BI-HAMILTONIAN STRUCTURE AND EXACTSOLUTIONS OF ONE BURGERS’ TYPE NONLINEARDYNAMICAL SYSTEM

Анотація

In the present work, we find the bi-Hamiltonian representation and three classes of ex￾act solutions for the dispersionful (Burgers’ type) nonlinear dynamical system introduced
by Szablikowski et al. [13]. In particular, for the above-mentioned system, we construct
the infinite hierarchy of functionally independent conservation laws utilizing the gradient
holonomic method [3]. Moreover, based on that hierarchy we find the implectic pair of
Noetherian operators and corresponding Hamiltonian functionals applying the differential￾algebraic algorithm [8, 12]. Furthermore, we construct three classes of exact traveling
wave solutions, in particular, solitary wave and periodic ones, using the
–expansion
method [18]. It is shown that for the case of the dynamical system under consideration,
degrees of the polynomials in  cannot be uniquely determined from the system of
algebraic equations of the homogeneous balance. Nevertheless, utilizing a more detailed
analysis, a general form of the solution is found uniquely. Further, we analyze the obtained
results, in particular, the analytical solution is verified by putting it back into original
equations. Finally, we anticipate future research objectives, especially finding the standard
Lax type representation of the above-mentioned dynamical system.