2025/11/23 by Gürses, Metin, Pekcan, Aslı
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Algebraic structures and combinatorial models #Holomorphic and Operator Theory
paper · doi:10.48550/arxiv.2511.18466
In this paper we study Hirota bilinear forms of the type P(D) \f⋅ f\=0. We prove that for P(D)=DxmDyrDtn the equations have three-soliton solutions if only if two of nonzero m,n,p are odd and the other one even. We explicitly derive the nonlinear partial differential equations corresponding to this form for m+n+p=4 and m+n+p=6. We show that the equations for P(D)=Dx(Dx3+α1 Dt+α2 Dy)2k+1 possess three-soliton solutions for any constants (α1,α2)≠ (0,0) and k∈ ℕ. We conjecture that these equations have four-soliton solution only for k=0. Finally, we consider the equations for P(D)=Dxm1Dym2Dtm3Dzm4. We prove that these equations have three-soliton solutions if only if one of mi=1, and all the other mi's are odd for i=1,2,3,4. We observe that the monomials DxmDyrDtn and Dxm1Dym2Dtm3Dzm4 do not result genuine four-soliton solutions. In addition, we obtain three-soliton, lump, and hybrid solutions of these three type of equations for particular powers of the Hirota D-operators.