×

Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour. (English) Zbl 1436.37080

Summary: The first main aim of this article is to derive an explicit solution formula for the scalar two-dimensional Toda lattice depending on three independent operator parameters, ameliorating work in [the second author, Glasg. Math. J. 47A, 177–189 (2005; Zbl 1076.37068)]. This is achieved by studying a noncommutative version of the 2d-Toda lattice, generalizing its soliton solution to the noncommutative setting.
The purpose of the applications part is to show that the family of solutions obtained from matrix data exhibits a rich variety of asymptotic behaviour. The first indicator is that web structures, studied extensively in the literature, see [G. Biondini and D. Wang, J. Phys. A, Math. Theor. 43, No. 43, Article ID 434007, 20 p. (2010; Zbl 1209.37074)] and references therein, are a subfamily. Then three further classes of solutions (with increasingly unusual behaviour) are constructed, and their asymptotics are derived.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81Q80 Special quantum systems, such as solvable systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aden, H.. Elementary Operators and Solutions of the Korteweg-de Vries Equation. Thesis, Jena 1996. · Zbl 0873.35083
[2] 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 · doi:10.1063/1.531482
[3] Biondini, G.; Chakravarty, S., Soliton solutions of the Kadomtsev-Petviashvili II equation, J. Math. Phys, 47, 033541 (2006) · Zbl 1111.35055 · doi:10.1063/1.2181907
[4] Biondini, G.; Wang, D., On the soliton solutions of the two-dimensional Toda lattice. J, Phys. A, 43, 434007 (2010) · Zbl 1209.37074 · doi:10.1088/1751-8113/43/43/434007
[5] Blohm, H., Solution of nonlinear equations by trace methods, Nonlinearity, 13, 1925-1964 (2000) · Zbl 0972.35132 · doi:10.1088/0951-7715/13/6/304
[6] Cauchy, A.. Exercises d’Analyse et de Physique Mathématique. Tome 2. Bachelier, Paris 1841.
[7] Carl, B.; Huang, S.-Z., On realizations of solutions of the KdV equation by the C0-semigroup method, Amer. J. Math, 122, 403-438 (2000) · Zbl 1059.37049 · doi:10.1353/ajm.2000.0010
[8] Carl, B.; Schiebold, C., Nonlinear equations in soliton physics and operator ideals, Nonlinearity, 12, 333-364 (1999) · Zbl 0940.35175 · doi:10.1088/0951-7715/12/2/012
[9] Carl, B.; Schiebold, C., Ein direkter Ansatz zur Untersuchung von Solitonengleichungen, Jahresber. Deutsch. Math.-Verein, 102, 102-148 (2000) · Zbl 0970.35121
[10] Darboux, G., Leçon sur la Théorie des Surfaces 2 (1915), Gauthier-Villars: Gauthier-Villars, Paris · JFM 45.0881.05
[11] Eschmeier, A., Tensor products and elementary operators, J. reine und angew. Mathematik, 390, 47-66 (1988) · Zbl 0639.47003
[12] Grothendieck, A.
[13] Hirota, R., The Direct Method in Soliton Theory (2004), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1099.35111
[14] Hirota, R.; Ito, M.; Kako, F., Two-dimensional Toda lattice equations, Progr. Theoret. Phys. Suppl., 94, 42-58 (1988) · doi:10.1143/PTPS.94.42
[15] Hirota, R.; Ohta, Y.; Satsuma, J., Solutions of the Kadomtsev-Petviashvili equation and ths two-dimensional Toda equations, J. Phys. Soc. Jpn, 57, 1901-1904 (1988) · doi:10.1143/JPSJ.57.1901
[16] Hirota, R.; Ohta, Y.; Satsuma, J., Wronskian structures of solitons for soliton equations, Prog. Theor. Phys. Suppl, 94, 59-72 (1988) · doi:10.1143/PTPS.94.59
[17] Kodama, Y., KP Solitons and the Grassmannians. Combinatorics and Geometry of Two-Dimensional Wave Patterns, 22 (2017), Springer: Springer, Singapore · Zbl 1372.35266
[18] Li, S.; Biondini, G.; Schiebold, C., On the degenerate soliton solutions of the focusing nonlinear Schrödinger equation, J. Math. Phys, 58, 033507 (2017) · Zbl 1365.35159 · doi:10.1063/1.4977984
[19] Maruno, K.; Biondini, G., Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete analogues, J. Phys. A, 36, 11819-11839 (2004) · Zbl 1071.37055 · doi:10.1088/0305-4470/37/49/005
[20] Mikhailov, A. V., Integrability of a two-dimensional generalization of the Toda chain, JETP Lett, 30, 443-448 (1979)
[21] Miles, J., Resonantly interacting solitary waves, J. Fluid Mech, 79, 171-179 (1976) · Zbl 0353.76015 · doi:10.1017/S0022112077000093
[22] Nijhoff, F. W.; Atkinson, J.; Hietarinta, J., Soliton solutions for ABS lattice equations: I Cauchy matrix approach, J. Phys. A, 42, 404005 (2009) · Zbl 1184.35281 · doi:10.1088/1751-8113/42/40/404005
[23] Novikov, S.; Manakov, S. V.; Pitaevskii, L. P.; Zakharov, V. E., Theory of Solitons. The Inverse Scattering Transform (1984), Plenum: Plenum, New York · Zbl 0598.35002
[24] Pietsch, A., Eigenvalues and s-Numbers, 13 (1987), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 0615.47019
[25] Pólya, G.; Szegö, G., Aufgaben und Lehrsätze zur Analysis II (1976), Springer: Springer, Berlin · Zbl 0311.00002
[26] Saksman, E. and Tylli, H.-O.. Multiplications and elementary operators in the Banach space setting. In: Methods in Banach Space Theory: Proceedings of the V Conference on Banach Spaces, Cáceres, Spain, 13-18 September 2004, edited by J.M.F. Castillo and W.B. Johnson (London Mathematical Society Lecture Notes Series, 337, Cambridge University Press 2006), pp. 253-292. · Zbl 1133.47029
[27] Schiebold, C., An operator theoretic approach to the Toda lattice equation, Physica D, 122, 37-61 (1998) · Zbl 0977.37042 · doi:10.1016/S0167-2789(98)00173-0
[28] Schiebold, C., Solitons of the sine-Gordon equation coming in clusters, Revista Mat. Complut, 15, 265-325 (2002) · Zbl 1059.35128
[29] Schiebold, C., On Negatons of the Toda Lattice, J. Nonlin. Math. Phys, 10, 181-193 (2003) · Zbl 1362.37135
[30] Schiebold, C.. Integrable Systems and Operator Equations. Habilitation Thesis, Jena 2004.
[31] Schiebold, C., From the non-abelian to the scalar two-dimensional Toda lattice, Glasgow Math. J, 47, 177-189 (2005) · Zbl 1076.37068 · doi:10.1017/S0017089505002387
[32] Schiebold, C., Explicit solution formulas for the matrix-KP, Glasgow Math. J, 51, 147-155 (2009) · Zbl 1215.37046 · doi:10.1017/S0017089508004862
[33] Schiebold, C., Noncommutative AKNS systems and multisoliton solutions to the matrix sine-Gordon equation. Discrete and Continuous Dynamical Systems, Supplement, 678-690 (2009) · Zbl 1187.35197
[34] Schiebold, C., Cauchy-type determinants and integrable systems, Linear Algebra Appl., 433, 447-475 (2010) · Zbl 1195.15006 · doi:10.1016/j.laa.2010.03.011
[35] Schiebold, C., Asymptotics for the multiple poles solutions of the Nonlinear Schrödinger equation, Nonlinearity, 30, 2930-2981 (2017) · Zbl 1368.35251 · doi:10.1088/1361-6544/aa6d9a
[36] Schiebold, C.; Euler, N., Matrix solutions for equations of the AKNS system, Nonlinear Systems and Their Remarkable Mathematical Structures, 256-293 (2018), CRC Press: CRC Press, Boca Raton, FL, USA
[37] Takasaki, K., Toda hierarchies and their applictions, J. Phys. A, 51, 203001 (2018) · Zbl 1406.37053 · doi:10.1088/1751-8121/aabc14
[38] White, M. C., Analytic multivalued functions and spectral trace, Math. Ann, 304, 669-683 (1996) · Zbl 0864.47007 · doi:10.1007/BF01446313
[39] Xu, D. D.; Zhang, D. J.; Zhao, S. L., The Sylvester equation and integrable equations: I. The Kortewegde Vries system and sine-Gordon equation, J. Nonlin. Math. Phys, 21, 382-406 (2014) · Zbl 1420.35299 · doi:10.1080/14029251.2014.936759
[40] Zhang, D. J.; Zhao, S. L., Solutions to ABS lattice equations via generalized Cauchy matrix approach, Stud. Appl. Math, 131, 72-103 (2013) · Zbl 1338.37113 · doi:10.1111/sapm.12007
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.