BI-HAMILTONIAN STRUCTURE AND EXACT SOLUTIONS OF ONE BURGERS� TYPE NONLINEAR DYNAMICAL SYSTEM

Abstract

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.