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Direct and Inverse Methods in Nonlinear Evolution Equations: Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 512, 1999 2003 ed. [Kietas viršelis]

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  • Formatas: Hardback, 279 pages, aukštis x plotis: 235x155 mm, weight: 1310 g, XI, 279 p., 1 Hardback
  • Serija: Lecture Notes in Physics 632
  • Išleidimo metai: 21-Oct-2003
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540200878
  • ISBN-13: 9783540200871
Kitos knygos pagal šią temą:
Direct and Inverse Methods in Nonlinear Evolution Equations: Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 512, 1999 2003 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 279 pages, aukštis x plotis: 235x155 mm, weight: 1310 g, XI, 279 p., 1 Hardback
  • Serija: Lecture Notes in Physics 632
  • Išleidimo metai: 21-Oct-2003
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540200878
  • ISBN-13: 9783540200871
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783540200871.html
Raktažodžiai:
Many physical phenomena are described by nonlinear evolution equation. Those that are integrable provide various mathematical methods, presented by experts in this tutorial book, to find special analytic solutions to both integrable and partially integrable equations. The direct method to build solutions includes the analysis of singularities ą la Painlevé, Lie symmetries leaving the equation invariant, extension of the Hirota method, construction of the nonlinear superposition formula. The main inverse method described here relies on the bi-hamiltonian structure of integrable equations. The book also presents some extension to equations with discrete independent and dependent variables. The different chapters face from different points of view the theory of exact solutions and of the complete integrability of nonlinear evolution equations. Several examples and applications to concrete problems allow the reader to experience directly the power of the different machineries involved.
Exact solutions of nonlinear partial differential equations by singularity analysis
1(85)
Robert Conte
Introduction
1(1)
Various levels of integrability for PDEs, definitions
2(7)
Importance of the singularities: a brief survey of the theory of Painleve
9(2)
The Painleve test for PDEs in its invariant version
11(7)
Singular manifold variable φ and expansion variable χ
11(3)
The WTC part of the Painleve test for PDEs
14(3)
The various ways to pass or fail the Painleve test for PDEs
17(1)
Ingredients of the ``singular manifold method''
18(6)
The ODE situation
19(1)
Transposition of the ODE situation to PDEs
19(1)
The singular manifold method as a singular part transformation
20(1)
The degenerate case of linearizable equations
21(1)
Choices of Lax pairs and equivalent Riccati pseudopotentials
21(1)
Second-order Lax pairs and their privilege
21(2)
Third-order Lax pairs
23(1)
The admissible relations between τ and ψ
24(1)
The algorithm of the singular manifold method
24(5)
Where to truncate, and with which variable?
27(2)
The singular manifold method applied to one-family PDEs
29(24)
Integrable equations with a second order Lax pair
29(1)
The Liouville equation
30(2)
The AKNS equation
32(1)
The KdV equation
33(2)
Integrable equations with a third order Lax pair
35(1)
The Boussinesq equation
35(2)
The Hirota-Satsuma equation
37(1)
The Tzitzeica equation
38(5)
The Sawada-Kotera and Kaup-Kupershmidt equations
43(1)
The Sawada-Kotera equation
44(1)
The Kaup-Kupershmidt equation
45(4)
Nonintegrable equations, second scattering order
49(1)
The Kuramoto-Sivashinsky equation
49(3)
Nonintegrable equations, third scattering order
52(1)
Two common errors in the one-family truncation
53(1)
The constant level term does not define a BT
53(1)
The WTC truncation is suitable iff the Lax order is two
54(1)
The singular manifold method applied to two-family PDEs
54(15)
Integrable equations with a second order Lax pair
55(1)
The sine-Gordon equation
55(2)
The modified Korteweg-de Vries equation
57(2)
The nonlinear Schrodinger equation
59(1)
Integrable equations with a third order Lax pair
59(1)
Nonintegrable equations, second and third scattering order
60(1)
The KPP equation
60(5)
The cubic complex Ginzburg-Landau equation
65(3)
The nonintegrable Kundu-Eckhaus equation
68(1)
Singular manifold method versus reduction methods
69(3)
Truncation of the unknown, not of the equation
72(2)
Birational transformations of the Painleve equations
74(2)
Conclusion, open problems
76(9)
References
77(8)
The method of Poisson pairs in the theory of nonlinear PDEs
85(52)
Franco Magri
Gregorio Falqui
Marco Pedroni
Introduction: The tensorial approach and the birth of the method of Poisson pairs
85(11)
The Miura map and the KdV equation
86(2)
Poisson pairs and the KdV hierarchy
88(2)
Invariant submanifolds and reduced equations
90(4)
The modified KdV hierarchy
94(2)
The method of Poisson pairs
96(5)
A first class of examples and the reduction technique
101(8)
Lie--Poisson manifolds
101(1)
Polynomial extensions
102(1)
Geometric reduction
103(1)
An explicit example
104(4)
A more general example
108(1)
The KdV theory revisited
109(11)
Poisson pairs on a loop algebra
109(1)
Poisson reduction
110(2)
The GZ hierarchy
112(1)
The central system
113(2)
The linearization process
115(2)
The relation with the Sato approach
117(3)
Lax representation of the reduced KdV flows
120(5)
Lax representation
120(2)
First example
122(2)
The generic stationary submanifold
124(1)
What more?
125(1)
Darboux-Nijenhuis coordinates and separability
125(12)
The Poisson pair
126(2)
Passing to a symplectic leaf
128(2)
Darboux--Nijenhuis coordinates
130(1)
Separation of variables
131(3)
References
134(3)
Nonlinear superposition formulae of integrable partial differential equations by the singular manifold method
137(34)
Micheline Musette
Introduction
137(1)
Integrability by the singularity approach
138(1)
Backlund transformation: definition and example
139(1)
Singularity analysis of nonlinear differential equations
139(4)
Nonlinear ordinary differential equations
139(3)
Nonlinear partial differential equations
142(1)
Lax Pair and Darboux transformation
143(4)
Second order scalar scattering problem
144(1)
Third order scalar scattering problem
145(1)
A third order matrix scattering problem
146(1)
Different truncations in Painleve analysis
147(2)
Method for a one-family equation
149(2)
Nonlinear superposition formula
151(1)
Results for PDEs possessing a second order Lax pair
151(5)
First example: KdV equation
151(2)
Second example: MKdV and sine-Gordon equations
153(3)
PDEs possessing a third order Lax pair
156(15)
Sawada-Kotera, KdV5, Kaup-Kupershmidt equations
156(1)
Painleve test
157(1)
Truncation with a second order Lax pair
158(1)
Truncation with a third order Lax pair
158(1)
Backlund transformation
159(1)
Nonlinear superposition formula for Sawada-Kotera
160(1)
Nonlinear superposition formula for Kaup-Kupershmidt
161(4)
Tzitzeica equation
165(1)
References
166(5)
Hirota bilinear method for nonlinear evolution equations
171(52)
Junkichi Satsuma
Introduction
171(1)
Soliton solutions
172(8)
The Burgers equation
172(1)
The Korteweg-de Vries equation
173(1)
The nonlinear Schrodinger equation
174(1)
The Toda equation
175(1)
Painleve equations
176(1)
Difference vs differential
177(3)
Multidimensional equations
180(7)
The Kadomtsev-Petviashvili equation
180(1)
The two-dimensional Toda lattice equation
181(3)
Two-dimensional Toda molecule equation
184(1)
The Hirota-Miwa equation
185(2)
Sato theory
187(23)
Micro-differential operators
187(2)
Introduction of an infinite number of time variables
189(3)
The Sato equation
192(2)
Generalized Lax equation
194(1)
Structure of tau functions
195(5)
Algebraic identities for tau functions
200(4)
Vertex operators and the KP bilinear identity
204(3)
Fermion analysis based on an infinite dimensional Lie algebra
207(3)
Extensions of the bilinear method
210(13)
q-discrete equations
210(2)
Special function solution for soliton equations
212(3)
Ultra discrete soliton system
215(3)
Trilinear equations
218(3)
References
221(2)
Lie groups, singularities and solutions of nonlinear partial differential equations
223(51)
Pavel Winternitz
Introduction
223(2)
The symmetry group of a system of differential equations
225(13)
Formulation of the problem
225(1)
Prolongation
226(1)
Symmetry group: Global approach, use the chain rule
227(1)
Symmetry group: Infinitesimal approach
227(1)
Reformulation
227(1)
Prolongation of vector fields and the symmetry algorithm
228(2)
First example: Variable coefficient KdV equation
230(2)
Symmetry reduction for the KdV
232(3)
Second example: Modified Kadomtsev-Petviashvili equation
235(3)
Classification of the subalgebras of a finite dimensional Lie algebra
238(14)
Formulation of the problem
238(1)
Subalgebras of a simple Lie algebra
239(1)
Example: Maximal subalgebras of o(4,2)
240(4)
Subalgebras of semidirect sums
244(4)
Example: All subalgebras of sl(3, R) classified under the group SL(3, R)
248(4)
Generalizations
252(1)
The Clarkson-Kruskal direct reduction method and conditional symmetries
252(11)
Formulation of the problem
252(1)
Symmetry reduction for Boussinesq equation
253(1)
The direct method
254(2)
Conditional symmetries
256(5)
General comments
261(2)
Concluding comments
263(11)
References on nonlinear superposition formulas
263(1)
References on continuous symmetries of difference equations
264(1)
References
264(10)
List of Participants 274(3)
Index 277