John’s interests are in Integrable Hamiltonian systems, particularly those which are dispersionless. Typical of these is the Benney hierarchy, originally derived as a model of long waves on shallow water. This system, with infinitely many dynamical variables, possesses reductions in which only finitely many of these are independent. Such reduced equations can be solved exactly, using the hodograph transformation.
John’s research programme is concerned with the classification and the description of such reductions, and the construction of explicit examples of them. Some families of these have been found explicitly in terms of Kleinian sigma functions on algebraic curves. This work has also led to explicit series expansions for such sigma functions, and improved understanding of them.
Gibbons J, Matsutani S, Onishi Y, 2013, Relationship between the prime form and the sigma function for some cyclic (<i>r</i>, <i>s</i>) curves, Journal of Physics A - Mathematical and Theoretical, Vol:46, ISSN:1751-8113
Eilbeck JC, Enolski VZ, Gibbons J, 2010, Sigma, tau and Abelian functions of algebraic curves, Journal of Physics A - Mathematical and Theoretical, Vol:43, ISSN:1751-8113
Gibbons J, Lorenzoni P, Raimondo A, 2010, Purely nonlocal Hamiltonian formalism for systems of hydrodynamic type, Journal of Geometry and Physics, Vol:60, ISSN:0393-0440, Pages:1112-1126
England M, Gibbons J, 2009, A genus six cyclic tetragonal reduction of the Benney equations, Journal of Physics A - Mathematical and Theoretical, Vol:42, ISSN:1751-8113
Gibbons J, Lorenzoni P, Raimondo A, 2009, Hamiltonian Structures of Reductions of the Benney System, Communications in Mathematical Physics, Vol:287, ISSN:0010-3616, Pages:291-322