Higher order Hirota bilinear forms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909919846858752 |
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| author | Gürses, Metin Pekcan, Aslı |
| author_facet | Gürses, Metin Pekcan, Aslı |
| contents | In this paper we study Hirota bilinear forms of the type $P(D) \{f\cdot f\}=0$. We prove that for $P(D)=D_x^mD_y^rD_t^n$ 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 $P(D)=D_x(D_x^3+α_1 D_t+α_2 D_y)^{2k+1}$ 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 $P(D)=D_x^{m_1}D_y^{m_2}D_t^{m_3}D_z^{m_4}$. 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 $D_x^mD_y^rD_t^n$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_18466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher order Hirota bilinear forms Gürses, Metin Pekcan, Aslı Exactly Solvable and Integrable Systems In this paper we study Hirota bilinear forms of the type $P(D) \{f\cdot f\}=0$. We prove that for $P(D)=D_x^mD_y^rD_t^n$ 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 $P(D)=D_x(D_x^3+α_1 D_t+α_2 D_y)^{2k+1}$ 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 $P(D)=D_x^{m_1}D_y^{m_2}D_t^{m_3}D_z^{m_4}$. 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 $D_x^mD_y^rD_t^n$ 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. |
| title | Higher order Hirota bilinear forms |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2511.18466 |