Multiplicity and parity in Hirota's N-soliton conditions: the bilinear families L (Dx3+Λ)2n+1
Abstract
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(ℓ)+Q containing the Kadomtsev--Petviashvili surface, and, with a prescribed one-parameter phase family, for the cubic graph.
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BibTeX
@misc{zhenhua2026multiplicity,
title = {{Multiplicity and parity in Hirota's $N$-soliton conditions: the bilinear families $L\,(D_x^3+Λ)^2n+1$}},
author = {Shi Zhenhua},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04953},
eprint = {2611.04953},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04953},
}
RIS
TY - UNPB
TI - Multiplicity and parity in Hirota's N-soliton conditions: the bilinear families L (D_x^3+Λ)^2n+1
AU - Shi Zhenhua
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04953
UR - https://arxiv.org/abs/2611.04953
AB - 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 (D_x^3+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=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≥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 D_x,Λ or not, five for n=2 and L=D_x, 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)+D_t), for every irreducible even quadric component in any dimension and with an arbitrary cofactor, for parabolic components, for the dispersions F(ℓ)+Q containing the Kadomtsev--Petviashvili surface, and, with a prescribed one-parameter phase family, for the cubic graph.
ER -