An Exact Solution to the Bateman-Burgers Equation Using Generalized First Integral Method

Abstract

The classical first integral method, while effective for many nonlinear partial differential equations, is inherently limited by its polynomial structure. This paper introduces the Generalized First Integral Method (GFIM), which extends the classical framework by operating within the
richer algebraic structure of Laurent polynomials. This generalization enables the method to handle equations with singularities, rational terms, and more complex nonlinearities where the classical approach fails. We establish the theoretical foundation by proving a Division Theorem, Remainder Theorem, and supporting lemmas within the ring of Laurent polynomials. These results allow nonlinear partial
differential equations to be reduced to autonomous differential systems, yielding exact closed-form solutions without perturbative techniques or infinite series expansions. The method’s effectiveness is demonstrated through the Bateman-Burgers equation, a canonical model for nonlinear wave propagation and shock formation in fluid dynamics. We derive new exact solutions and analyze their bifurcation and stability properties, showing how the parameters A, λ, and ν govern the transition between shock-forming and diffusion-dominated regimes. A comparison with classical FIM highlights cases where only the generalized approach succeeds. This work advances the theory of first integrals while providing a practical tool for solving nonlinear PDEs arising in fluid mechanics, traffic flow, and material science.