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