The associativity equations in the two-dimensional topological field theory as integrable Hamiltonian nondiagonalizable systems of hydrodynamic type (Q1357909)

The authors mainly treat here the strictly hyperbolic systems of hydrodynamic type \(u^i_t= v^i_j(u)u^j_x\); \(i,j=1,2,3\). Especially they treat two systems \[ (a,b,c)_t= (b,c,(b^2- ac))_x,\tag{1} \] \[ (a,b,c)_t= (b,c,(1+ bc)/a)_x,\tag{2} \] derived from associativity of an algebra in topological field theory. First of all, by connecting with a spectral problem, the Hamiltonian property of system (1) is established. Both systems (1) and (2) are nondiagonalizable (i.e. not possessing Riemann invariants) and linearly degenerate. Next, they show that any such Hamiltonian \(3\times 3\) system can be reduced to an integrable 3-wave system by some standard chain of transformations. Finally, the explicit Bäcklund type transformation connecting solutions of systems (1) and (2) is given.