From infinite symmetries to multi-soliton manifolds: dispersion-driven reduction for the KP and mKP equations

The infinite-dimensional symmetry algebras of integrable systems in $(2+1)$ dimensions exhibit a remarkable reduction property when restricted to finite-dimensional multi-soliton manifolds. In this paper, we systematically investigate this phenomenon for the Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili equations. For the Kadomtsev-Petviashvili equation, we analyze both the $K$-symmetry and the $\tau$-symmetry hierarchies. It is shown that for any $n$-soliton solution, each higher-order $K$-symmetry generator reduces to a linear combination of the $n$ independent translational symmetries ${\sigma }_{{\xi }_{m0}}$ associated with the individual soliton phase shifts, where the coefficients are precisely the dispersion relations evaluated at the soliton parameters. Closed-form expressions for odd- and even-order $K$-symmetries are derived, and determinant representations are provided to characterize the reduction systematically. For the $\tau$-symmetries, each ${\tau }_n$ becomes a linear combination of the three fundamental soliton symmetries ${\sigma }_{k_m}$, ${\sigma }_{l_m}$, and ${\sigma }_{{\xi }_{m0}}$, with coefficients that are rational functions of soliton parameters $k_m$ and $l_m$. These results are further extended to the modified Kadomtsev-Petviashvili equation, where analogous reduction patterns are established. This work establishes a direct and intrinsic link between the algebraic structure of infinite symmetries and the spectral properties of soliton solutions, offering a unified physical interpretation: the infinite set of higher-order symmetries generates linear superpositions of elementary soliton translations, weighted by powers of the wave numbers and dispersion coefficients.