The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

References

• 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.
   Web of Science ®Google Scholar
• T. Aktosun, F. Demontis and C. van der Mee, Exact solutions to the focusing nonlinear Schrödinger equation, Inverse Problems 23 (2007) 2171–2195.
   Google Scholar
• T. Aktosun, F. Demontis and C. van der Mee, Exact solutions to the sine-Gordon equation, J. Math. Phys 51 Article ID: 123521, 27pp.
   Google Scholar
• T. Aktosun and M. Klaus, Inverse theory: problem on the line, in Scattering, eds. E.R. Pike and P.C. Sabatier, (Academic, London, 2001), pp.770–785.
   Google Scholar
• T. Aktosun and C. van der Mee, Explicit solutions to the Korteweg-de Vries equation on the half line, Inverse Problems 22 (2006) 2165–2174.
   Web of Science ®Google Scholar
• M. Bergvelt, M. Gekhtman and A. Kasman, Spin Calogero particles and bispectral solutions of the matrix KP hierarchy, Math. Phys. Anal. Geome 12 (2009) 181–200.
   Google Scholar
• R. Bhatia and P. Rosenthal, How and why to solve the operator equation AX - X B = Y, Bull. London Math. Soc 29 (1997) 1–21.
   Google Scholar
• H. Blohm, Solution of nonlinear equations by trace methods, Nonlinearity 13 (2000) 1925–1964.
   Google Scholar
• B. Carl and C. Schiebold, Nonlinear equations in soliton physics and operator ideals, Nonlinearity 12 (1999) 333–364.
   Web of Science ®Google Scholar
• B. Carl and C. Schiebold, A direct approach to the study of soliton equations, J. Deutsch. Math.-Verein 102 102–148. English version is avilable on http://apachepersonal.miun.se/corsch/.
   Google Scholar
• S. Carillo and C. Schiebold, Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods, J. Math. Phys 50 (2000) (2009), Article ID: 073510, 14pp.
   Google Scholar
• S. Carillo and C. Schiebold, Matrix KdV and mKdV hierarchies: Noncommutative soliton solutions and explicit formulae, J. Math. Phys 52 (2011) Article ID: 053507, 21pp.
   Google Scholar
• S. Carillo and C. Schiebold, On the recursion operator for the noncommutative Burgers hierarchy, J. Nonlin. Math. Phys 19 (2012), Article ID: 1250003, 11pp.
   Google Scholar
• D.Y. Chen, D.J. Zhang and S.F. Deng, The novel multi-soliton solutions of the mKdV-SineGordon equations, J. Phys. Soc. Jpn 71 (2002) 658–659.
   Google Scholar
• D.Y. Chen, D.J. Zhang and S.F. Deng, Remarks on some solutions of soliton equations, J. Phys. Soc. Jpn 71 (2002) 2072–2073.
   Google Scholar
• E.A. Coddington and N. Levinson, Theory of ordinary differential equations (McGraw-Hill, New York, 1955)
   Google Scholar
• F. Demontis, Exact solutions of the modified Korteweg-de Vries equation, Theore. Math. Phys 168 (2011) 886–897.
   Google Scholar
• P. Deift and E. Trubowitz, Inverse scattering on the line, Commun. Pure Appl. Math 32 (1979) 121–251.
   Web of Science ®Google Scholar
• A. Dimakis and F. Müller-Hoissen, Bidifferential graded algebras and integrable systems, Discr. Cont. Dyn. Syst. Suppl 2009 (2009) 208–219.
   Google Scholar
• A. Dimakis and F. Müller-Hoissen, Solutions of matrix NLS systems and their discretizations: a unified treatment, Inverse Problems 26 (2010), Article ID: 095007, 55pp.
   Google Scholar
• A. Dimakis and F. Müller-Hoissen, Bidifferential calculus approach to AKNS hierarchies and their solutions, SIGMA 6 (2010) 2010055 (27pp).
   Google Scholar
• A.S. Fokas and M.J. Ablowitz, Linearization of the Korteweg-de Vries and Painlevé II equations, Phys. Rev. Lett 47 (1981) 1096–1100.
   Google Scholar
• M. Gekhtman and A. Kasman, Integrable systems and rank one conditions for rectangular matrices, Theore. Math. Phys 133 (2002) 1498–1503.
   Google Scholar
• M. Gekhtman and A. Kasman, On KP generators and the geometry of the HBDE, J. Geome. Phys 56 (2006) 282–309.
   Google Scholar
• W.T. Han and Y.S. Li, Remarks on the solutions of the Kadomtsev-Petviashvili equational, Phys. Lett. A 283 (2001) 185–194.
   Web of Science ®Google Scholar
• M. Jaworski, Breather-like solutions of the Korteweg-de Vries equation, Phys. Lett. A 104 (1984) 245–247.
   Google Scholar
• P. Jennings and F.W. Nijhoff, On an elliptic extension of the Kadomtsev-Petviashvili equation, J. Phys. A: Math. Theor 47 (2014), Article ID: 055205, 14pp.
   Google Scholar
• A. Kasman and M. Gekhtman, Solitons and almost-intertwining matrices, J. Math. Phys 42 (2001) 3540–3553.
   Google Scholar
• W.X. Ma, Complexiton solution to the KdV equation, Phys. Lett. A 301 (2002) 35–44.
   Web of Science ®Google Scholar
• V.A. Marchenko, Nonlinear Equations and Operator Algebras(D. Reidel, Dordrecht, 1988)
   Google Scholar
• V.B. Matveev, Generalized Wronskian formula for solutions of the KdV equations: first applications, Phys. Lett. A 166 (1992) 205–208.
   Web of Science ®Google Scholar
• V.B. Matveev, Positon-positon and soliton-positon collisions: KdV case, Phys. Lett. A 166 (1992) 209–212.
   Web of Science ®Google Scholar
• T. Miwa, M. Jimbo and E. Date, Solitons: Differential equations, symmetries and infinite dimensional algebras, (Cambridge University Press, Cambridge, 2000)
   Google Scholar
• F.W. Nijhoff, Discrete Systems and Integrability, Math4490, University of Leeds, preprint (2004)
   Google Scholar
• F.W. Nijhoff, A higher-rank version of the Q3 equation, preprint (2011) arXiv:1104.1166.
   Google Scholar
• F.W. Nijhoff, private communication (2013)
   Google Scholar
• F.W. Nijhoff and J. Atkinson, Elliptic N-soliton solutions of ABS lattice equations, Inter. Math. Res. Not 2010(2010), No.20, 3837–3895.
   Google Scholar
• F.W. Nijhoff, J. Atkinson and J. Hietarinta, Soliton solutions for ABS lattice equations: I: Cauchy matrix approach, J. Phys. A: Math. Theor 42 (2009) Article ID: 404005, 34pp.
   Google Scholar
• F.W. Nijhoff and H. Capel, The discrete Korteweg-de Vries equation, Acta Appl. Math 39 (1995) 133–158.
   Web of Science ®Google Scholar
• F.W. Nijhoff and S.E. Puttock, On a two-parameter extension of the lattice KdV system associated with an elliptic curve, J. Nonlin. Math. Phys 10 (Suppl.1) (2003) 107–123.
   Web of Science ®Google Scholar
• F.W. Nijhoff, G.R.W. Quispel and H.W. Capel,, Direct linearization of non-linear difference-difference equations, Phys. Lett. A 97 (1983) 125–128.
   Google Scholar
• F.W. Nijhoff, G.R.W. Quispel, J. Ven der Linden and H.W. Capel, On some linear integral-equations generating solutions of non-linear partial-differential equations, Physica A 119 (1983) 101–142.
   Google Scholar
• F.W. Nijhoff, D.J. Zhang and S.L. Zhao, The Sylvester equation and matrix discrete systems, preprint (2013)
   Google Scholar
• Y. Ohta, J. Satsuma, D. Takahashi and T. Tokihiro, An elementary introduction to Sato theory, Prog. Theor. Phys. Suppl 94 (1988) 210–241.
   Google Scholar
• C. Schiebold, An operator-theoretic approach to the Toda lattice equation, Physica D 122 (1998) 37–61.
   Google Scholar
• C. Schiebold, Noncommutative AKNS systems and multisoliton solutions to the matrix sine-Gordon equation, Discre. Continu. Dyn. Syst. Suppl 2009 (2009) 678–90.
   Google Scholar
• C. Schiebold, Cauchy-type determinants and integrable systems, Linear Algebra Appl 433 (2010) 447–75.
   Google Scholar
• C. Schiebold, Structural properties of the noncommutative KdV recursion operator, J. Math. Phys 52 (2011), Article ID: 113504, 16pp.
   Google Scholar
• J. Sylvester, Sur l'equation en matrices px = xq, C. R. Acad. Sci. Paris 99 (1884) 115–116 67–71.
   Google Scholar
• M. Wadati and K. Ohkuma, Multiple-pole solutions of the modified Korteweg de Vries equation, J. Phys. Soc. Jpn 51 (1982) 2029–2035.
   Google Scholar
• G. Wilson, Collisions of Calogero-Moser particals and adelic Grassmannian, Invent. Math 133 (1998) 1–41.
   Web of Science ®Google Scholar
• S. Yoo-Kong and F.W. Nijhoff, Elliptic (N, N')-soliton solutions of the lattice Kadomtsev-Petviashvili equation, J. Math. Phys 54 Article ID: 043511, 20pp.
   Google Scholar
• D.J. Zhang, Notes on solutions in Wronskian form to soliton equations: KdV-type, preprint (2006), arXiv:nlin.SI/0603008.
   Google Scholar
• D.J. Zhang, The Sylvester equation, Cauchy matrices and matrix discrete integrable systems, collaborated with F.W. Nijhoff and S.L. Zhao, web: http://www.newton.ac.uk/programmes/DIS/disw05p.html.
   Google Scholar
• D.J. Zhang, J.B. Zhang and Q. Shen, A limit symmetry of the KdV equation and its applications, Theore. Math. Phys 163 (2010) 634–643.
   Google Scholar
• D.J. Zhang and S.L. Zhao, Solutions to the ABS lattice equations via generalized Cauchy matrix approach, Stud. Appl. Math 131 (2013) 72–103.
   Google Scholar
• D.J. Zhang, S.L. Zhao and F.W. Nijhoff, Direct linearization of an extended lattice BSQ system, Stud. Appl. Math 129 (2012) 220–248.
   Google Scholar
• S.L. Zhao and D.J. Zhang, The Sylvester equation and integrable equations: II, in preparation (2014)
   Google Scholar