Soliton solutions of the Toda hierarchy on quasi-periodic backgrounds revisited. (English) Zbl 1166.37028

The authors after introducing the Toda hierarchy and recalling necessary facts on algebra-geometric quasi-periodic finite-gap solutions briefly review the single and double commutation methods. The phase shift (in the Jacobian variety) caused by inserting one eigenvalue for both methods is computed. The authors also review the direct scattering theory for Jacobi operators with different (quasi)-periodic asymptotics in the same isospectral class. As main result they give a complete description of the effect of the double commutation method on the scattering data. In addition they provide some detailed asymptotic formulas for the Jost functions \(\psi_{\pm}(z,n)\) (which are normalized as \(n\to\pm\infty\)) at the other side, that is, as \(n\to\mp\infty\). Finally they establishe the inverse scattering transform for this setting. The main results here are the time dependence of both the scattering data and the kernel of the Gel’fand-Levitan-Marchenko equation.


37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
34L25 Scattering theory, inverse scattering involving ordinary differential operators
Full Text: DOI arXiv


[1] W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, Algebro-geometric Quasi-periodic Finite-gap Solutions of the Toda and Kac-van Moerbeke Hierarchies, Memoirs of the American Mathematical Society Vol. 135, No. 641 (Amer. Math. Soc., Providence, RI, 1998). · Zbl 0906.35099
[2] Egorova, Scattering theory for Jacobi operators with quasi-periodic background, Comm. Math. Phys. 264 (3) pp 811– (2006) · Zbl 1115.39025
[3] Egorova, Inverse scattering transform for the Toda hierarchy with quasi-periodic background, Proc. Amer. Math. Soc. 135 pp 1817– (2007) · Zbl 1118.37034
[4] Egorova, Scattering theory for Jacobi operators with steplike quasi-periodic background, Inverse Problems 23 pp 905– (2007) · Zbl 1152.47023
[5] Forest, Spectral theory, J. Math. Phys. 23 (7) pp 1248– (1982)
[6] Gesztesy, Commutation methods applied to the mKdV-equation, Trans. Amer. Math. Soc. 324 (2) pp 465– (1991) · Zbl 0728.35106
[7] F. Gesztesy, and R. Svirsky, (m)KdV solitons on the background of quasi-periodic finite-gap solutions, Memoirs of the American Mathematical Society Vol. 118, No. 563 (Amer. Math. Soc., Providence, RI, 1995). · Zbl 0855.35109
[8] Gesztesy, Commutation methods for Jacobi operators, J. Differential Equations 128 pp 252– (1996) · Zbl 0854.34079
[9] Kamvissis, Stability of periodic soliton equations under short range perturbations, Phys. Lett. A 364-6 pp 480– (2007) · Zbl 1203.35226
[10] S. Kamvissis, and G. Teschl, Stability of the periodic Toda lattice under short range perturbations, arXiv:0705. 0346 · Zbl 1301.37055
[11] Kuznetsov, Stability of stationary waves in nonlinear weakly dispersive media, Soviet Phys. JETP 40 (5) pp 855– (1975)
[12] Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 pp 467– (1968) · Zbl 0162.41103
[13] Alonso, Soliton interaction and prime forms on Riemann surfaces, Phys. Lett. A 188 (1) pp 32– (1994) · Zbl 0941.37535
[14] J. Michor, and G. Teschl, Trace formulas for Jacobi operators in connection with scattering theory for quasi-periodic background, in: Operator Theory, Analysis, and Mathematical Physics, edited by J. Janas et al., Operator Theory: Advances and Applications Vol. 174 (Birkhäuser, Basel, 2007), pp. 69-76. · Zbl 1133.47024
[15] Pełka, Effective velocity of soliton in the presence of a periodic background, Act. Phys. Polon. B 100 (6) pp 871– (2001)
[16] Rubenstein, Sine-Gordon equation, J. Math. Phys. 11 (1) pp 258– (1969)
[17] Shin, Soliton on a cnoidal wave background in the coupled nonlinear Schrödinger equation, J. Phys. A, Math. Gen. 37 pp 8017– (2004) · Zbl 1065.35220
[18] Shin, The dark soliton on a cnoidal wave background, J. Phys. A, Math. Gen. 38 pp 3307– (2005) · Zbl 1065.81050
[19] Teschl, On the Toda and Kac-van Moerbeke hierarchies, Math. Z. 231 pp 325– (1999) · Zbl 0935.37045
[20] Teschl, Inverse scattering transform for the Toda hierarchy, Math. Nachr. 202 pp 163– (1999) · Zbl 1120.37315
[21] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs Vol. 72 (Amer. Math. Soc., Providence, RI, 2000). · Zbl 1056.39029
[22] Teschl, Algebro-geometric constraints on solitons with respect to quasi-periodic backgrounds, Bull. Lond. Math. Soc. 39 (4) pp 677– (2007) · Zbl 1131.34012
[23] M. Toda, Theory of Nonlinear Lattices, 2nd edition (Springer, Berlin, 1989). · Zbl 0694.70001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.