an:05784658
Zbl 1194.37114
Fuchssteiner, B.; Fokas, A. S.
Symplectic structures, their B??cklund transformations and hereditary symmetries
EN
Physica D 4, No. 1, 47-66 (1981).
00048154
1981
j
37K05 37K35
Summary: It is shown that compatible symplectic structures lead in a natural way to hereditary symmetries. (We recall that a hereditary symmetry is an operator-valued function which immediately yields a hierarchy of evolution equations, each having infinitely many commuting symmetries all generated by this hereditary symmetry. Furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations). This result then provides us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations. In addition, we show how symplectic structures transform under B??cklund transformations. This leads to a method for generating a whole class of symplectic structures from a given one. Several examples and applications are given illustrating the above results. Also the connection of our results with those of Gel'fand and Dikii, and of Magri is briefly pointed out.