Preface |
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xi | |
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1 Introduction to difference equations |
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1 | (30) |
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1.1 A first look at discrete equations |
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1 | (15) |
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16 | (5) |
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1.3 Partial difference equations |
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21 | (6) |
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27 | (4) |
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28 | (3) |
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2 Discrete equations from transformations of continuous equations |
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31 | (36) |
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2.1 Special functions and linear equations |
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32 | (9) |
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41 | (3) |
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2.3 The Painleve equations |
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44 | (6) |
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2.4 Backlund transformations for nonlinear PDEs |
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50 | (3) |
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2.5 Infinite sequence of conservation laws and KdV hierarchy |
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53 | (8) |
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61 | (6) |
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62 | (5) |
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67 | (52) |
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68 | (3) |
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3.2 Consistency-around-the-cube as integrability |
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71 | (4) |
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3.3 Lax pairs and Backlund transformation from CAC |
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75 | (7) |
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82 | (4) |
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3.5 Classification of quadrilateral PΔEs |
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86 | (4) |
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3.6 Different equations on different faces of the consistency cube |
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90 | (4) |
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3.7 CAC for multi-component equations |
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94 | (9) |
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3.8 Lattice KdV, SKdV and mKdV equations |
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103 | (7) |
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3.9 Higher-dimensional equations: the KP class |
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110 | (4) |
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114 | (5) |
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116 | (3) |
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4 Interlude: Lattice equations and numerical algorithms |
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119 | (17) |
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119 | (7) |
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4.2 Convergence acceleration algorithm |
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126 | (2) |
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4.3 Rutishauser's QD algorithm |
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128 | (5) |
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133 | (3) |
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134 | (2) |
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5 Continuum limits of lattice PΔE |
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136 | (23) |
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5.1 How to take a continuum limit |
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136 | (1) |
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5.2 Plane-wave factors and linearization |
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137 | (2) |
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5.3 The semi-continuous limits |
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139 | (5) |
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5.4 Semi-discrete Lax pairs |
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144 | (3) |
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147 | (4) |
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5.6 All at once, or the double continuum limit |
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151 | (1) |
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5.7 Continuum limits of the 9-point BSQ |
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152 | (2) |
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154 | (5) |
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155 | (4) |
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6 One-dimensional lattices and maps |
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159 | (38) |
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6.1 Integrability of maps |
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159 | (9) |
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6.2 The Kahan--Hirota--Kimura discretization |
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168 | (1) |
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169 | (8) |
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177 | (8) |
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6.5 Lax pair for the periodic reductions and construction of invariants |
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185 | (4) |
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6.6 Pole reduction of the semi-discrete KP equation |
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189 | (4) |
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193 | (4) |
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194 | (3) |
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7 Identifying integrable difference equations |
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197 | (26) |
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7.1 Singularity analysis of differential and difference equations |
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197 | (10) |
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207 | (6) |
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7.3 Singularities from a geometric point of view |
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213 | (6) |
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219 | (4) |
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221 | (2) |
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8 Hirota's bilinear method |
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223 | (27) |
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223 | (3) |
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226 | (3) |
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8.3 Hirota's and Miwa's equations |
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229 | (4) |
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8.4 Reductions of the Hirota-Miwa equation |
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233 | (5) |
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8.5 Bilinearization of a lattice equation |
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238 | (3) |
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8.6 Solutions in matrix form |
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241 | (4) |
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245 | (5) |
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246 | (4) |
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9 Multi-soliton solutions and the Cauchy matrix scheme |
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250 | (30) |
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9.1 Cauchy matrix structure for KdV-type equations |
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250 | (5) |
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9.2 Closed-form lattice equations |
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255 | (2) |
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9.3 Derivation of Lax pairs |
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257 | (4) |
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9.4 Bilinear form from soliton solutions |
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261 | (5) |
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9.5 The NQC and Q3 equations |
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266 | (2) |
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9.6 Proof of the Q3 N-soliton solution |
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268 | (4) |
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9.7 Higher-dimensional soliton systems: the KP class |
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272 | (5) |
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277 | (3) |
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277 | (3) |
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10 Similarity reductions of integrable PΔEs |
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280 | (24) |
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10.1 Introduction to dimensional reductions |
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280 | (4) |
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10.2 Compatibility of lattice constraint with quad equations |
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284 | (1) |
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285 | (4) |
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10.4 Similarity constraints for the lattice KdV family |
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289 | (11) |
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300 | (4) |
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301 | (3) |
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11 Discrete Painleve equations |
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304 | (30) |
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11.1 Early discoveries of discrete Painleve equations |
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305 | (2) |
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11.2 Discrete Painleve equations from Sakai's classification |
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307 | (3) |
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11.3 Coalescences and degeneracies of the discrete Painleve equations |
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310 | (1) |
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11.4 Backlund and other transformations of discrete Painleve equations |
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311 | (4) |
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315 | (7) |
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322 | (4) |
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11.7 Linearization of discrete Painleve equations |
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326 | (2) |
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11.8 Sakai's elliptic discrete Painleve equation |
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328 | (1) |
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329 | (5) |
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331 | (3) |
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12 Lagrangian multiform theory |
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334 | (29) |
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12.1 Conventional Lagrange theory and its discrete analogue |
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335 | (8) |
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12.2 Lagrangian 2-form structure |
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343 | (9) |
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12.3 Lagrangian 1-form structure |
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352 | (7) |
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359 | (4) |
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360 | (3) |
Appendix A Elementary difference calculus and difference equations |
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363 | (21) |
Appendix B Theta functions and elliptic functions |
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384 | (20) |
Appendix C The continuous Painleve equations and the Garnier system |
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404 | (3) |
Appendix D Some determinantal identities |
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407 | (4) |
References |
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411 | (29) |
Index |
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440 | |