Nonlinear Systems and Their Remarkable Mathematical Structures-Volume 2

541 Pages·2019·16.98 MB·\541

by  Norbert Euler (Editor), Maria Clara Nucci (Editor)

Nonlinear Systems and Their
Remarkable Mathematical
Structures
Volume 2

Nonlinear Systems and Their
Remarkable Mathematical
Structures
Volume 2
Edited by
Norbert Euler
Centro Internacional de Ciencias AC
and
Maria Clara Nucci
University of Perugia
Front cover image: acrylic painting titled “Arbelos” by Nora Euler (May 2019).
CRC Press
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Contents
Preface........................................................................xi
The Authors ................................................................. xiii
Part A: Integrability, Lax Pairs and Symmetry
A1. Reciprocal transformations and their role in the integrability
and classification of PDEs ............................................1
P Albares, P G Est´evez and C Sard´on
1. Introduction .........................................................1
2. Fundamentals .......................................................3
3. Reciprocal transformations as a way to identify and
classify PDEs ........................................................7
4. Reciprocal transformations to derive Lax pairs ......................17
5. A Miura-reciprocal transformation ..................................21
6. Conclusions ........................................................25
A2. Contact Lax pairs and associated (3+1)-dimensional integrable
dispersionless systems................................................29
M Bl(cid:32)aszak and A Sergyeyev
1. Introduction ........................................................29
2. Isospectral versus nonisospectral Lax pairs ..........................30
3. Lax representations for dispersionless systems
in (1+1)D and (2+1)D .............................................31
4. Lax representations for dispersionless systems in (3+1)D ........... 37
5. R-matrix approach for dispersionless systems with nonisospectral
Lax representations ................................................ 42
A3. Lax pairs for edge-constrained Boussinesq systems of partial
difference equations ..................................................59
T J Bridgman and W Hereman
1. Introduction ........................................................59
2. Gauge equivalence of Lax pairs for PDEs and P∆Es ................62
3. Derivation of Lax pairs for Boussinesq systems ..................... 65
v
vi Contents
4. Gauge and gauge-like equivalences of Lax pairs .....................72
5. Application to generalized Hietarinta systems .......................75
6. Summary of results .................................................76
7. Software implementation and conclusions ...........................85
A4. Lie point symmetries of delay ordinary differential equations ..89
V A Dorodnitsyn, R Kozlov, S V Meleshko and P Winternitz
1. Introduction ........................................................89
2. Illustrating example ................................................91
3. Formulation of the problem for first-order DODEs ..................91
4. Construction of invariant first-order DODSs ........................93
5. First-order linear DODSs ...........................................95
6. Lie symmetry classification of first-order nonlinear DODSs .........100
7. Exact solutions of the DODSs .....................................105
8. Higher order DODSs ..............................................110
9. Traffic flow micro-model equation ................................. 112
10. Conclusions .......................................................114
A5. The symmetry approach to integrability: recent advances .....119
R Hern´andez Heredero and V Sokolov
1. Introduction ...................................................... 119
2. The symmetry approach to integrability ...........................124
3. Integrable non-abelian equations .................................. 135
4. Non-evolutionary systems .........................................143
A6. Evolution of the concept of λ−symmetry and main
applications .........................................................158
C Muriel and J L Romero
1. Introduction ...................................................... 158
2. Basic notions on Lie point symmetries and C∞− symmetries .......160
3. Analytical applications of C∞−symmetries ........................ 167
4. Extensions and geometric interpretations of C∞−symmetries ...... 175
Contents vii
A7. Heir-equations for partial differential equations:
a 25-year review ....................................................188
M C Nucci
1. Introduction ...................................................... 188
2. Constructing the heir-equations ...................................190
3. Symmetry solutions of heir-equations ..............................192
4. Zhdanov’s conditional Lie-B¨acklund symmetries and heir-equations 193
5. Nonclassical symmetries as special solutions of heir-equations ......195
6. Final remarks .....................................................198
Part B: Algebraic and Geometric Methods
B1. Coupled nonlinear Schr¨odinger equations: spectra and
instabilities of plane waves .........................................206
A Degasperis, S Lombardo and M Sommacal
1. Introduction ...................................................... 206
2. Spectra ...........................................................211
3. Dispersion relation and instability .................................228
4. Conclusions .......................................................235
Appendix A: Case r = 0 ...........................................236
Appendix B: Polynomials: a tool box ..............................239
B2. Rational solutions of Painlev´e systems ...........................249
D G´omez-Ullate, Y Grandati and R Milson
1. Introduction ...................................................... 249
2. Dressing chains and Painlev´e systems ..............................252
3. Hermite τ-functions ...............................................257
4. Hermite-type rational solutions ....................................265
5. Cyclic Maya diagrams .............................................273
6. Examples of Hermite-type rational solutions .......................278
B3. Cluster algebras and discrete integrability .......................294
A N W Hone, P Lampe and T E Kouloukas
1. Introduction ...................................................... 294
2. Cluster algebras: definition and examples ..........................294
viii Contents
3. Cluster algebras with periodicity ..................................302
4. Algebraic entropy and tropical dynamics ..........................304
5. Poisson and symplectic structures .................................309
6. Discrete Painlev´e equations from coefficient mutation ..............318
7. Conclusions .......................................................321
B4. A review of elliptic difference Painlev´e equations................326
N Joshi and N Nakazono
1. Introduction ...................................................... 326
(1)
2. E -lattice ........................................................331
8
3. The initial value space of the RCG equation .......................335
4. Cremona isometries ...............................................337
5. Birational actions of the Cremona isometries for the
Jacobi setting .....................................................339
6. Special solutions of the RCG equation .............................342
(1)
Appendix A: A -lattice ..........................................344
4
Appendix B: General elliptic difference equations ..................350
B5. Linkage mechanisms governed by integrable deformations
of discrete space curves ............................................356
S Kaji, K Kajiwara and H Park
1. Introduction ...................................................... 356
2. A mathematical model of linkage ..................................358
3. Hinged network and discrete space curve ..........................363
4. Deformation of discrete curves .....................................367
5. Extreme Kaleidocycles ............................................374
B6. The Cauchy problem of the Kadomtsev-Petviashvili hierarchy
and infinite-dimensional groups ...................................382
J-P Magnot and E G Reyes
1. Introduction ...................................................... 382
2. Diffeologies, Fr¨olicher spaces, and the Ambrose-Singer theorem ....383
3. Infinite-dimensional groups and pseudodifferential operators .......393
4. The Cauchy problem for the KP hierarchy .........................406
5. A non-formal KP hierarchy ........................................410
Contents ix
B7. Wronskian solutions of integrable systems .......................415
D-j Zhang
1. Introduction ...................................................... 415
2. Preliminary .......................................................416
3. The KdV equation ................................................420
4. The mKdV equation ..............................................425
5. The AKNS and reductions ........................................432
6. Discrete case: the lpKdV equation ................................ 436
7. Conclusions .......................................................440
Part C: Applications
C1. Global gradient catastrophe in a shallow water model:
evolution unfolding by stretched coordinates ....................445
R Camassa
1. Introduction ...................................................... 445
2. Exact solutions ....................................................446
3. Evolution beyond the gradient catastrophe ........................451
4. Discussion and conclusions ........................................456
C2. Vibrations of an elastic bar, isospectral deformations,
and modified Camassa-Holm equations ...........................459
X K Chang and J Szmigielski
1. Introduction ...................................................... 459
2. The Lax formalism: the boundary value problem ..................461
3. Liouville integrability ............................................. 465
4. Forward map: spectrum and spectral data .........................469
5. Inverse problem ...................................................471
6. Multipeakons for N = 2K .........................................479
7. Multipeakons for N = 2K +1 .....................................483
8. Reductions of multipeakons .......................................487
Appendix A: Lax pair for the 2-mCH peakon ODEs ...............491
Appendix B: Proof of Theorem 13 .................................493
Appendix C: Proof of Theorem 17 .................................495