Darboux transformations and integrable discrete systems

Groups of discrete transformations between the solution spaces of an equation or related equations have numerous applications and originate, in essence, in the works of Darboux and other geometers of the last part of the nineteenth century. They have been particularly fruitful in soliton theory where they are a machine for generating multi-soliton solutions. However, they also generate lattices of equations which themselves have integrable structure.

- Willox, R.; Ohta, Y.; Gilson, C. R.; Tokihiro, T.; Satsuma, J. Phys. Lett. A 252 (1999), no. 3-4, 163--172.
- Nimmo, J. J. C.; Schief, W. K. Stud. Appl. Math. 100 (1998), no. 3, 295--309.
- Nimmo, J. J. C. Darboux transformations and the discrete KP equation. J. Phys. A 30 (1997), no. 24, 8693--8704.
- Nimmo, J. J. C.; Willox, R. Darboux transformations for the two-dimensional Toda system. Proc. Roy. Soc. London Ser. A 453 (1997), no. 1967, 2497--2525.
- Nimmo, J. J. C.; Schief, W. K. Superposition principles associated with the Moutard transformation: an integrable discretization of a (2+1)-dimensional sine-Gordon system. Proc. Roy. Soc. London Ser. A 453 (1997), no. 1957, 255--279.
- Athorne, C. Phys. Lett. A 206 (1995), no. 3-4, 162--166.
- Athorne, C. On the characterization of Moutard transformations. Inverse Problems 9 (1993), no. 2, 217--232.
- Nimmo, J. J. C. Darboux transformations for a two-dimensional Zakharov-Shabat/AKNS spectral problem. Inverse Problems 8 (1992), no. 2, 219--243.
- Athorne, C.; Nimmo, J. J. C. On the Moutard transformation for integrable partial differential equations. Inverse Problems 7 (1991), no. 6, 809--826.
- Athorne, C.; Nimmo, J. J. C. Darboux theorems and factorization of second- and third-order ordinary differential operators. Inverse Problems 7 (1991), no. 5, 645--654.