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 $τ$-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 $σ_{ξ_{m0}}$ 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 \(τ\)-symmetries, each \(τ_n\) becomes a linear combination of the three fundamental soliton symmetries \(σ_{k_m}\), \(σ_{l_m}\), and \(σ_{ξ_{m0}}\), 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.