KP-hierarchy reduction and non-autonomous localized waves in a variable-coefficient BLMP equation

In this paper, we study a time-dependent variable-coefficient form of the Boiti–Leon–Manna–Pempinelli equation, which models nonlinear waves in nonuniform media. The Painlevé analysis shows that the proposed equation possesses the Painlevé property under the stated local nondegeneracy assumptions, thereby providing strong evidence for an underlying integrable structure rather than, by itself, a complete proof of integrability. By applying the Hirota bilinear method together with the KP hierarchy reduction, we construct exact multi-soliton and breather-type solutions and derive rational-degenerate and semi-rational limits from the same Gram determinant. The breather reduction is obtained from complex-conjugate spectral pairs and the same spectral constraint as the soliton determinant. The corrected long-wave degeneration produces meromorphic rational and semi-rational solutions whose tau functions may vanish, so they are not claimed to be globally nonsingular lumps or rogue waves. Based on our literature survey, the KP hierarchy reduction has not previously been applied to this time-dependent variable-coefficient form of the BLMP equation. The results reveal non-autonomous kink motion, oscillatory breather dynamics, and algebraically localized meromorphic degenerations. In the constant-coefficient limit, the construction also yields corresponding exact solutions of the classical Boiti–Leon–Manna–Pempinelli equation. These findings extend the model and offer useful insight into nonlinear wave behavior in realistic physical systems.