Higher order Hirota bilinear forms

Abstract: In this paper we study Hirota bilinear forms of the type $P(D) {f\cdot f}=0$. We prove that for $P(D)=D_x<sup>mD_y<sup>rD_t<sup>n$ 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 $$P(D)=D_x(D_x^3+α_1 D_t+α_2 D_y)<sup>{2k+1}$$ possess three-soliton solutions for any constants $(α_1,α_2)\neq (0,0)$ and $k\in \mathbb{N}$. We conjecture that these equations have four-soliton solution only for $k=0$. Finally, we consider the equations for $$P(D)=D_x<sup>{m_1}D_y<sup>{m_2}D_t<sup>{m_3}D_z<sup>{m_4}$$. We prove that these equations have three-soliton solutions if only if one of $m_i=1$, and all the other $m_i$'s are odd for $i=1,2,3,4$. We observe that the monomials $D_x<sup>mD_y<sup>rD_t<sup>n$ and $$D_x<sup>{m_1}D_y<sup>{m_2}D_t<sup>{m_3}D_z<sup>{m_4}$$ 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.