The Darboux coordinates for a new family of Hamiltonian operators and linearization of associated evolution equations

Abstract

u is the dependent variable and D is the total derivative with respect to the independent variable. We present a differential substitution that reduces any linear combination of these operators to an operator with constant coefficients and linearizes any evolution equation which is bi-Hamiltonian with respect to a pair of any nontrivial linear combinations of the operators H (N,0) . We also give the Darboux coordinates for H (N,0) for any odd N 3.