Multiplicity and parity in Hirota's N-soliton conditions: the bilinear families L\,(Dₓ³+Λ)²ⁿ⁺¹

Hirota's three-soliton condition is the standard test for the integrability of a bilinear equation P(D)f· f=0, and the folklore implication ``three-soliton condition ⇒ N-soliton solutions for all N'' underlies the classifications of Hietarinta. We study this implication in two directions. The first is the family L\,(Dₓ³+Λ)²ⁿ⁺¹f· f=0 with L, $Λ$ linear forms in the Hirota operators, which for $n=0$ contains the Korteweg--de Vries, Ito and Boiti--Leon--Manna--Pempinelli equations and for L=Dₓ is the family of Gürses and Pekcan. On the principal component of the dispersion variety the N-soliton condition reduces to a sign sum of two bialternants; counting odd exponents shows that for $n=0$ the class is Hirota integrable and that for n≥1 the three-soliton condition holds for every L, and an explicit formula for the four-soliton residual proves the Gürses--Pekcan conjecture that the four-soliton condition fails for every n≥1. The formula yields sum-of-squares certificates excluding four-soliton solutions at every real configuration for the Gürses--Pekcan and Ito equations, classifies the real members of the class, and determines maximal numbers of nondegenerate solitons: exactly four or five for $n=1$ according as L lies in the span of Dₓ,Λ or not, five for $n=2$ and L=Dₓ, and finite for every n≥1 and every L. The second direction is a reformulation of the implication for a single non-hyperplane component of multiplicity one, which we prove, with complete classifications, for all graph-type dispersions L(q(D)+Dₜ), for every irreducible even quadric component in any dimension and with an arbitrary cofactor, for parabolic components, for the dispersions F()+Q containing the Kadomtsev--Petviashvili surface, and, with a prescribed one-parameter phase family, for the cubic graph.