Abstract:
A gauge-covariant formulation of the theory of the generating operator $\Lambda$
for a quadratic bundle is found. On this basis, the method of expansion
with respect to “squared solutions” is applied to the auxiliary linear problem
$$
\left\{iS_0(x)\frac{d}{dx}+\lambda S_1(x)-\lambda^2\right\}\tilde v(x,\lambda)=0.
$$
Thus, for nonlinear evolution equations associated with this problem
a hierarchy of Hamiltonian structures is obtained and their complete
integrability is proved. Some examples, including equations of
Landau–Lifshitz type, are considered for suitable reduction.