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Integrable Hierarchy Covering the Lattice Burgers Equation in Fluid Mechanics: N-fold Darboux Transformation and Conservation Laws

Burgers-type equations can describe some phenomena in fluids, plasmas, gas dynamics, traffic, etc. In this paper, an integrable hierarchy covering the lattice Burgers equation is derived from a discrete spectral problem. N-fold Darboux transformation (DT) and conservation laws for the lattice Burgers equation are constructed based on its Lax pair. N-soliton solutions in the form of Vandermonde-like determinant are derived via the resulting DT with symbolic computation, structures of which are shown graphically. Coexistence of the elastic-inelastic interaction among the three solitons is firstly reported for the lattice Burgers equation, even if the similar phenomenon for certern continuous systems is known. Results in this paper might be helpful for understanding some ecological problems describing the evolution of competing species and the propagation of nonlinear waves in fluids.