B. A. Dubrovin | Searcharxiv

arXiv2014

Linearly Degenerate Hamiltonian PDEs and a New Class of Solutions to the WDVV Associativity Equations

We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of classifying such solutions of the WDVV equations to the particular case of the so-called algebraic Riccati equation and, in this way, arrive at a complete classification of irreducible solutions.

nlin.SI↗

arXiv1999

Recurrent procedure for the determination of the Free Energy $ε^{2}$-expansion in the Topological String Theory

We present here the iteration procedure for the determination of free energy $ε^{2}$-expansion using the theory of KdV - type equations. In our approach we use the conservation laws for KdV - type equations depending explicitly on times $t_{1}, t_{2}, ...$ to find the $ε^{2}$-expansion of $u(x,t_{1},t_{2},...)$ after the infinite number of shifts of $u(x,0,0,...) \equiv x$ along $t_{1}, t_{2}, ...$ in recurrent form. The formulas for the free energy expansion are just obtained then as a result of quite simple integration procedure applied to $u_{n}(x)$.

solv-int↗