Abstract
We present a 2-component equation with exactly two nontrivial generalized symmetries, a counterexample to Fokas' conjecture that equations with as many symmetries as components are integrable. Furthermore we prove the existence of infinitely many evolution equations with finitely many symmetries. We introduce the concept of almost integrability to describe the situation where one has a finite number of symmetries. The symbolic calculus of Gel'fand-Dikiî andp-adic analysis are used to prove our results.
Similar content being viewed by others
References
I.M. Bakirov. On the symmetries of some system of evolution equations. Technical report, Akad. Nauk SSSR Ural. Otdel. Bashkir. Nauchn Tsentr, Ufa, 1991.
F. Beukers, On a sequence of polynomials.Journal of Pure and Applied Algebra 117, and118 (1997), 97–103.
F. Beukers, J.A. Sanders and J.P. Wang. One symmetry does not imply integrability.J. Differential Equations 146 (1998), no. 1, 251–260.
F. Beukers, J.A. Sanders and J.P. Wang. On integrability of systems of evolution equations.Journal of Differential Equations 172 (2001), no. 2, 396–408.
F. Beukers and C.J. Smyth. Cyclotomic points on curves. Millennial Conference on Number Theory, May 21–26, 2000, Urbana-Champaign, A.K. Peters, 2001.
B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan and S.M. Watt.Maple V Language Reference Manual. Springer-Verlag, Berlin, 1991.
A.S. Fokas. A symmetry approach to exactly solvable evolution equations.Journal of Mathematical Physics 21 (1980), no. 6, 1318–1325.
A.S. Fokas. Symmetries and integrability.Studies in Applied Mathematics 77 (1987), 253–299.
C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura. Korteweg-de Vries equation and generalizations. VI. Methods for exact solution.Communications on Pure and Applied Mathematics 27 (1974), 97–133.
I.M. Gel'fand and L.A. Dikiî. Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations.Uspehi Mat. Nauk 30(5(185)) (1975), 67–100; English translation.Russian Math. Surveys 30 (1975), no. 5, 77–113.
N.H. Ibragimov and A.B. Šabat. Evolution equations with a nontrivial Lie-Bäcklund group.Funktsional Anal. i Prilozhen 14 (1980), no. 1, 25–36, 96.
D.J. Korteweg and G. de Vries. On the change of form of long waves advancing in a rectangular canal, and a new type of long stationary waves.Philos. mag. 39 (1895), no. 5, 422–443.
R.M. Miura, C.S. Gardner and M.D. Kruskal. Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion.Journal of Mathematical Physics 9 (1968), no. 8, 1204–1209.
E. Noether. Invariante Variationsprobleme.Nachr. v.d. Ges. d. Wiss. zu Göttingen, Math.-phys. Kl. 2 (1918), 235–257.
P.J. Olver. Evolution equations possessing infinitely many symmetries.J. Mathematical Phys. 18 (1977), no. 6, 1212–1215.
P.J. Olver.Applications of Lie groups to differentiatial equations, volume 107 of Graduate Texts in Mathematics,. second edition. Springer-Verlag, New York, 1993.
P.J. Olver and V.V. Sokolov. Integrable evolution equations on associative algebras.Comm. Math. Phys. 193 (1998), no. 2, 245–268.
J.A. Sanders and J.P. Wang. On the integrability of homogeneous scalar evolution equations.J. Differential Equations 147 (1998), no. 2, 410–434.
J.A. Sanders and J.P. Wang. On the integrability of systems of second order evolution equations with two components. Technical Report WS-557, Vrije Universiteit Amsterdam, Amsterdam, 2001, Submitted to Journal of Differential Equations.
T. Tsuchida and M. Wadati. Complete integrability of derivative nonlinear Schrödinger-type equations.Inverse Problems 15 (1999), no. 5, 1363–1373.
P.H. van der Kamp. Classification of Integrable β-equations. Technical Report WS-566, Vrije Universiteit Amsterdam, Amsterdam, 2002.
P.H. van der Kamp. Integrable Evolution Equations: a Diophantine Approach, 2002.
P.H. van der Kamp and J.A. Sanders. On testing integrability.J. Nonlinear Math. Phys. 8 (2001), no. 4, 561–574.
G.Z. Tu and M.Z. Qin. The invariant groups and conservation laws of nonlinear evolution equations—an approach of symmetric function.Sci. Sinica 24 (1981), no. 1, 13–26.
As was shown by V.V. Sokolov and A.G. Meshkov at the summerschool “What is Integrability?”, August 2001, Cambridge.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van der Kamp, P.H., Sanders, J.A. Almost integrable evolution equations. Selecta Mathematica, New Series 8, 705–719 (2002). https://doi.org/10.1007/BF02637315
Issue Date:
DOI: https://doi.org/10.1007/BF02637315