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December  2019, 6(2): 449-467. doi: 10.3934/jcd.2019023

Linear degree growth in lattice equations

Dinh T. Tran 1, and  John A. G. Roberts 2,,

1. 

School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
2. 

School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia

* Corresponding author: John A. G. Roberts

To Reinout Quispel on his 66th birthday, in friendship and with gratitude

Received  July 2019 Published  November 2019

We conjecture recurrence relations satisfied by the degrees of some linearizable lattice equations. This helps to prove linear degree growth of these equations. We then use these recurrences to search for lattice equations that have linear growth and hence are linearizable.

Keywords: Linear growth, linearizable, lattice equations, integrable, algebraic entropy, degree relations.
Mathematics Subject Classification: Primary: 37K10; Secondary: 39A14.
Citation: Dinh T. Tran, John A. G. Roberts. Linear degree growth in lattice equations. Journal of Computational Dynamics, 2019, 6 (2) : 449-467. doi: 10.3934/jcd.2019023
References:
[1]

V. E. Adler, A. I. Bobenko and Y. B. Suris, Classification of integrable equations on quad-graphs. The consistency approach, Communications in Mathematical Physics, 233 (2003), 513-543.  doi: 10.1007/s00220-002-0762-8.  Google Scholar

[2]

V. É. Adler and S. Y. Startsev, Discrete analogues of the Liouville equation, Theoretical and Mathematical Physics, 121 (1999), 1484-1495.  doi: 10.1007/BF02557219.  Google Scholar

[3]

M. P. Belon, Algebraic entropy of birational maps with invariant curves, Lett. Math. Phys., 50 (1999), 79-90.  doi: 10.1023/A:1007634406786.  Google Scholar

[4]

J. Blanc and J. Déserti, Degree growth of birational maps of the plane, Ann. Sc. Norm. Super. Pisa Cl. Sci., 14 (2015), 507-533.   Google Scholar

[5]

P. Galashin and P. Pylyavskyy, Quivers with additive labelings: Classification and algebraic entropy, preprint, arXiv: 1704.05024v2. Google Scholar

[6]

R. N. Garifullin and R. I. Yamilov, Generalized symmetry classification of discrete equations of a class depending on twelve parameters, J. Phys. A: Math. Theor., 45 (2012), 345205, 23 pp. doi: 10.1088/1751-8113/45/34/345205.  Google Scholar

[7]

B. Grammaticos, R. G. Halburd, A. Ramani and C.-M. Viallet, How to detect the integrability of discrete systems, J. Phys. A: Math. Theor., 42 (2009), 454002, 30 pp. doi: 10.1088/1751-8113/42/45/454002.  Google Scholar

[8]

B. Grammaticos and A. Ramani, Singularity confinement property for the (non-autonomous) Adler-Bobenko-Suris integrable lattice equations, Lett. Math. Phys., 92 (2010), 33-45.  doi: 10.1007/s11005-010-0378-4.  Google Scholar

[9]

B. Grammaticos, A. Ramani and C.-M. Viallet, Solvable chaos, Physics Letters A, 336 (2005), 152-158.  doi: 10.1016/j.physleta.2005.01.026.  Google Scholar

[10]

G. Gubbiotti, C. Scimiterna and D. Levi, Algebraic entropy, symmetries and linearization of quad equations consistent on the cube, J. Non. Math. Phys., 23 (2016), 507-543.  doi: 10.1080/14029251.2016.1237200.  Google Scholar

[11]

J. Hietarinta, A new two-dimensional lattice model that is 'consistent around a cube', J. Phys. A: Math. Gen., 37 (2004), L67–L73. doi: 10.1088/0305-4470/37/6/L01.  Google Scholar

[12]

J. Hietarinta and C. Viallet, Searching for integrable lattice maps using factorisation, J. Phys. A: Math. Theor., 40 (2007), 12629-12643.  doi: 10.1088/1751-8113/40/42/S09.  Google Scholar

[13]

P. E. Hydon and C.-M. Viallet, Asymmetric integrable quad-graph equations, Applicable Analysis, 89 (2010), 493-506.  doi: 10.1080/00036810903329951.  Google Scholar

[14]

P. H. van der Kamp, Growth of degrees of integrable mapping, J. Difference Equ. Appl., 18 (2012), 447-460.  doi: 10.1080/10236198.2010.510137.  Google Scholar

[15]

M. Kanki, T. Mase and T. Tokihiro, Algebraic entropy of an extended Hietarinta-Viallet equation, J. Phys. A: Math. Theor., 48 (2015), 355202, 19 pp. doi: 10.1088/1751-8113/48/35/355202.  Google Scholar

[16]

D. Levi and C. Scimiterna, Linearizability of nonlinear equations on a quad-graph by a Point, two points and generalized Hopf-Cole transformations, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), Paper 079, 24 pp. doi: 10.3842/SIGMA.2011.079.  Google Scholar

[17]

D. Levi and C. Scimiterna, Linearization through symmetries for discrete equations, J. Phys. A: Math. Theor., 46 (2013), 325204, 18 pp. doi: 10.1088/1751-8113/46/32/325204.  Google Scholar

[18]

D. Levi and C. Scimiterna, Four points linearizable lattice schemes, JGSP, 31 (2013), 93-104.   Google Scholar

[19]

C. U. Maheswari and R. Sahadevan, On the conservation laws for nonlinear partial difference equations, J. Phys. A: Math. Theor., 44 (2011), 275203, 16 pp. doi: 10.1088/1751-8113/44/27/275201.  Google Scholar

[20]

T. Mase, Investigation into the role of the Laurent property in integrability, Journal of Mathematical Physics, 57 (2016), 022703, 21 pp. doi: 10.1063/1.4941370.  Google Scholar

[21]

G. R. W. Quispel, H. W. Capel, V. G. Papageorgiou and F. W. Nijhoff, Integrable mappings derived from soliton equations, Physica A, 173 (1991), 243-266.  doi: 10.1016/0378-4371(91)90258-E.  Google Scholar

[22]

A. Ramani, N. Joshi, B. Grammaticos and T. Tamizhmani, Deconstructing an integrable lattice equation, J. Phys. A: Math. Gen., 39 (2006), L145–L149. doi: 10.1088/0305-4470/39/8/L01.  Google Scholar

[23]

J. A. G. Roberts and D. T. Tran, Algebraic entropy of (integrable) lattice equations and their reductions, Nonlinearity, 32 (2019), 622-653.  doi: 10.1088/1361-6544/aaecda.  Google Scholar

[24]

C. Scimiterna and D. Levi, Classification of discrete equations linearizable by point transformation on a square lattice, Front. Math. China, 8 (2013), 1067-1076.  doi: 10.1007/s11464-013-0280-3.  Google Scholar

[25]

T. Takenawa, M. Eguchi, B. Gramaticos, Y. Ohta, A. Ramani and J. Satsuma, The space of initial conditions for linearizable mappings, Nonlinearity, 16 (2003), 457-477.  doi: 10.1088/0951-7715/16/2/306.  Google Scholar

[26]

S. Tremblay, B. Grammaticos and A. Ramani, Integrable lattice equations and their growth properties, Phys. Lett. A, 278 (2001), 319-324.  doi: 10.1016/S0375-9601(00)00806-9.  Google Scholar

[27]

C. Viallet, Algebraic entropy for lattice equations, preprint, arXiv: 0609043v2. Google Scholar

[28]

C. M. Viallet, Integrable lattice maps: $Q_V$, a rational version of $Q_4$, Glasg. Math. J., 51 (2009), 157-163.  doi: 10.1017/S0017089508004874.  Google Scholar

[29]

C.-M. Viallet, On the algebraic structure of rational discrete dynamical systems, J. Phys. A: Math. Theor., 48 (2015), 16FT01, 21 pp. doi: 10.1088/1751-8113/48/16/16FT01.  Google Scholar

show all references

References:
[1]

V. E. Adler, A. I. Bobenko and Y. B. Suris, Classification of integrable equations on quad-graphs. The consistency approach, Communications in Mathematical Physics, 233 (2003), 513-543.  doi: 10.1007/s00220-002-0762-8.  Google Scholar

[2]

V. É. Adler and S. Y. Startsev, Discrete analogues of the Liouville equation, Theoretical and Mathematical Physics, 121 (1999), 1484-1495.  doi: 10.1007/BF02557219.  Google Scholar

[3]

M. P. Belon, Algebraic entropy of birational maps with invariant curves, Lett. Math. Phys., 50 (1999), 79-90.  doi: 10.1023/A:1007634406786.  Google Scholar

[4]

J. Blanc and J. Déserti, Degree growth of birational maps of the plane, Ann. Sc. Norm. Super. Pisa Cl. Sci., 14 (2015), 507-533.   Google Scholar

[5]

P. Galashin and P. Pylyavskyy, Quivers with additive labelings: Classification and algebraic entropy, preprint, arXiv: 1704.05024v2. Google Scholar

[6]

R. N. Garifullin and R. I. Yamilov, Generalized symmetry classification of discrete equations of a class depending on twelve parameters, J. Phys. A: Math. Theor., 45 (2012), 345205, 23 pp. doi: 10.1088/1751-8113/45/34/345205.  Google Scholar

[7]

B. Grammaticos, R. G. Halburd, A. Ramani and C.-M. Viallet, How to detect the integrability of discrete systems, J. Phys. A: Math. Theor., 42 (2009), 454002, 30 pp. doi: 10.1088/1751-8113/42/45/454002.  Google Scholar

[8]

B. Grammaticos and A. Ramani, Singularity confinement property for the (non-autonomous) Adler-Bobenko-Suris integrable lattice equations, Lett. Math. Phys., 92 (2010), 33-45.  doi: 10.1007/s11005-010-0378-4.  Google Scholar

[9]

B. Grammaticos, A. Ramani and C.-M. Viallet, Solvable chaos, Physics Letters A, 336 (2005), 152-158.  doi: 10.1016/j.physleta.2005.01.026.  Google Scholar

[10]

G. Gubbiotti, C. Scimiterna and D. Levi, Algebraic entropy, symmetries and linearization of quad equations consistent on the cube, J. Non. Math. Phys., 23 (2016), 507-543.  doi: 10.1080/14029251.2016.1237200.  Google Scholar

[11]

J. Hietarinta, A new two-dimensional lattice model that is 'consistent around a cube', J. Phys. A: Math. Gen., 37 (2004), L67–L73. doi: 10.1088/0305-4470/37/6/L01.  Google Scholar

[12]

J. Hietarinta and C. Viallet, Searching for integrable lattice maps using factorisation, J. Phys. A: Math. Theor., 40 (2007), 12629-12643.  doi: 10.1088/1751-8113/40/42/S09.  Google Scholar

[13]

P. E. Hydon and C.-M. Viallet, Asymmetric integrable quad-graph equations, Applicable Analysis, 89 (2010), 493-506.  doi: 10.1080/00036810903329951.  Google Scholar

[14]

P. H. van der Kamp, Growth of degrees of integrable mapping, J. Difference Equ. Appl., 18 (2012), 447-460.  doi: 10.1080/10236198.2010.510137.  Google Scholar

[15]

M. Kanki, T. Mase and T. Tokihiro, Algebraic entropy of an extended Hietarinta-Viallet equation, J. Phys. A: Math. Theor., 48 (2015), 355202, 19 pp. doi: 10.1088/1751-8113/48/35/355202.  Google Scholar

[16]

D. Levi and C. Scimiterna, Linearizability of nonlinear equations on a quad-graph by a Point, two points and generalized Hopf-Cole transformations, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), Paper 079, 24 pp. doi: 10.3842/SIGMA.2011.079.  Google Scholar

[17]

D. Levi and C. Scimiterna, Linearization through symmetries for discrete equations, J. Phys. A: Math. Theor., 46 (2013), 325204, 18 pp. doi: 10.1088/1751-8113/46/32/325204.  Google Scholar

[18]

D. Levi and C. Scimiterna, Four points linearizable lattice schemes, JGSP, 31 (2013), 93-104.   Google Scholar

[19]

C. U. Maheswari and R. Sahadevan, On the conservation laws for nonlinear partial difference equations, J. Phys. A: Math. Theor., 44 (2011), 275203, 16 pp. doi: 10.1088/1751-8113/44/27/275201.  Google Scholar

[20]

T. Mase, Investigation into the role of the Laurent property in integrability, Journal of Mathematical Physics, 57 (2016), 022703, 21 pp. doi: 10.1063/1.4941370.  Google Scholar

[21]

G. R. W. Quispel, H. W. Capel, V. G. Papageorgiou and F. W. Nijhoff, Integrable mappings derived from soliton equations, Physica A, 173 (1991), 243-266.  doi: 10.1016/0378-4371(91)90258-E.  Google Scholar

[22]

A. Ramani, N. Joshi, B. Grammaticos and T. Tamizhmani, Deconstructing an integrable lattice equation, J. Phys. A: Math. Gen., 39 (2006), L145–L149. doi: 10.1088/0305-4470/39/8/L01.  Google Scholar

[23]

J. A. G. Roberts and D. T. Tran, Algebraic entropy of (integrable) lattice equations and their reductions, Nonlinearity, 32 (2019), 622-653.  doi: 10.1088/1361-6544/aaecda.  Google Scholar

[24]

C. Scimiterna and D. Levi, Classification of discrete equations linearizable by point transformation on a square lattice, Front. Math. China, 8 (2013), 1067-1076.  doi: 10.1007/s11464-013-0280-3.  Google Scholar

[25]

T. Takenawa, M. Eguchi, B. Gramaticos, Y. Ohta, A. Ramani and J. Satsuma, The space of initial conditions for linearizable mappings, Nonlinearity, 16 (2003), 457-477.  doi: 10.1088/0951-7715/16/2/306.  Google Scholar

[26]

S. Tremblay, B. Grammaticos and A. Ramani, Integrable lattice equations and their growth properties, Phys. Lett. A, 278 (2001), 319-324.  doi: 10.1016/S0375-9601(00)00806-9.  Google Scholar

[27]

C. Viallet, Algebraic entropy for lattice equations, preprint, arXiv: 0609043v2. Google Scholar

[28]

C. M. Viallet, Integrable lattice maps: $Q_V$, a rational version of $Q_4$, Glasg. Math. J., 51 (2009), 157-163.  doi: 10.1017/S0017089508004874.  Google Scholar

[29]

C.-M. Viallet, On the algebraic structure of rational discrete dynamical systems, J. Phys. A: Math. Theor., 48 (2015), 16FT01, 21 pp. doi: 10.1088/1751-8113/48/16/16FT01.  Google Scholar

Figure 2.  Illustration of the degree recurrence relation (10)
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Figure 3.  Equations equivalent to equation (10)
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Figure 1.  Initial values $ I_1 $ (left) and $ I_2 $ (right) for lattice equations
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