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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[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. Google Scholar

[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. Google Scholar

[33]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

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

[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. Google Scholar

[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. Google Scholar

[41]

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

[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. Google Scholar

[43]

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

[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. Google Scholar

[45]

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

[46]

A. V. MikhailovA. B. Shabat and R. I. Yamilov, The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems, Russian Math. Surveys, 42 (1987), 3-53. doi: 10.1070/RM1987v042n04ABEH001441. Google Scholar

[47]

A. V. MikhailovV. S. Novikov and J. P. Wang, On classification of integrable nonevolutionary equations, Stud. in Appl. Math., 118 (2007), 419-457. doi: 10.1111/j.1467-9590.2007.00376.x. Google Scholar

[48]

R. M. Miura, Korteweg-de Vries equation and generalizations. Ⅰ. A remarkable explicit non-linear transformation, J. Math. Phys., 9 (1968), 1202-1204. doi: 10.1063/1.1664700. Google Scholar

[49]

R. M. MiuraC. S. Gardner and M. D. Kruskal, Korteweg-de Vries equation and generalizations. Ⅱ. Existence of conservation laws and constants of motion, Journal of Math. Phys., 9 (1968), 1204-1209. doi: 10.1063/1.1664701. Google Scholar

[50]

W. Oevel and S. Carillo, Squared eigenfunction symmetries for soliton equations: Part Ⅰ, J. Math. Anal. and Appl., 217 (1998), 161-178. doi: 10.1006/jmaa.1997.5707. Google Scholar

[51]

W. Oevel and S. Carillo, Squared eigenfunction symmetries for soliton equations: Part Ⅱ, J. Math. Anal. and Appl., 217 (1998), 179-199. doi: 10.1006/jmaa.1997.5708. Google Scholar

[52]

P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys., 18 (1977), 1212-1215. doi: 10.1063/1.523393. Google Scholar

[53]

P. J. Olver and V. V. Sokolov. Integrable evolution equations on associative algebras, Integrable evolution equations on associative algebras, Comm. Math. Phys., 193 (1998), 245-268. doi: 10.1007/s002200050328. Google Scholar

[54]

C. Rogers, The Harry Dym equation in 2+1-dimensions: A reciprocal link with the Kadomtsev-Petviashvili equation, Phys. Lett., 120 (1987), 15-18. doi: 10.1016/0375-9601(87)90256-8. Google Scholar

[55]

C. Rogers and W. F. Ames, Nonlinear Boundary Value Problems in Science and Engineering, Academic Press, Boston, 1989. Google Scholar

[56]

C. Rogers and S. Carillo, On reciprocal properties of the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt hierarchies, Phys. Scripta, 36 (1987), 865-869. doi: 10.1088/0031-8949/36/6/001. Google Scholar

[57]

C. Rogers and M. C. Nucci, On reciprocal Bäcklund transformations and the Korteweg- de Vries hierarchy, Phys. Scr., 33 (1986), 289-292. doi: 10.1088/0031-8949/33/4/001. Google Scholar

[58]

C. Rogers and W. F. Shadwick. Bäcklund Transformations and Their Applications, Mathematics in Science and Engineering, Vol. 161, Academic Press, Inc., New York-London, 1982. Google Scholar

[59]

C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511606359. Google Scholar

[60]

C. Rogers and P. Wong, On reciprocal Bäcklund transformations of inverse scattering schemes, Phys. Scripta, 30 (1984), 10-14. doi: 10.1088/0031-8949/30/1/003. Google Scholar

[61]

A. K. Sawada and A. T. Kotera, A method for finding N-soliton solutions of the KdV and KdV-like equation, J. Progr. Theor. Phys., 51 (1974), 1355-1367. doi: 10.1143/PTP.51.1355. Google Scholar

[62]

C. Schiebold, Cauchy-type determinants and integrable systems, Linear Algebra and its Applications, 433 (2010), 447-475. doi: 10.1016/j.laa.2010.03.011. Google Scholar

[63]

C. Schiebold, Noncommutative AKNS systems and multisoliton solutions to the matrix sine-Gordon equation, Discr. Cont. Dyn. Systems Suppl., 2009 (2009), 678-690. doi: 10.3934/proc.2009.2009.678. Google Scholar

[64]

C. Schiebold, The noncommutative AKNS system: Projection to matrix systems, countable superposition and soliton-like solutions, J. Phys. A, 43 (2010), 434030, 18pp. doi: 10.1088/1751-8113/43/43/434030. Google Scholar

[65]

C. Schiebold, Structural properties of the noncommutative KdV recursion operator, J. Math. Phys., 52 (2011), 113504, 16pp. doi: 10.1063/1.3656271. Google Scholar

[66]

S. I. Svinolupov and V. V. Sokolov, Evolution equations with nontrivial conservation laws, Funct Anal Its Appl, 16 (1982), 317-319. doi: 10.1007/BF01077866. Google Scholar

[67]

J. P. Wang, A list of 1 + 1 dimensional integrable equations and their properties, J. Nonlinear Math. Phys., 9 (2002), 213-233. doi: 10.2991/jnmp.2002.9.s1.18. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[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. Google Scholar

[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. Google Scholar

[33]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

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

[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. Google Scholar

[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. Google Scholar

[41]

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

[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. Google Scholar

[43]

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

[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. Google Scholar

[45]

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

[46]

A. V. MikhailovA. B. Shabat and R. I. Yamilov, The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems, Russian Math. Surveys, 42 (1987), 3-53. doi: 10.1070/RM1987v042n04ABEH001441. Google Scholar

[47]

A. V. MikhailovV. S. Novikov and J. P. Wang, On classification of integrable nonevolutionary equations, Stud. in Appl. Math., 118 (2007), 419-457. doi: 10.1111/j.1467-9590.2007.00376.x. Google Scholar

[48]

R. M. Miura, Korteweg-de Vries equation and generalizations. Ⅰ. A remarkable explicit non-linear transformation, J. Math. Phys., 9 (1968), 1202-1204. doi: 10.1063/1.1664700. Google Scholar

[49]

R. M. MiuraC. S. Gardner and M. D. Kruskal, Korteweg-de Vries equation and generalizations. Ⅱ. Existence of conservation laws and constants of motion, Journal of Math. Phys., 9 (1968), 1204-1209. doi: 10.1063/1.1664701. Google Scholar

[50]

W. Oevel and S. Carillo, Squared eigenfunction symmetries for soliton equations: Part Ⅰ, J. Math. Anal. and Appl., 217 (1998), 161-178. doi: 10.1006/jmaa.1997.5707. Google Scholar

[51]

W. Oevel and S. Carillo, Squared eigenfunction symmetries for soliton equations: Part Ⅱ, J. Math. Anal. and Appl., 217 (1998), 179-199. doi: 10.1006/jmaa.1997.5708. Google Scholar

[52]

P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys., 18 (1977), 1212-1215. doi: 10.1063/1.523393. Google Scholar

[53]

P. J. Olver and V. V. Sokolov. Integrable evolution equations on associative algebras, Integrable evolution equations on associative algebras, Comm. Math. Phys., 193 (1998), 245-268. doi: 10.1007/s002200050328. Google Scholar

[54]

C. Rogers, The Harry Dym equation in 2+1-dimensions: A reciprocal link with the Kadomtsev-Petviashvili equation, Phys. Lett., 120 (1987), 15-18. doi: 10.1016/0375-9601(87)90256-8. Google Scholar

[55]

C. Rogers and W. F. Ames, Nonlinear Boundary Value Problems in Science and Engineering, Academic Press, Boston, 1989. Google Scholar

[56]

C. Rogers and S. Carillo, On reciprocal properties of the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt hierarchies, Phys. Scripta, 36 (1987), 865-869. doi: 10.1088/0031-8949/36/6/001. Google Scholar

[57]

C. Rogers and M. C. Nucci, On reciprocal Bäcklund transformations and the Korteweg- de Vries hierarchy, Phys. Scr., 33 (1986), 289-292. doi: 10.1088/0031-8949/33/4/001. Google Scholar

[58]

C. Rogers and W. F. Shadwick. Bäcklund Transformations and Their Applications, Mathematics in Science and Engineering, Vol. 161, Academic Press, Inc., New York-London, 1982. Google Scholar

[59]

C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511606359. Google Scholar

[60]

C. Rogers and P. Wong, On reciprocal Bäcklund transformations of inverse scattering schemes, Phys. Scripta, 30 (1984), 10-14. doi: 10.1088/0031-8949/30/1/003. Google Scholar

[61]

A. K. Sawada and A. T. Kotera, A method for finding N-soliton solutions of the KdV and KdV-like equation, J. Progr. Theor. Phys., 51 (1974), 1355-1367. doi: 10.1143/PTP.51.1355. Google Scholar

[62]

C. Schiebold, Cauchy-type determinants and integrable systems, Linear Algebra and its Applications, 433 (2010), 447-475. doi: 10.1016/j.laa.2010.03.011. Google Scholar

[63]

C. Schiebold, Noncommutative AKNS systems and multisoliton solutions to the matrix sine-Gordon equation, Discr. Cont. Dyn. Systems Suppl., 2009 (2009), 678-690. doi: 10.3934/proc.2009.2009.678. Google Scholar

[64]

C. Schiebold, The noncommutative AKNS system: Projection to matrix systems, countable superposition and soliton-like solutions, J. Phys. A, 43 (2010), 434030, 18pp. doi: 10.1088/1751-8113/43/43/434030. Google Scholar

[65]

C. Schiebold, Structural properties of the noncommutative KdV recursion operator, J. Math. Phys., 52 (2011), 113504, 16pp. doi: 10.1063/1.3656271. Google Scholar

[66]

S. I. Svinolupov and V. V. Sokolov, Evolution equations with nontrivial conservation laws, Funct Anal Its Appl, 16 (1982), 317-319. doi: 10.1007/BF01077866. Google Scholar

[67]

J. P. Wang, A list of 1 + 1 dimensional integrable equations and their properties, J. Nonlinear Math. Phys., 9 (2002), 213-233. doi: 10.2991/jnmp.2002.9.s1.18. Google Scholar

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
[1]

M. Euler, N. Euler, M. C. Nucci. On nonlocal symmetries generated by recursion operators: Second-order evolution equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4239-4247. doi: 10.3934/dcds.2017181

[2]

Sara Cuenda, Niurka R. Quintero, Angel Sánchez. Sine-Gordon wobbles through Bäcklund transformations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1047-1056. doi: 10.3934/dcdss.2011.4.1047

[3]

Hermann Brunner, Stefano Maset. Time transformations for delay differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751

[4]

Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23

[5]

Masahiro Kubo. Quasi-subdifferential operators and evolution equations. Conference Publications, 2013, 2013 (special) : 447-456. doi: 10.3934/proc.2013.2013.447

[6]

Svetlana Katok, Ilie Ugarcovici. Theory of $(a,b)$-continued fraction transformations and applications. Electronic Research Announcements, 2010, 17: 20-33. doi: 10.3934/era.2010.17.20

[7]

Svetlana Katok, Ilie Ugarcovici. Structure of attractors for $(a,b)$-continued fraction transformations. Journal of Modern Dynamics, 2010, 4 (4) : 637-691. doi: 10.3934/jmd.2010.4.637

[8]

Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure & Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039

[9]

Goro Akagi. Doubly nonlinear evolution equations and Bean's critical-state model for type-II superconductivity. Conference Publications, 2005, 2005 (Special) : 30-39. doi: 10.3934/proc.2005.2005.30

[10]

Jin Liang, James H. Liu, Ti-Jun Xiao. Condensing operators and periodic solutions of infinite delay impulsive evolution equations. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 475-485. doi: 10.3934/dcdss.2017023

[11]

A. V. Bobylev, Vladimir Dorodnitsyn. Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 35-57. doi: 10.3934/dcds.2009.24.35

[12]

Risei Kano, Yusuke Murase. Solvability of nonlinear evolution equations generated by subdifferentials and perturbations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 75-93. doi: 10.3934/dcdss.2014.7.75

[13]

Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331

[14]

Akisato Kubo. Nonlinear evolution equations associated with mathematical models. Conference Publications, 2011, 2011 (Special) : 881-890. doi: 10.3934/proc.2011.2011.881

[15]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353

[16]

Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442

[17]

Aleksander Ćwiszewski, Piotr Kokocki. Krasnosel'skii type formula and translation along trajectories method for evolution equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 605-628. doi: 10.3934/dcds.2008.22.605

[18]

Tong Tang, Hongjun Gao. On the compressible Navier-Stokes-Korteweg equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2745-2766. doi: 10.3934/dcdsb.2016071

[19]

Irena Pawłow, Wojciech M. Zajączkowski. The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 859-880. doi: 10.3934/cpaa.2014.13.859

[20]

Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253

2018 Impact Factor: 1.048

Metrics

  • PDF downloads (31)
  • HTML views (392)
  • Cited by (0)

[Back to Top]