Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher order Hirota bilinear forms

Metin Gürses, Aslı Pekcan

TL;DR

The work addresses the problem of classifying higher-order Hirota bilinear forms that admit multi-soliton solutions, establishing precise parity conditions for the exponents in $D$-operators and deriving explicit nonlinear PDEs via bilinearization. It introduces a universal three-soliton form $D_x(D_x^3+\alpha_1 D_t+\alpha_2 D_y)^{2k+1}$ with three-soliton existence for all nonzero $(\alpha_1,\alpha_2)$ and $k$, and analyzes $(3+1)$-D generalizations, showing no genuine four-soliton solutions for the studied cases. The paper also constructs lump and hybrid solutions for several specific operator forms, including higher-order $(2+1)$-D and $(3+1)$-D equations, and provides explicit examples and reductions to lower dimensions. The results expand the catalog of Hirota-bilinear equations that sustain three-soliton, lump, and hybrid solutions, with implications for integrable and near-integrable dynamics.

Abstract

In this paper we study Hirota bilinear forms of the type $P(D) \{f\cdot f\}=0$. We prove that for $P(D)=D_x^mD_y^rD_t^n$ 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.

Higher order Hirota bilinear forms

TL;DR

The work addresses the problem of classifying higher-order Hirota bilinear forms that admit multi-soliton solutions, establishing precise parity conditions for the exponents in -operators and deriving explicit nonlinear PDEs via bilinearization. It introduces a universal three-soliton form with three-soliton existence for all nonzero and , and analyzes -D generalizations, showing no genuine four-soliton solutions for the studied cases. The paper also constructs lump and hybrid solutions for several specific operator forms, including higher-order -D and -D equations, and provides explicit examples and reductions to lower dimensions. The results expand the catalog of Hirota-bilinear equations that sustain three-soliton, lump, and hybrid solutions, with implications for integrable and near-integrable dynamics.

Abstract

In this paper we study Hirota bilinear forms of the type . We prove that for the equations have three-soliton solutions if only if two of nonzero are odd and the other one even. We explicitly derive the nonlinear partial differential equations corresponding to this form for and . We show that the equations for possess three-soliton solutions for any constants and . We conjecture that these equations have four-soliton solution only for . Finally, we consider the equations for . We prove that these equations have three-soliton solutions if only if one of , and all the other 's are odd for . We observe that the monomials and 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 -operators.

Paper Structure

This paper contains 26 sections, 4 theorems, 108 equations, 17 figures.

Table of Contents

  1. Introduction
  2. $(2+1)$-dimensional Hirota bilinear forms
  3. Equations of the form $D_x^mD_y^nD_t^p\{f\cdot f\}=0$
  4. Equations of the form $D_x(D_x^3+\alpha_1 D_t+\alpha_2 D_y)^{2k+1}\{f\cdot f\}=0$
  5. $(3+1)$-dimensional Hirota bilinear forms
  6. Equations of the form $D_x^{m_1}D_y^{m_2}D_t^{m_3}D_z^{m_4}\{f\cdot f\}=0$
  7. Soliton solutions
  8. Three-soliton solutions of (\ref{['Dx^mD_y^nD_t^p']})
  9. Three- and four-soliton solutions of (\ref{['bilinearProb2']})
  10. Three-soliton solution of the equation (\ref{['bilinearDxDyDtDz']})
  11. Lump solutions
  12. A. Lump solutions with two functions
  13. Lump solutions of the equation (\ref{['Dx^mD_y^nD_t^p']})
  14. Lump solutions of the equation (\ref{['bilinearProb2']})
  15. Lump solutions of the equation (\ref{['bilinearDxDyDtDz']})
  16. ...and 11 more sections

Key Result

Lemma 2.1

The Hirota bilinear equation possesses three-soliton solutions if only if two of nonzero $m,n,p$ are odd and the other one even.

Figures (17)

  • Figure 1: Breather-type wave solution of the equation (\ref{['m=3p=1n=2']}) at $t=0$ (a) 3D graph, (b) contour plot.
  • Figure 2: Three-soliton solution (\ref{['threesolex1']}) of the equation (\ref{['exk=0']}) at $t=0$ (a) 3D graph, (b) contour plot.
  • Figure 3: Three-soliton solution (\ref{['threesolex2']}) of the equation (\ref{['exk=1']}) (a) 3D graph, (b) contour plot.
  • Figure 4: Four-soliton solution of the equation (\ref{['exk=0']}) at $t=0$ (a) 3D graph, (b) contour plot.
  • Figure 5: Three-soliton solution of (\ref{['bilinearDxDyDtDz']}) for the parameters $k_1=4, k_2=2, k_3=1$, $\omega_1=2, \omega_2=-1$, $l_1=3, l_3=-1$, $s_2=2, s_3=-1$, $\delta_1=1, \delta_2=-1, \delta_3=2$, and $s_1=l_2=\omega_3=0$ at $y=z=0$ (a) 3D graph, (b) contour plot.
  • ...and 12 more figures

Theorems & Definitions (5)

  • Definition 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1