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Original Articles

Finite genus solutions for Geng hierarchy

Pages 54-65 | Received 25 Oct 2016, Accepted 18 Aug 2017, Published online: 19 Feb 2018

References

  • M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform (SIAM, Philadelphia, 1981).
  • Y. Matsuno, Bilinear Transformation Method (Academic Press, New York, 1984).
  • V. B. Matveev and M. A. Salle, Darboux transformations and solitons (Springer, Berlin, 1991).
  • C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory (Cambridge University Press, Cambridge, 2002).
  • E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its and V. B. Matveev, Algebro-geometric approach to nonlinear integrable equations (Springer, Berlin, 1994).
  • C. W. Cao, Y. T. Wu and X. G. Geng, Relation between the Kadomtsev-Petviashvili equation and the confocal involutive system, J. Math. Phys. 40 (1999) 3948–3970. doi: 10.1063/1.532936
  • M. L. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A, 199 (1995) 169–172. doi: 10.1016/0375-9601(95)00092-H
  • E. Date and S. Tanaka, Periodic multi-soliton solutions of Korteweg-de Vries equation and Toda lattice, Progr. Theoret. Phys. Suppl. 59 (1976) 107–125. doi: 10.1143/PTPS.59.107
  • H. H. Dong, T. T. Chen, L. F. Chen and Y. Zhang, A new integrable symplectic map and the lie point symmetry associated with nonlinear lattice equations, J. Nonlinear Sci. Appl. 9 (2016) 5107–5118. doi: 10.22436/jnsa.009.07.13
  • X. G. Geng and Y. T. Wu, Finite-band solutions of the classical Boussinesq-Burgers equations, J. Math. Phys. 40 (1999) 2971–2982. doi: 10.1063/1.532739
  • W. X. Ma and J. H. Lee, A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation, Chaos Solitons & Fractals 42 (2009) 1356–1363. doi: 10.1016/j.chaos.2009.03.043
  • X. G. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions, Adv. Math. 226 (2011) 827–839. doi: 10.1016/j.aim.2010.07.009
  • G. Bluman and S. Kumei, On the remarkable nonlinear diffusion equation (∂ /∂ x)[a(u +b)−2(∂ u/∂ x)] − (∂ u/∂ t) = 0, J. Math. Phys. 21 (1980) 1019–1023. doi: 10.1063/1.524550
  • H. Wilhelmsson, Explosive instabilities of reaction-diffusion equations, Phys. Rev. A 36 (1987) 965–966. doi: 10.1103/PhysRevA.36.965
  • O. P. Bhutani and K. Vijayakumar, On the isogroups of the generalised diffusion equation, Int. J. Eng. Sci. 28 (1990) 375–387. doi: 10.1016/0020-7225(90)90003-2
  • X. G. Geng, A new hierarchy of nonlinear evolution equations and corresponding finite-dimensional completely integrable systems, Phys. Lett. A 162 (1992) 375–380. doi: 10.1016/0375-9601(92)90057-S
  • F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Vol. I: (1+1)- Dimensional Continuous Models (Cambridge University Press, Cambridge, 2003).
  • X. G. Geng and B. Xue, Soliton solutions and quasiperiodic solutions of modified Korteweg-de Vries type equations, J. Math. Phys. 51 (2010) 063516. doi: 10.1063/1.3409345
  • X. G. Geng, L. H. Wu and G. L. He, Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions, Physica D 240 (2011) 1262–1288. doi: 10.1016/j.physd.2011.04.020
  • Y. Y. Zhai and X. G. Geng, Straightening out of the flows for the Hu hierarchy and its algebro-geometric solutions, J. Math. Anal. Appl. 397 (2013) 561–576. doi: 10.1016/j.jmaa.2012.08.023
  • X. G. Geng, X. Zeng and B. Xue, Algebro-geometric solutions of the TD Hierarchy, Math. Phys. Anal. Geom. 16 (2013) 229–251. doi: 10.1007/s11040-013-9129-y
  • G. L. He, X. G. Geng and L. H. Wu, Algebro-geometric quasi-periodic solutions to the three-wave resonant interaction hierarchy, SIAM J. Math. Anal. 46 (2014) 1348–1384. doi: 10.1137/130918794
  • F. Gesztesy, H. Holden, J. Michor and G. Teschl, Soliton Equations and Their Algebro-Geometric Solutions. Vol. II: (1+1)-Dimensional Discrete Models (Cambridge University Press, Cambridge, 2008).
  • M. Antonowicz and A. P. Fordy, Coupled Harry Dym equations with multi-Hamiltonian structures, J. Phys. A 21 (1988) L269–L275. doi: 10.1088/0305-4470/21/5/001
  • M. Antonowicz and A. P. Fordy, Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems, Comm. Math. Phys. 124 (1989) 465–486. doi: 10.1007/BF01219659
  • M.S. Alber, G.G. Luther and J.E. Marsden, Energy dependent Schrödinger operators and complex Hamiltonian systems on Riemann surfaces, Nonlinearity 10 (1997) 223–241. doi: 10.1088/0951-7715/10/1/015
  • R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Acta Math. Sinica 6 (1990) 35–41.
  • C. W. Cao, Stationary Harry-Dym’s equation and its relation with geodesics on ellipsoid, Phys. Rev. A 36 (1987) 965–966. doi: 10.1103/PhysRevA.36.965
  • M. S. Alber and Y. N. Fedorov, Wave solutions of evolution equations and Hamiltonian flows on non-linear subvarieties of generalized Jacobians, J. Phys. A 33 (2000) 8409–8425. doi: 10.1088/0305-4470/33/47/307
  • M. S. Alber and Y. N. Fedorov, Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians, Inverse Problems 17 (2001) 1017–1042. doi: 10.1088/0266-5611/17/4/329
  • S. Abenda and Y. Fedorov, On the weak Kowalevski-Painlevé property for hyperelliptically separable systems, Acta Appl. Math. 60 (2000) 137–178. doi: 10.1023/A:1006425609939
  • P. Vanhaecke, Integrable systems and symmetric products of curves, Math. Z. 227 (1998) 93–127. doi: 10.1007/PL00004370
  • P. Griffiths and J. Harris, Principles of Algebraic Geometry (Wiley, New York, 1994).
  • D. Mumford, Tata Lectures on Theta II (Birkhäuser, Boston, 1984).
  • B. A. Dubrovin, Theta functions and nonlinear equations, Russian Math. Surveys (36) (1981) 11–92. doi: 10.1070/RM1981v036n02ABEH002596

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