Multiplicity and parity in Hirota's $N$-soliton conditions: the bilinear families $L\,(D_x^{3}+Λ)^{2n+1}$

Hirota's three-soliton condition is the standard test for the integrability of a bilinear equation $P(D)f\cdot f=0$, and the folklore implication ``three-soliton condition $\Rightarrow$ $N$-soliton solutions for all $N$'' underlies the classifications of Hietarinta. We study this implication in two directions. The first is the family $L\,(D_x^3+Λ)^{2n+1}f\cdot f=0$ 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 $L=D_x$ 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 $n\ge1$ 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 $n\ge1$. 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 $D_x,Λ$ or not, five for $n=2$ and $L=D_x$, and finite for every $n\ge1$ 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 $L(q(D)+D_t)$, for every irreducible even quadric component in any dimension and with an arbitrary cofactor, for parabolic components, for the dispersions $F(\ell)+Q$ containing the Kadomtsev–Petviashvili surface, and, with a prescribed one-parameter phase family, for the cubic graph.