The Sylvester equation and integrable equations. I: The Korteweg-de Vries system and sine-Gordon equation. (English) Zbl 1420.35299
Summary: The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation \(KM + MK = r s^T\) we introduce a scalar function \(S^{(i, j)} = s^TK^j(I + M)^{-1}K^ir\) which is defined as same as in discrete case. \(S^{(i, j)}\) satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on \(r\) and \(s\), we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of \(S^{(i, j)}\) defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property \(S^{(i, j)} = S^{(j,i)}\). The solution \(M\) provides \(t\) function by \(t = \mid I + M\mid\). We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.
MSC:
| 35Q51 | Soliton equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
| 39A99 | Difference equations |
References:
| [1] | Aden, H.; Carl, B., On realizations of solutions of the KdV equation by determinants on operator ideals, J. Math. Phys, 37, 1833-1857 (1996) · Zbl 0863.35094 |
| [2] | Aktosun, T.; Demontis, F.; Van Der Mee, C., Exact solutions to the focusing nonlinear Schrödinger equation, Inverse Problems, 23, 2171-2195 (2007) · Zbl 1126.35058 |
| [3] | Aktosun, T.; Demontis, F.; Van Der Mee, C., Exact solutions to the sine-Gordon equation, J. Math. Phys, 51, 27 · Zbl 1314.35158 |
| [4] | Aktosun, T.; Klaus, M.; Pike, E. R.; Sabatier, P. C., Scattering, Inverse theory: problem on the line, 770-785 (2001), Academic: Academic, London |
| [5] | Aktosun, T.; Van Der Mee, C., Explicit solutions to the Korteweg-de Vries equation on the half line, Inverse Problems, 22, 2165-2174 (2006) · Zbl 1105.35099 |
| [6] | Bergvelt, M.; Gekhtman, M.; Kasman, A., Spin Calogero particles and bispectral solutions of the matrix KP hierarchy, Math. Phys. Anal. Geome, 12, 181-200 (2009) · Zbl 1186.37072 |
| [7] | Bhatia, R.; Rosenthal, P., How and why to solve the operator equation AX - X B = Y, Bull. London Math. Soc, 29, 1-21 (1997) · Zbl 0909.47011 |
| [8] | Blohm, H., Solution of nonlinear equations by trace methods, Nonlinearity, 13, 1925-1964 (2000) · Zbl 0972.35132 |
| [9] | Carl, B.; Schiebold, C., Nonlinear equations in soliton physics and operator ideals, Nonlinearity, 12, 333-364 (1999) · Zbl 0940.35175 |
| [10] | Carl, B.; Schiebold, C., A direct approach to the study of soliton equations, J. Deutsch. Math.-Verein, 102, 102-148 · Zbl 0970.35121 |
| [11] | Carillo, S.; Schiebold, C., Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods, J. Math. Phys, 50, 14 (2009) · Zbl 1256.37036 |
| [12] | Carillo, S.; Schiebold, C., Matrix KdV and mKdV hierarchies: Noncommutative soliton solutions and explicit formulae, J. Math. Phys, 52, 21 (2011) · Zbl 1317.35220 |
| [13] | Carillo, S.; Schiebold, C., On the recursion operator for the noncommutative Burgers hierarchy, J. Nonlin. Math. Phys, 19, 11 (2012) · Zbl 1279.37043 |
| [14] | Chen, D. Y.; Zhang, D. J.; Deng, S. F., The novel multi-soliton solutions of the mKdV-SineGordon equations, J. Phys. Soc. Jpn, 71, 658-659 (2002) · Zbl 1080.35529 |
| [15] | Chen, D. Y.; Zhang, D. J.; Deng, S. F., Remarks on some solutions of soliton equations, J. Phys. Soc. Jpn, 71, 2072-2073 (2002) |
| [16] | Coddington, E. A.; Levinson, N., Theory of ordinary differential equations (1955), McGraw-Hill: McGraw-Hill, New York · Zbl 0064.33002 |
| [17] | Demontis, F., Exact solutions of the modified Korteweg-de Vries equation, Theore. Math. Phys, 168, 886-897 (2011) |
| [18] | Deift, P.; Trubowitz, E., Inverse scattering on the line, Commun. Pure Appl. Math, 32, 121-251 (1979) · Zbl 0388.34005 |
| [19] | Dimakis, A.; Müller-Hoissen, F., Bidifferential graded algebras and integrable systems, Discr. Cont. Dyn. Syst. Suppl, 2009, 208-219 (2009) · Zbl 1184.37052 |
| [20] | Dimakis, A.; Müller-Hoissen, F., Solutions of matrix NLS systems and their discretizations: a unified treatment, Inverse Problems, 26, 55 (2010) · Zbl 1205.35291 |
| [21] | Dimakis, A.; Müller-Hoissen, F., Bidifferential calculus approach to AKNS hierarchies and their solutions, SIGMA, 6, 27 (2010) · Zbl 1218.37076 |
| [22] | Fokas, A. S.; Ablowitz, M. J., Linearization of the Korteweg-de Vries and Painlevé II equations, Phys. Rev. Lett, 47, 1096-1100 (1981) |
| [23] | Gekhtman, M.; Kasman, A., Integrable systems and rank one conditions for rectangular matrices, Theore. Math. Phys, 133, 1498-1503 (2002) |
| [24] | Gekhtman, M.; Kasman, A., On KP generators and the geometry of the HBDE, J. Geome. Phys, 56, 282-309 (2006) · Zbl 1080.37072 |
| [25] | Han, W. T.; Li, Y. S., Remarks on the solutions of the Kadomtsev-Petviashvili equational, Phys. Lett. A, 283, 185-194 (2001) · Zbl 0981.35063 |
| [26] | Jaworski, M., Breather-like solutions of the Korteweg-de Vries equation, Phys. Lett. A, 104, 245-247 (1984) |
| [27] | Jennings, P.; Nijhoff, F. W., On an elliptic extension of the Kadomtsev-Petviashvili equation, J. Phys. A: Math. Theor, 47, 14 (2014) · Zbl 1300.37044 |
| [28] | Kasman, A.; Gekhtman, M., Solitons and almost-intertwining matrices, J. Math. Phys, 42, 3540-3553 (2001) · Zbl 1005.37036 |
| [29] | Ma, W. X., Complexiton solution to the KdV equation, Phys. Lett. A, 301, 35-44 (2002) · Zbl 0997.35066 |
| [30] | Marchenko, V. A., Nonlinear Equations and Operator Algebras (1988), D. Reidel: D. Reidel, Dordrecht · Zbl 0644.47053 |
| [31] | Matveev, V. B., Generalized Wronskian formula for solutions of the KdV equations: first applications, Phys. Lett. A, 166, 205-208 (1992) |
| [32] | Matveev, V. B., Positon-positon and soliton-positon collisions: KdV case, Phys. Lett. A, 166, 209-212 (1992) |
| [33] | Miwa, T.; Jimbo, M.; Date, E., Solitons: Differential equations, symmetries and infinite dimensional algebras (2000), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 0986.37068 |
| [34] | Nijhoff, F. W., Discrete Systems and Integrability (2004) |
| [35] | Nijhoff, F. W., A higher-rank version of the Q3 equation, preprint (2011) |
| [36] | Nijhoff, F. W., private communication (2013) |
| [37] | Nijhoff, F. W.; Atkinson, J., Elliptic N-soliton solutions of ABS lattice equations, Inter. Math. Res. Not, 2010, 3837-3895 (2010) · Zbl 1217.37068 |
| [38] | Nijhoff, F. W.; Atkinson, J.; Hietarinta, J., Soliton solutions for ABS lattice equations: I: Cauchy matrix approach, J. Phys. A: Math. Theor, 42, 34 (2009) · Zbl 1184.35281 |
| [39] | Nijhoff, F. W.; Capel, H., The discrete Korteweg-de Vries equation, Acta Appl. Math, 39, 133-158 (1995) · Zbl 0841.58034 |
| [40] | Nijhoff, F. W.; Puttock, S. E., On a two-parameter extension of the lattice KdV system associated with an elliptic curve, J. Nonlin. Math. Phys, 10, 107-123 (2003) · Zbl 1362.35271 |
| [41] | Nijhoff, F. W.; Quispel, G. R.W.; Capel, H. W., Direct linearization of non-linear difference-difference equations, Phys. Lett. A, 97, 125-128 (1983) |
| [42] | Nijhoff, F. W.; Quispel, G. R.W.; Ven Der Linden, J.; Capel, H. W., On some linear integral-equations generating solutions of non-linear partial-differential equations, Physica A, 119, 101-142 (1983) · Zbl 0511.35074 |
| [43] | Nijhoff, F. W.; Zhang, D. J.; Zhao, S. L., The Sylvester equation and matrix discrete systems, preprint (2013) |
| [44] | Ohta, Y.; Satsuma, J.; Takahashi, D.; Tokihiro, T., An elementary introduction to Sato theory, Prog. Theor. Phys. Suppl, 94, 210-241 (1988) |
| [45] | Schiebold, C., An operator-theoretic approach to the Toda lattice equation, Physica D, 122, 37-61 (1998) · Zbl 0977.37042 |
| [46] | Schiebold, C., Noncommutative AKNS systems and multisoliton solutions to the matrix sine-Gordon equation, Discre. Continu. Dyn. Syst. Suppl, 2009, 678-90 (2009) · Zbl 1187.35197 |
| [47] | Schiebold, C., Cauchy-type determinants and integrable systems, Linear Algebra Appl, 433, 447-75 (2010) · Zbl 1195.15006 |
| [48] | Schiebold, C., Structural properties of the noncommutative KdV recursion operator, J. Math. Phys, 52, 16 (2011) · Zbl 1272.35170 |
| [49] | Sylvester, J., Sur l’equation en matrices px = xq, C. R. Acad. Sci. Paris, 99, 115-116 67-71 (1884) · JFM 16.0108.03 |
| [50] | Wadati, M.; Ohkuma, K., Multiple-pole solutions of the modified Korteweg de Vries equation, J. Phys. Soc. Jpn, 51, 2029-2035 (1982) |
| [51] | Wilson, G., Collisions of Calogero-Moser particals and adelic Grassmannian, Invent. Math, 133, 1-41 (1998) · Zbl 0906.35089 |
| [52] | Yoo-Kong, S.; Nijhoff, F. W., Elliptic (N, N’)-soliton solutions of the lattice Kadomtsev-Petviashvili equation, J. Math. Phys, 54, 20 · Zbl 1417.37241 |
| [53] | Zhang, D. J., Notes on solutions in Wronskian form to soliton equations: KdV-type, preprint (2006) |
| [54] | Zhang, D. J., The Sylvester equation, Cauchy matrices and matrix discrete integrable systems |
| [55] | Zhang, D. J.; Zhang, J. B.; Shen, Q., A limit symmetry of the KdV equation and its applications, Theore. Math. Phys, 163, 634-643 (2010) · Zbl 1252.35244 |
| [56] | Zhang, D. J.; Zhao, S. L., Solutions to the ABS lattice equations via generalized Cauchy matrix approach, Stud. Appl. Math, 131, 72-103 (2013) · Zbl 1338.37113 |
| [57] | Zhang, D. J.; Zhao, S. L.; Nijhoff, F. W., Direct linearization of an extended lattice BSQ system, Stud. Appl. Math, 129, 220-248 (2012) · Zbl 1256.35188 |
| [58] | Zhao, S. L.; Zhang, D. J., The Sylvester equation and integrable equations: II, in preparation (2014) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
