Abstract
In this note we give an overview of results concerning the Korteweg-de Vries equation
and small perturbations of it. All the technical details will be contained in our forthcoming book [27].
Keywords
- Vries Equation
- Invariant Torus
- Quasiperiodic Solution
- Birkhoff Normal Form
- Nonlinear Schr6dinger Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
D. BÄttig, A. M. Bloch, J.-C. Guillot & T. Kappeler, On the symplectic structure of the phase space for periodic KdV, Toda, and defocusing NLS. Duke Math. J. 79 (1995), 549–604.
D. BÄttig, T. Kappeler & B. Mityagin, On the Korteweg-de Vries equation: Convergent Birkhoff normal form. J. Fund. Anal. 140 (1996), 335–358.
R. F. BIKBAEV & S. B. Kuksin, On the parametrization of finite gap solutions by frequency vector and wave number vector and a theorem of I. Krichever. Lett. Math. Phys. 28(1993), 115–122.
A. BOBENKO & S. B. Kuksin, Finite-gap periodic solutions of the KdV equation are non-degenerate. Physics Letters A 161 (1991), 274–276.
A. I. BOBENKO & S. B. Kuksin, The nonlinear Klein-Gordon equation on an interval as a perturbed sine-Gordon equation. Comm. Math. Helv. 70 (1995), 63–112.
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation. Geom. Funct. Anal. 3 (1993), 209–262.
J. Bourgain, On the Cauchy problem for periodic KdV-type equations. Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). J. Fourier Anal. Appl. (1995), Special Issue, 17–86.
J. Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations. Geom. Funct. Anal. 6 (1996), 201–230.
J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations. Colloquium Publications, American Mathematical Society, 1999.
J. Boussinesq, Théorie de l’intumescence liquid appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. Comptes Rend. Acad. Sci. (Paris) 72 (1871), 755–759.
W. CRAIG & C. E. Wayne, Newton’s method and periodic solutions of nonlinear wave equations. Comm. Pure Appl. Math. 46 (1993), 1409–1498.
B. A. Dubrovin, Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials. Funct. Anal. Appl. 9 (1975), 215–223.
B. A. DUBROVIN, I. M. KRICHEVER & S. P. Novikov, The Schrödinger equation in a periodic field and Riemann surfaces. Sov. Math. Dokl. 17 (1976), 947–951.
B. A. Dubrovin, I. M. Krichever & S. P. Novikov, Integrable Systems I. In: Dynamical Systems IV. Encyclopedia of Mathematical Sciences vol. 4, V. I. ARNOLD & S.P. NOVIKOV (eds.). Springer, 1990, 173–280.
B. A. DUBROVIN, V. B. MATVEEV & S. P. Novikov, Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators and Abelian varieties. Russ. Math. Surv. 31 (1976), 59–146.
B. A. DUBROVIN & S. P. Novikov, Periodic and conditionally periodic analogues of the many-soliton solutions of the Kortweg-de Vries equation. Sov. Phys.-JETP 40 (1974), 1058–1063.
L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems. Ann. Sc. Norm. Sup. Pisa 15 (1988), 115–147.
L. D. FADDEEV & V.E. Zakharov, Kortweg-de Vries equation: a completely integrable Hamiltonian system. Funct. Anal. Appl. 5 (1971), 280–287.
H. FLASCHKA & D. Mclaughlin, Canonically conjugate variables for the Korteweg-de Vries equation and Toda lattices with periodic boundary conditions. Progress Theor. Phys. 55 (1976), 438–456.
J. P. Françoise, The Amol’d formula for algebraically completely integrable systems. Bull. Amer. Math. Soc. (N.S.) 17 (1987), 301–303.
C. S. Gardner, Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries equation as a Hamiltonian system. J. Math. Phys. 12 (1971), 1548–1551.
J. GARNETT & E. T. Trubowitz,Gaps and bands of one dimensional periodic Schrödinger operators. Comment. Math. Helv. 59 (1984), 258–312.
B. GrÉbert, T. Kappeler & J. Pöschel, Normal form theory for the NLS equation. In preparation.
PH. GRIFFITHS & J. Harris, Principles of Algebraic Geometry. John Wiley & Sons, New York, 1978.
A. R. ITS & V. B. Matveev, A class of solutions of the Korteweg-de Vries equation. Probl. Mat. Fiz. 8. Leningrad State University, Leningrad, 1976, 70–92.
T. KAPPELER & M. Makarov, On Birkhoff coordinates for KdV. Ann. Henri Poincaré 2 (2001), 807–856.
T. Kappeler & J. Pöschel, KdV & KAM. Springer, to appear.
C. Kenig, G. Ponce & L. Vega, A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9 (1996), 573–603.
D. J. Korteweg & G. de. Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. Ser. 5 39 (1895), 422–443.
I. Krichever, Integration of nonlinear equations by methods of algebraic geometry. Funkt. Anal. Appl. 11 (1977), 15–31.
I. Krichever, “Hessians” of integrals of the Korteweg-de Vries equation and perturbations of finite-gap solutions. Dokl. Akad. Nauk SSSR 270 (1983), 1312–1317 [Russian]. English translation in Sov. Math. Dokl.27(1983), 757-761.
M. D. KRUSKAL & N. J. Zabusky, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15 (1965), 240–243.
S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funts. Anal. Prilozh. 21 (1987), 22–37 [Russian]. English translation in Fund. Anal. Appl. 21 (1987), 192-205.
S. B. Kuksin, Perturbation theory for quasiperiodic solutions of infinite-dimensional Hamiltonian systems, and its application to the Korteweg-de Vries equation. Matem. Sbornik 136 (1988) [Russian]. English translation in Math. USSR Sbornik 64 (1989), 397–413.
S. B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems. Lecture Notes in Mathematics 1556, Springer, 1993.
S. B. Kuksin, On small-denominator equations with large variable coefficients. Preprint, 1995.
S. B. Kuksin, A KAM-theorem for equations of the Korteweg-de Vries type. Rev. Math. Phys. 10 (1998), 1–64.
S. B. Kuksin, Analysis of Hamiltonian PDEs. Oxford University Press, Oxford, 2000.
S. B. KUKSIN & J. PöSCHEL, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 143 (1996), 149–179.
P. Lax, Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21 (1968), 467–490.
P. Lax, Periodic solutions of the KdV equation. Comm. Pure Appl Math. 28 (1975), 141–188.
W. MAGNUS & S. Winkler, Hill’s Equation. Second edition, Dover, New York, 1979.
V. A. MarČhenko, Sturm-Liouville Operators and Applications. Birkhäuser, Basel, 1986.
H. P. MCKEAN & P. VAN Moerbecke, The spectrum of Hill’s equation. Invent. Math. 30 (1975), 217–274.
H. P. Mckean & E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points. Comm. Pure Appl. Math. 29 (1976), 143–226.
H. P. MCKEAN & E. Trubowitz, Hill’s surfaces and their theta functions. Bull. Am. Math. Soc. 84 (1978), 1042–1085.
H. P. MCKEAN & K. L. Vaninsky, Action-angle variables for the cubic Schroedinger equation.Comm. Pure Appl. Math. 50 (1997), 489–562.
H. P. MCKEAN & K. L. Vaninsky, Cubic Schroedinger: The petit canonical ensemble in action-angle variables. Comm. Pure Appl. Math. 50 (1997), 594–622.
V. K. Melnikov, On some cases of conservation of conditionally periodic motions under a small change of the Hamilton function. Soviet Math. Doklady 6 (1965), 1592–1596.
R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9 (1968), 1202–1204.
R. M. MIURA, C. S. GARDNER & M. D. Kruskal, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys. 9 (1968), 1204–1209.
J. PÖSCHEL, On elliptic lower dimensional tori in Hamiltonian systems. Math. Z. 202 (1989), 559–608.
J. PÖSCHEL, Small divisors with spatial structure in infinite dimensional Hamiltonian systems. Comm. Math. Phys. 127 (1990), 351–393.
J. PÖSCHEL, A KAM-theorem for some nonlinear partial differential equations. Ann. Sc. Norm. Sup. Pisa 23 (1996), 119–148.
J. PÖSCHEL, Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helv. 71 (1996), 269–296.
J. PÖSCHEL, On the construction of almost periodic solutions for a nonlinear Schrödinger equation. Preprint, 1996.
LORD Rayleigh, On waves. Phil. Mag. Ser. 5 1 (1876), 257–279.
J. S. Russell, Report on waves. In: Report of the Fourteenth Meeting of the British Association for the Advancement of Sciences. John Murray, London, 1844, 311–390.
J. SAUT & R. Temam, Remarks on the Korteweg-de Vries equation. Israel J. Math. 24 (1976), 78–87.
A. SJöBERG, On the Korteweg-de Vries equation: existence and uniqueness. J. Math. Anal. Appl. 29 (1970),569–579.
R. Temam, Sur un probllème non linéaire. J. Math. Pures Appl. 48 (1969), 159–172.
P. van Moerbeke, The spectrum of Jacobi matrices. Invent. Math. 37 (1976), 45–81.
G. B. Witham, Linear and Nonlinear Waves. Wiley, New York, 1974.
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Kappeler, T., Pöschel, J. (2003). On the Korteweg — de Vries Equation and KAM Theory. In: Hildebrandt, S., Karcher, H. (eds) Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55627-2_20
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