Bäcklund Transformations and Integrable Discrete Equations.

(See also: resources and, in particular, research topics.)

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.

1. Willox, R.; Ohta, Y.; Gilson, C. R.; Tokihiro, T.; Satsuma, J. Quadrilateral lattices and eigenfunction potentials for $N$-component KP hierarchies. Phys. Lett. A 252 (1999), no. 3-4, 163--172. MathSciNet Review
2. Nimmo, J. J. C.; Schief, W. K. An integrable discretization of a $(2+1)$-dimensional sine-Gordon equation. Stud. Appl. Math. 100 (1998), no. 3, 295--309. MathSciNet Review
3. Nimmo, J. J. C. Darboux transformations and the discrete KP equation. J. Phys. A 30 (1997), no. 24, 8693--8704. MathSciNet Review
4. 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. MathSciNet Review
5. 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. MathSciNet Review
6. Athorne, C. A $\bold Z\sp 2\times\bold R\sp 3$ Toda system. Phys. Lett. A 206 (1995), no. 3-4, 162--166. MathSciNet Review
7. Gilson, C. R.; Nimmo, J. J. C.; Willox, R. A $(2+1)$-dimensional generalization of the AKNS shallow water wave equation. Phys. Lett. A 180 (1993), no. 4-5, 337--345. MathSciNet Review
8. Athorne, C. On the characterization of Moutard transformations. Inverse Problems 9 (1993), no. 2, 217--232. MathSciNet Review
9. Nimmo, J. J. C. A class of solutions of the Konopel\cprime chenko-Rogers equations. Phys. Lett. A 168 (1992), no. 2, 113--119. MathSciNet Review
10. Nimmo, J. J. C. Darboux transformations for a two-dimensional Zakharov-Shabat/AKNS spectral problem. Inverse Problems 8 (1992), no. 2, 219--243. MathSciNet Review
11. Athorne, C.; Nimmo, J. J. C. On the Moutard transformation for integrable partial differential equations. Inverse Problems 7 (1991), no. 6, 809--826. MathSciNet Review
12. 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. MathSciNet Review