Higher order Hirota bilinear forms

Metin GÃ¼rses; AslÄ± Pekcan

paper

Higher order Hirota bilinear forms

arXiv:2511.18466

Abstract

In this paper we study Hirota bilinear forms of the type $P (D) {f \cdot f} = 0$ . We prove that for $P (D) = D_{x}^{m} D_{y}^{r} D_{t}^{n}$ 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 $P (D) = D_{x} (D_{x}^{3} + I^\pm_{1} D_{t} + I^\pm_{2} D_{y})^{2 k + 1}$ possess three-soliton solutions for any constants $(I^\pm_{1}, I^\pm_{2}) \neq = (0, 0)$ and $k \in N$ . We conjecture that these equations have four-soliton solution only for $k = 0$ . Finally, we consider the equations for $P (D) = D_{x}^{m_{1}} D_{y}^{m_{2}} D_{t}^{m_{3}} D_{z}^{m_{4}}$ . 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 $D_{x}^{m} D_{y}^{r} D_{t}^{n}$ and $D_{x}^{m_{1}} D_{y}^{m_{2}} D_{t}^{m_{3}} D_{z}^{m_{4}}$ 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.

A contribution to Metin GÃ¼rses' Festschrift (GURSES-FS-2025)