Integrable Structure and Explicit Traveling-Wave Solutions of the Mikhailov–Novikov–Wang Equation via a Polynomial First-Integral Approach

The Mikhailov–Novikov–Wang (MNW) equation is a nonlinear dispersive equation of higher order that occurs in the theory of integrable systems and nonlinear wave propagation. In spite of analytic methods having been applied to build traveling-wave solutions of this equation, little attention has been paid to systematically exploiting its integrability structure through polynomial first integrals and phase-space reduction. Specifically, a geometrical understanding of the relationship of invariant manifolds to explicit analytic solutions has not been fully investigated. This study investigates the MNW equation from the perspective of dynamical systems. A traveling-wave transformation is used to simplify the equation to a nonlinear ordinary differential system. This is then followed by the construction of a polynomial first integral, such that the system can be reduced to a lower dimensional invariant manifold, with explicit finding of the solutions in closed form. The improved Kudryashov method, the modified improved Kudryashov method, and the 1/G′-expansion method are separately applied in order to further expand the solution space and to check analytical consistency. The analysis yields several exact solutions, such as kink-type, antikink, smooth soliton, and singular wave profiles. The consistency between solutions obtained using various analytical frameworks indicates the strength of the proposed technique and emphasizes the integrability of the MNW equation. This work also provides a systematic technique that can be used with other higher order integrable nonlinear evolution equations.