral problem. J. Theor. Math. Phys. 137, 1396–1405 (2003)
Cieśliński, J.L.: Pseudospherical surfaces on time scales: a geometric deformation and the spectral approach. J. Phys. A 40, 12525–12538 (2007)
David, F., Ginsparg, P., Zinn-Justin, J.: Fluctuating Geometries in Statistical Mechanics and Field Theory. North-Holland, Amsterdam (1996)
Davydov, A.S.: Solitons in Molecular Systems. Kluwer, New York (1991)
Dillen, F.J.E., Verstraelen, L.C.A.: Handbook of Differential Geometry. North-Holland, Amsterdam (2000)
Din, A.M., Horváth, Z., Zakrzewski, W.J.: The Riemann-Hilbert problem and finite action \({\mathbb{C}}P^{N-1}\) solutions. Nucl. Phys. B 233, 269–288 (1984)
Din, A.M., Zakrzewski, W.: General class of solutions in the \({\mathbb{C}}P^{N-1}\) model. Nucl. Phys. B 174, 397–406 (1980)
Doliwa, A., Sym, A.: Constant mean curvature surfaces in \(E^3\) as an example of soliton surfaces. In: Nonlinear Evolution Equations and Dynamical Systems. World Scientific, River Edge, pp. 111–117 (1992)
Eichenherr, H.: \(SU(N)\) invariant nonlinear \(\sigma \) models. Nucl. Phys. B 146, 215–223 (1978)
Fokas, A.S., Gel’fand, I.M.: Surfaces on Lie groups, on Lie algebras, and their integrability. Commun. Math. Phys. 177, 203–220 (1996)
Fokas, A.S., Gel’fand, I.M., Finkel, F., Liu, Q.M.: A formula for constructing infinitely many surfaces on Lie algebras and integrable equations. Sel. Math. 6, 347–375 (2000)
Giachetta, G., Mangiarotti, L., Sardanashvily, G.: Cohomology of the infinite-order jet space and the inverse problem. J. Math. Phys. 42, 4272–4282 (2001)
Goldstein, P.P., Grundland, A.M.: Invariant recurrence relations for \(CP^{N-1}\) models. J. Phys. A 43, 265206 (2010)
Golo, V.L., Perelomov, A.M.: Solution of the duality equations for the two-dimensional \(SU(N)\)-invariant chiral model. Phys. Lett. B 79, 112–113 (1978)
Gross, D.J., Piran, T., Weinberg, S.: Two-Dimensional Quantum Gravity and Random Surfaces. World Scientific, Singapore (1992)
Grundland, A.M.: Soliton surfaces in the generalized symmetry approach. Theor. Math. Phys. 188, 1322–1333 (2016)
Grundland, A.M., Levi, D., Martina, L.: On immersion formulas for soliton surfaces. Acta Polytech. 56, 180–192 (2016)
Grundland, A.M., Post, S.: Soliton surfaces associated with generalized symmetries of integrable equations. J. Phys. A 44, 165203 (2011)
Grundland, A.M., Post, S.: Surfaces immersed in Lie algebras associated with elliptic integrals. J. Phys. A 45, 015204 (2012)
Grundland, A.M., Post, S., Riglioni, D.: Soliton surfaces and generalized symmetries of integrable systems. J. Phys. A 47, 015201 (2014)
Grundland, A.M., Strasburger, A., Dziewa–Dawidczyk, D.: \({\mathbb{C}}P^N\) sigma models via the \(SU(2)\) coherent states approach, Banach Center Publications, Polish Academy of Sciences, 50th seminar ‘Sophus Lie’ 113 (2018)
Grundland, A.M., Strasburger, A., Zakrzewski, W.J.: Surfaces immersed in \({{\mathfrak{s}}}{{\mathfrak{u}}}(N+1)\) Lie algebras obtained from the \({\mathbb{C}}P^N\) sigma models. J. Phys. A 39, 9187 (2006)
Grundland, A.M., Yurduşen, I.: On analytic descriptions of two-dimensional surfaces associated with the \({\mathbb{C}}P^{N-1}\) sigma model. J. Phys. A 42, 172001 (2009)
Guo, X.R.: Three new \((2+1)-\)dimensional integrable systems and some related Darboux transformations. Commun. Theor. Phys. 65, 735–742 (2016)
Hélein, F.: Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2001)
Hopf, H.: Über die Abbildungen der Dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 637–665 (1931)
Igonin, S., Krasilshchik, J.: On one-parametric families of Bcklund transformations. Adv. Stud. Pure Math. 37, 99–114 (2002)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. (Wiley Classics Library). Wiley, New York (1996)
Konopelchenko, B.G., Landolfi, G.: Generalized Weierstrass representation for surfaces in multi-dimensional Riemann spaces. J. Geom. Phys. 29, 319–333 (1999)
Konopelchenko, B.G.: Induced surfaces and their integrable dynamics. Stud. Appl. Math. 96, 9–51 (1996)
Krasil’shchik, J., Verbovetsky, A.: Geometry of jet spaces and integrable systems. J. Geom. Phys. 61, 1633–1674 (2011)
Kruglikov, B., Lychagin, V.V.: Geometry of differential equations. In: Krupka, D., Saunders, D. (eds.) Handbook of Global Analysis, vol. 1214, pp. 725–771. Elsevier, Amsterdam (2008)
Landolfi, G.: New results on the Canham–Helfrich membrane model via the generalized Weierstrass representation. J. Phys. A 36, 11937–11954 (2003)
Manakov, S.V., Santini, P.M.: Inverse scattering problem for vectors fields and the Cauchy problem for the heavenly equations. Phys. Lett. A 359, 613–619 (2006)
Manton, N., Sutcliffe, P.: Topological Solitons (Cambridge Monographs on Mathematical Physics). Cambridge University Press, Cambridge (2004)
Marvan, M.: On the horizontal gauge cohomology and nonremovability of the spectral parameter. Acta Appl. Math. 72, 51–65 (2002)
May, J.P.: A Concise Course in Algebraic Topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1999)
Mikhailov, A.V.: Integrable Magnetic Models Soliton. Modern Problems in Condensed Matter, vol. 17, pp. 623–690. North-Holland, Amsterdam (1986)
Mikhailov, A.V., Shabat, A.B., Sokolov, V.V.: The symmetry approach to classification of integrable equations. What is integrability? In: Zakharov, V.E. (ed.) Nonlinear Dynamics, pp. 115–184. Springer, Berlin (1991)
Nelson, D., Piran, T., Weinberg, S.: Statistical Mechanics of Membranes and Surfaces. World Scientific, Singapore (1992)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)
Ou-Yang, Z., Liu, J., Xie, Y.: Geometric Methods in Elastic Theory of Membranes in Liquid Crystal Phases. World Scientific, Singapore (1999)
Polchinski, J., Strominger, A.: Effective string theory. Phys. Rev. Lett. 67, 1681–1684 (1991)
Rogers, C., Schief, W.K.: Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2000)
Safran, S.A.: Statistical Thermodynamics of Surfaces, Interfaces, and Membranes. Frontiers of Physics, vol. 90. Westview Press, Boulder (2003)
Sasaki, J.R.: General class of solutions of the complex Grassmannian and \({\mathbb{C}}P^{N-1}\) model. Phys. Lett. B 130, 69–72 (1983)
Sommerfeld, A.: Lectures on Theoretical Physics. Academic Press, New York (1952)
Sym, A.: Soliton surfaces. Lett. Nuovo Cimento 33, 394–400 (1982)
Sym, A.: Soliton surfaces and their applications (soliton geometry from spectral problems). In: Geometric Aspect of the Einstein Equation and Integrable Systems. Lectures Notes in Physics, vol. 239, pp. 154–231. Springer, Berlin (1995)
Tafel, J.: Surfaces in \({\mathbb{R}}^3\) with prescribed curvature. J. Geom. Phys. 17, 381–390 (1995)
Urbantke, H.K.: The Hopf fibration-seven times in physics. J. Geom. Phys. 46, 125–150 (2003)
Vinogradov, A.M.: Cohomological Analysis of Partial Differential Equations and Secondary Calculus. Translations of Mathematical Monographs, vol. 204. American Mathematical Society, Providence (2001)
Vinogradov, A.M., Krasil’shchik, I.S.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Translations of Mathematical Monographs, vol. 182. American Mathematical Society, Providence (1999)
Zakharov, V.E.: Dispersionless limit of integrable systems in 2+1 dimensions. In: Ercolani, N.M., Gabitov, I.R., Levermore, C.D., Serre, D. (eds.) Singular Limits of Dispersive Waves. NATO Advanced Study Institute, Series B: Physics, vol. 320. Plenum, New York (1994)
Zakharov, V.E., Mikhailov, A.V.: Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Sov. Phys. JETP 74, 1953–1973 (1978)
Zakrzewski, W.J.: Low-Dimensional Sigma Models. Adam Hilger, Bristol (1989)
Acknowledgements
A.M. Grundland was partially supported by the research Grant ANR-11LABX-0056-LMHLabEX LMH (France) and from the NSERC (Canada). J. de Lucas and A.M. Grundland acknowledge partial support from Project MAESTRO DEC-2012/06/A/ST1/00256 of the National Science Center (Poland). This work was partially accomplished during the stay of A.M. Grundland and J. de Lucas at the École Normale Superieure de Cachan (CMLA). The authors would also like to thank CMLA for its hospitality and attention during their stay. Finally, we thank an anonymous referee for valuable comments to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
de Lucas, J., Grundland, A.M. A cohomological approach to immersed submanifolds via integrable systems. Sel. Math. New Ser. 24, 4749–4780 (2018). https://doi.org/10.1007/s00029-018-0434-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-018-0434-y
Keywords
- Cohomology
- \({\mathbb {C}}P^{N-1}\) sigma model
- Generalized symmetries
- \({\mathfrak {g}}\)-valued differential forms
- \({\mathfrak {g}}\)-valued de Rham cohomology
- Integrable systems
- Immersion formulas
- Soliton surfaces