Bilinear Methods

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

The group has a long history of involvement in Hirota's bilinear formalism and its application to equations in 1+1 and 2+1 dimensions.

In this approach certain, special equations are found, under a transformation of the dependent variable, to reduce to a form bilinear (over the constants) in the new dependent variable. Derivatives occur in a special combinations called Hirota derivatives which make it relatively easy to obtain exact solutions. Much of the theory comes down to identities of the Jacobi-Trudi kind and involves Wronskians and Pfaffians.

There are also connections with representation theory.

- Gilson, Claire R.; Ratter, Mark C. Three-dimensional three-wave interactions: a bilinear approach. J. Phys. A 31 (1998) 349-67. MathSciNet Review
- Gilson, C.; Lambert, F.; Nimmo, J.; Willox, R. On the combinatorics of the Hirota D-operators. Proc. Roy. Soc. London Ser. A 452 (1996), no. 1945, 223--234. MathSciNet Review
- Gilson, C. R.; Nimmo, J. J. C. A direct method for dromion solutions of the Davey-Stewartson equations and their asymptotic properties. Proc. Roy. Soc. London Ser. A 435 (1991), no. 1894, 339--357. MathSciNet Review
- Nimmo, J. J. C. Hall-Littlewood symmetric functions and the BKP equation. J. Phys. A 23 (1990), no. 5, 751--760. MathSciNet Review
- Gilson, C. R.; Nimmo, J. J. C.; Freeman, N. C. Rational solutions to the two-component K-P hierarchies. Nonlinear evolution equations and dynamical systems (Kolymbari, 1989), 32--35, Res. Rep. Phys., Springer, Berlin, 1990. MathSciNet Review
- Nimmo, J. J. C. Hirota's method. Soliton theory: a survey of results, 75--96, Nonlinear Sci. Theory Appl., Manchester Univ. Press, Manchester, 1990. MathSciNet Review
- Nimmo, J. J. C. Wronskian determinants, the KP hierarchy and supersymmetric polynomials. J. Phys. A 22 (1989), no. 16, 3213--3221. MathSciNet Review
- Nimmo, J. J. C. Symmetric functions and the KP hierarchy. Nonlinear evolutions (Balaruc-les-Bains, 1987), 245--261, World Sci. Publishing, Teaneck, NJ, 1988. MathSciNet Review