March  2019, 8(1): 43-55. doi: 10.3934/eect.2019003

Abelian versus non-Abelian Bäcklund charts: Some remarks

1. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università di Roma La Sapienza, Via A. Scarpa 16, I-00161, Italy

2. 

I.N.F.N. - Sez. Roma1, Gr.Ⅳ: Mathematical Methods in NonLinear Physics, Rome, Italy

3. 

Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza, Via Eudossiana 18, I-00100, Rome, Italy

4. 

Department of Mathematics and Science Education, Mid Sweden University, S-851 70 Sundsvall, Sweden

5. 

Instytut Matematyki, Uniwersytet Jana Kochanowskiego w Kielcach, Poland

Received  May 2018 Revised  May 2018 Published  January 2019

Fund Project: The authors are supported by G.N.F.M.-I.N.d.A.M., I.N.F.N. and La Sapienza Università di Roma, Italy

Connections via Bäcklund transformations among different nonlinear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Bäcklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Bäcklund charts in the case of the Burgers -heat equation, on one side, and KdV-type equations, on the other, are considered. The Bäcklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.

Citation: Sandra Carillo, Mauro Lo Schiavo, Cornelia Schiebold. Abelian versus non-Abelian Bäcklund charts: Some remarks. Evolution Equations & Control Theory, 2019, 8 (1) : 43-55. doi: 10.3934/eect.2019003
References:
[1]

M. J. AblowitzD. J. KaupA. C. Newell and H. Segur, The inverse scattering transform - Fourier analysis for nonlinear problems, Studies in Applied Mathematics, 53 (1974), 249-315. doi: 10.1002/sapm1974534249.

[2]

H. Aden and B. Carl, On realizations of solutions of the KdV equation by determinants on operator ideals, J. Math. Phys., 37 (1996), 1833-1857. doi: 10.1063/1.531482.

[3]

C. Athorne and A. Fordy, Generalised KdV and MKdV equations associated with symmetric spaces, J. Phys. A: Math. Gen., 20 (1987), 1377-1386. doi: 10.1088/0305-4470/20/6/021.

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P. Basarab-Horwath and F. Güngör, Linearizability for third order evolution equations, J. Math. Phys., 58 (2017), 081507, 13pp. doi: 10.1063/1.4997558.

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F. Calogero and A. Degasperis, A modified modified Korteweg-de Vries equation, Inverse Problems, 1 (1985), 57-66. doi: 10.1088/0266-5611/1/1/006.

[6]

F. Calogero and S. De Lillo, The Burgers equation on the semiline with general boundary conditions at the origin, J. Math. Phys., 32 (1991), 99-105. doi: 10.1063/1.529101.

[7]

F. Calogero and A. Degasperis, Spectral Transform and Solitons I, Studies in Mathematics and its Application, Vol. 13, North Holland, Amsterdam, 1982. doi: 10.1007/978-3-642-82135-6_2.

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S. Carillo, Nonlinear evolution equations: Bäcklund transformations and Bäcklund charts, Acta Applicandae Math., 122 (2012), 93-106. doi: 10.1007/s10440-012-9729-8.

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S. Carillo, KdV-type Equations Linked via Bäcklund Transformations: Remarks and Perspectives, arXiv: 1702.06874, Applied Numerical Mathematics,(2018), in press. doi: 10.1016/j.apnum.2018.10.002.

[10]

S. Carillo and B. Fuchssteiner, The abundant symmetry structure of hierarchies of nonlinear equations obtained by reciprocal links, J. Math. Phys., 30 (1989), 1606-1613. doi: 10.1063/1.528604.

[11]

S. Carillo and C. Schiebold, A non-commutative operator-hierarchy of Burgers equations and Bäcklund transformations, in Applied and Industrial Mathematics in Italy III, (ed.s E. De Bernardis, R. Spigler, V. Valente), Ser. Adv. Math. Appl. Sci., Vol.82, World Scientific Publ., Singapore, (2010), 175-185. doi: 10.1142/9789814280303_0016.

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S. Carillo and C. Schiebold, Non-commutative KdV and mKdV hierarchies via recursion methods, J. Math. Phys., 50 (2009), 073510, 14pp. doi: 10.1063/1.3155080.

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S. Carillo and C. Schiebold, On the recursion operator for the non-commutative Burgers hierarchy, J. Nonlin. Math. Phys., 19 (2012), 1250003, 11 pp. doi: 10.1142/S1402925112500039.

[15]

S. Carillo, M. Lo Schiavo and C. Schiebold, Bäcklund Transformations and Non Abelian Nonlinear Evolution Equations: a novel Bäcklund chart, SIGMA, 12 (2016), Paper No. 087, 17 pp. doi: 10.3842/SIGMA.2016.087.

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S. Carillo, M. Lo Schiavo, E. Porten and C. Schiebold, A novel noncommutative KdV-type equation, its recursion operator, and solitons, J. Math. Phys., 59 (2018), 043501, 14 pp. doi: 10.1063/1.5027481.

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J. D. Cole, On a quasi-linear parabolic equation occuring in aerodynamics, Quart. App. Math., 9 (1951), 225-236. doi: 10.1090/qam/42889.

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A. S. Fokas and B. Fuchssteiner, Bäcklund transformation for hereditary symmetries, Nonlin. Anal., Theory Methods Appl., 5 (1981), 423-432. doi: 10.1016/0362-546X(81)90025-0.

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B. Fuchssteiner, Application of hereditary symmetries to nonlinear evolution equations, Nonlin. Anal., Th. Meth. Appl., 3 (1979), 849-862. doi: 10.1016/0362-546X(79)90052-X.

[23]

B. Fuchssteiner, The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems, Progr. Theor. Phys., 68 (1982), 1082-1104. doi: 10.1143/PTP.68.1082.

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B. Fuchssteiner and S. Carillo, Soliton structure versus singularity analysis: Third order completely integrable nonlinear equations in 1+1 dimensions, Phys. A, 154 (1989), 467-510. doi: 10.1016/0378-4371(89)90260-4.

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B. Fuchssteiner and S. Carillo, The action-angle transformation for soliton equations, Physica A, 166 (1990), 651-675. doi: 10.1016/0378-4371(90)90078-7.

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B. FuchssteinerT. Schulze and S. Carillo, Explicit solutions for the Harry Dym equation, J. Phys. A, 25 (1992), 223-230. doi: 10.1088/0305-4470/25/1/025.

[27]

B. Fuchssteiner and W. Oevel, The bi-Hamiltonian structure of some nonlinear fifth- and seventh-order differential equations and recursion formulas for their symmetries and conserved covariants, J. Math. Phys, 23 (1982), 358-363. doi: 10.1063/1.525376.

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B. Yu Guo and S. Carillo, Infiltration in soils with prescribed boundary concentration: A Burgers model, Acta Appl. Math. Sinica, 6 (1990), 365-369. doi: 10.1007/BF02015343.

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B. Yu Guo and C. Rogers, On Harry-Dym equation and its solution, Science in China, 32 (1989), 283-295.

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M. GürsesA. Karasu and V. V. Sokolov, On construction of recursion operators from Lax representation, J. Math. Phys., 40 (1999), 6473-6490. doi: 10.1063/1.533102.

[32]

M. GürsesA. Karasu and R. Turhan. On non-commutative integrable Burgers equations, On non-commutative integrable Burgers equations, J. Nonlinear Math. Phys., 17 (2010), 1-6. doi: 10.1142/S1402925110000532.

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M. Hamanaka and K. Toda, Noncommutative Burgers equation, J. Phys. A: Math. Gen., 36 (2003), 11981-11998. doi: 10.1088/0305-4470/36/48/006.

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E. Hopf, The partial differential equation $ u_{t} + u u_{x} = \mu u_{xx}$, Comm. Pure Appl. Math., 3 (1950), 201-230. doi: 10.1002/cpa.3160030302.

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S. Kawamoto, An exact transformation from the Harry Dym equation to the modified KdV equation, J. Phys. Soc. Japan, 54 (1985), 2055-2056. doi: 10.1143/JPSJ.54.2055.

[37]

F. A. Khalilov and E. Ya. Khruslov. Matrix generalisation of the modified Korteweg-de Vries equation, Matrix generalisation of the modified Korteweg-de Vries equation, Inv. Problems, 6 (1990), 193-204. doi: 10.1088/0266-5611/6/2/004.

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B. G. Konopelchenko, Soliton eigenfunction equations: The IST integrability and some properties, Rev. Math. Phys., 2 (1990), 399-440. doi: 10.1142/S0129055X90000120.

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B. A. Kupershmidt., On a group of automorphisms of the non-commutative Burgers hierarchy, J. Nonlinear Math. Phys., 12 (2005), 539-549. doi: 10.2991/jnmp.2005.12.4.8.

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M. LeoR. A. LeoG. Soliani and L. Solombrino, On the isospectral-eigenvalue problem and the recursion operator of the Harry Dym equation, Lett. Nuovo Cimento, 38 (1983), 45-51. doi: 10.1007/BF02782775.

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D. LeviO. Ragnisco and M. Bruschi., Continuous and discrete matrix Burgers' hierarchies, Il Nuovo Cimento, 74 (1983), 33-51. doi: 10.1007/BF02721683.

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S. Y. Lou, Symmetries and similarity reductions of the Dym equation, Phys. Scripta, 54 (1996), 428-435. doi: 10.1088/0031-8949/54/5/002.

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show all references

References:
[1]

M. J. AblowitzD. J. KaupA. C. Newell and H. Segur, The inverse scattering transform - Fourier analysis for nonlinear problems, Studies in Applied Mathematics, 53 (1974), 249-315. doi: 10.1002/sapm1974534249.

[2]

H. Aden and B. Carl, On realizations of solutions of the KdV equation by determinants on operator ideals, J. Math. Phys., 37 (1996), 1833-1857. doi: 10.1063/1.531482.

[3]

C. Athorne and A. Fordy, Generalised KdV and MKdV equations associated with symmetric spaces, J. Phys. A: Math. Gen., 20 (1987), 1377-1386. doi: 10.1088/0305-4470/20/6/021.

[4]

P. Basarab-Horwath and F. Güngör, Linearizability for third order evolution equations, J. Math. Phys., 58 (2017), 081507, 13pp. doi: 10.1063/1.4997558.

[5]

F. Calogero and A. Degasperis, A modified modified Korteweg-de Vries equation, Inverse Problems, 1 (1985), 57-66. doi: 10.1088/0266-5611/1/1/006.

[6]

F. Calogero and S. De Lillo, The Burgers equation on the semiline with general boundary conditions at the origin, J. Math. Phys., 32 (1991), 99-105. doi: 10.1063/1.529101.

[7]

F. Calogero and A. Degasperis, Spectral Transform and Solitons I, Studies in Mathematics and its Application, Vol. 13, North Holland, Amsterdam, 1982. doi: 10.1007/978-3-642-82135-6_2.

[8]

S. Carillo, Nonlinear evolution equations: Bäcklund transformations and Bäcklund charts, Acta Applicandae Math., 122 (2012), 93-106. doi: 10.1007/s10440-012-9729-8.

[9]

S. Carillo, KdV-type Equations Linked via Bäcklund Transformations: Remarks and Perspectives, arXiv: 1702.06874, Applied Numerical Mathematics,(2018), in press. doi: 10.1016/j.apnum.2018.10.002.

[10]

S. Carillo and B. Fuchssteiner, The abundant symmetry structure of hierarchies of nonlinear equations obtained by reciprocal links, J. Math. Phys., 30 (1989), 1606-1613. doi: 10.1063/1.528604.

[11]

S. Carillo and C. Schiebold, A non-commutative operator-hierarchy of Burgers equations and Bäcklund transformations, in Applied and Industrial Mathematics in Italy III, (ed.s E. De Bernardis, R. Spigler, V. Valente), Ser. Adv. Math. Appl. Sci., Vol.82, World Scientific Publ., Singapore, (2010), 175-185. doi: 10.1142/9789814280303_0016.

[12]

S. Carillo and C. Schiebold, Non-commutative KdV and mKdV hierarchies via recursion methods, J. Math. Phys., 50 (2009), 073510, 14pp. doi: 10.1063/1.3155080.

[13]

S. Carillo and C. Schiebold, Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Non-commutative soliton solutions, J. Math. Phys., 52 (2011), 053507, 21pp. doi: 10.1063/1.3576185.

[14]

S. Carillo and C. Schiebold, On the recursion operator for the non-commutative Burgers hierarchy, J. Nonlin. Math. Phys., 19 (2012), 1250003, 11 pp. doi: 10.1142/S1402925112500039.

[15]

S. Carillo, M. Lo Schiavo and C. Schiebold, Bäcklund Transformations and Non Abelian Nonlinear Evolution Equations: a novel Bäcklund chart, SIGMA, 12 (2016), Paper No. 087, 17 pp. doi: 10.3842/SIGMA.2016.087.

[16]

S. CarilloM. Lo Schiavo and C. Schiebold, Recursion operators admitted by non-Abelian Burgers equations: Some remarks, Math. and Comp. in Simul., 147 (2018), 40-51. doi: 10.1016/j.matcom.2017.02.001.

[17]

S. Carillo, M. Lo Schiavo, E. Porten and C. Schiebold, A novel noncommutative KdV-type equation, its recursion operator, and solitons, J. Math. Phys., 59 (2018), 043501, 14 pp. doi: 10.1063/1.5027481.

[18]

B. Carl and C. Schiebold, Nonlinear equations in soliton physics and operator ideals, Nonlinearity, 12 (1999), 333-364. doi: 10.1088/0951-7715/12/2/012.

[19]

P. J. CaudreyR. K. Dodd and J. D. Gibbon, A new hierarchy of Korteweg-de Vries equations, Proc. Roy. Soc. London Ser. A, 351 (1976), 407-422. doi: 10.1098/rspa.1976.0149.

[20]

J. D. Cole, On a quasi-linear parabolic equation occuring in aerodynamics, Quart. App. Math., 9 (1951), 225-236. doi: 10.1090/qam/42889.

[21]

A. S. Fokas and B. Fuchssteiner, Bäcklund transformation for hereditary symmetries, Nonlin. Anal., Theory Methods Appl., 5 (1981), 423-432. doi: 10.1016/0362-546X(81)90025-0.

[22]

B. Fuchssteiner, Application of hereditary symmetries to nonlinear evolution equations, Nonlin. Anal., Th. Meth. Appl., 3 (1979), 849-862. doi: 10.1016/0362-546X(79)90052-X.

[23]

B. Fuchssteiner, The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems, Progr. Theor. Phys., 68 (1982), 1082-1104. doi: 10.1143/PTP.68.1082.

[24]

B. Fuchssteiner and S. Carillo, Soliton structure versus singularity analysis: Third order completely integrable nonlinear equations in 1+1 dimensions, Phys. A, 154 (1989), 467-510. doi: 10.1016/0378-4371(89)90260-4.

[25]

B. Fuchssteiner and S. Carillo, The action-angle transformation for soliton equations, Physica A, 166 (1990), 651-675. doi: 10.1016/0378-4371(90)90078-7.

[26]

B. FuchssteinerT. Schulze and S. Carillo, Explicit solutions for the Harry Dym equation, J. Phys. A, 25 (1992), 223-230. doi: 10.1088/0305-4470/25/1/025.

[27]

B. Fuchssteiner and W. Oevel, The bi-Hamiltonian structure of some nonlinear fifth- and seventh-order differential equations and recursion formulas for their symmetries and conserved covariants, J. Math. Phys, 23 (1982), 358-363. doi: 10.1063/1.525376.

[28]

C. Gu, H. Hu and Z. Zhou, Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry, Mathematical Physics Studies, Vol. 26, Springer, Dordrecht, 2005. doi: 10.1007/1-4020-3088-6.

[29]

B. Yu Guo and S. Carillo, Infiltration in soils with prescribed boundary concentration: A Burgers model, Acta Appl. Math. Sinica, 6 (1990), 365-369. doi: 10.1007/BF02015343.

[30]

B. Yu Guo and C. Rogers, On Harry-Dym equation and its solution, Science in China, 32 (1989), 283-295.

[31]

M. GürsesA. Karasu and V. V. Sokolov, On construction of recursion operators from Lax representation, J. Math. Phys., 40 (1999), 6473-6490. doi: 10.1063/1.533102.

[32]

M. GürsesA. Karasu and R. Turhan. On non-commutative integrable Burgers equations, On non-commutative integrable Burgers equations, J. Nonlinear Math. Phys., 17 (2010), 1-6. doi: 10.1142/S1402925110000532.

[33]

M. Hamanaka, Non-commutative solitons and quasi-determinants, Phys. Scr., 89 (2014), 038006. doi: 10.1088/0031-8949/89/03/038006.

[34]

M. Hamanaka and K. Toda, Noncommutative Burgers equation, J. Phys. A: Math. Gen., 36 (2003), 11981-11998. doi: 10.1088/0305-4470/36/48/006.

[35]

E. Hopf, The partial differential equation $ u_{t} + u u_{x} = \mu u_{xx}$, Comm. Pure Appl. Math., 3 (1950), 201-230. doi: 10.1002/cpa.3160030302.

[36]

S. Kawamoto, An exact transformation from the Harry Dym equation to the modified KdV equation, J. Phys. Soc. Japan, 54 (1985), 2055-2056. doi: 10.1143/JPSJ.54.2055.

[37]

F. A. Khalilov and E. Ya. Khruslov. Matrix generalisation of the modified Korteweg-de Vries equation, Matrix generalisation of the modified Korteweg-de Vries equation, Inv. Problems, 6 (1990), 193-204. doi: 10.1088/0266-5611/6/2/004.

[38]

B. G. Konopelchenko, Soliton eigenfunction equations: The IST integrability and some properties, Rev. Math. Phys., 2 (1990), 399-440. doi: 10.1142/S0129055X90000120.

[39]

B. A. Kupershmidt., On a group of automorphisms of the non-commutative Burgers hierarchy, J. Nonlinear Math. Phys., 12 (2005), 539-549. doi: 10.2991/jnmp.2005.12.4.8.

[40]

M. LeoR. A. LeoG. Soliani and L. Solombrino, On the isospectral-eigenvalue problem and the recursion operator of the Harry Dym equation, Lett. Nuovo Cimento, 38 (1983), 45-51. doi: 10.1007/BF02782775.

[41]

D. LeviO. Ragnisco and M. Bruschi., Continuous and discrete matrix Burgers' hierarchies, Il Nuovo Cimento, 74 (1983), 33-51. doi: 10.1007/BF02721683.

[42]

S. Y. Lou, Symmetries and similarity reductions of the Dym equation, Phys. Scripta, 54 (1996), 428-435. doi: 10.1088/0031-8949/54/5/002.

[43]

F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), 1156-1162. doi: 10.1063/1.523777.

[44]

V. A. Marchenko, Nonlinear Equations and Operator Algebras, Mathematics and its Applications (Soviet Series), Vol. 17, D. Reidel Publishing Co., Dordrecht, 1988. doi: 10.1007/978-94-009-2887-9.

[45]

A. G. Meshkov and V. V. Sokolov, Integrable evolution equations with a constant separant, Ufimsk. Mat. Zh., 4 (2012), 104-153.

[46]

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Figure 1.  Burgers and mirror Burgers equations and their Bäcklund links: the non-commutative case.
Figure 2.  Burgers and mirror Burgers hierarchies and their Bäcklund links: the non-commutative case.
Figure 3.  KdV-type Bäcklund chart: the Abelian case
Figure 4.  Abelian KdV-type hierarchies Bäcklund chart: induced invariances
Figure 5.  KdV-type hierarchies Bäcklund chart: the non-Abelian case
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