A new Whitham-type nonlinear evolution equation describing short-wave perturbations in a relaxing medium is studied. Making use of the gradientholonomic analysis, the bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated. An infinite hierarchy of dispersive conservation laws which commute with each other is constructed. The twoand four-dimensional invariant reductions are studied in detail.

The algebraic structure of the gradient-holonomic algorithm for Lax integrable dynamical systems is discussed. A generalization of the ℛ-structure approach for the case of operator-valued affine Lie algebras is used to prove the bi-Hamiltonian formulation of nonlinear integrable dynamical systems in multidimensions. The monodromy transfer matrix is constructed to describe the operator manifold in relation to canonical Lie–Poisson bracket on initial affine Lie algebra with gauge central extension. As an illustration, the two-dimensional operator Benney–Kaup integrable hierarchy is considered and their bi-Hamiltonicity is proved.

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A new modification of the the Lie-algebraic scheme for solving partial differential equations with initial and boundary conditions based on constructing quasirepresentations of the Heisenberg-Weyl algebra operators involving boundary conditions is proposed. Approximation errors for the modified scheme are evaluated. Quasirepresentations involving boundary conditionsIn this section I intend to construct appropriate quasirepresentations of the Heisenberg-Weyl algebra operators with the domain Similarly to the case with no boundary condition [2] let us define mappings Φ (n),0 : B 0 → B (n),0 as the composition, where u (n) = {u (i) } (i)∈ = {u(x (i) )} (i)∈ ∈ B (n),0 and e (i) (x) = φ(x) φ (i)

We consider the general Lie-algebraic scheme of construction of integrable nonlinear dynamical systems on extended functional manifolds. We obtain an explicit expression for consistent Poisson structures and write explicitly nonlinear equations generated by the spectrum of a periodic problem for an operator of Lax-type representation.