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General soliton solutions to the coupled Hirota equation via the Kadomtsev-Petviashvili reduction

Changyan Shi, Bingyuan Liu, Bao-Feng Feng

TL;DR

This work develops general $N$-soliton solutions for the coupled Hirota equation and its Sasa-Satsuma reduction via Kadomtsev-Petviashvili hierarchy reduction, casting solutions in determinant ($\tau$-function) form. It systematically derives bright-bright, dark-dark, and bright-dark solitons for the $c$Hirota equation and provides corresponding bright and dark solitons for the SS equation, including new $N\times N$ determinant representations that extend previous $2N\times2N$ forms. The analysis covers first- and second-order dynamics, revealing a rich taxonomy of soliton shapes (oscillating, single-hump, two-hump, Mexican hat, double-hole, etc.) and collision-induced phase shifts, with additional phenomena such as Y-shaped solitons under degeneracy. The results highlight the power of KP reduction and bilinear methods to generate and classify multi-component, higher-order solitons in integrable systems, with implications for nonlinear optics and related fields. Overall, the paper provides a unified, determinant-based framework for multi-soliton solutions and their dynamic interactions in the cHirota and SS equations.

Abstract

In this paper, we are concerned with various soliton solutions to the coupled Hirota equation, as well as to the Sasa-Satsuma equation which can be viewed as one reduction case of the coupled Hirota equation. First, we derive bright-bright, dark-dark, and bright-dark soliton solutions of the coupled Hirota equation by using the Kadomtsev-Petviashvili reduction method. Then, we present the bright and dark soliton solutions to the Sasa-Satsuma equation which are expressed by determinants of $N \times N$ instead of $2N \times 2N$ in the literature. The dynamics of first-, second-order solutions are investigated in detail. It is intriguing that, for the SS equation, the bright soliton for \(N=1\) is also the soliton to the complex mKdV equation while the amplitude and velocity of dark soliton for \(N=1\) are determined by the background plane wave. For \(N=2\), the bright soliton can be classified into three types: oscillating, single-hump, and two-hump ones while the dark soliton can be classified into five types: dark (single-hole), anti-dark, Mexican hat, anti-Mexican hat and double-hole. Moreover, the types of bright solitons for the Sasa-Satsuma equation can be changed due to collision.

General soliton solutions to the coupled Hirota equation via the Kadomtsev-Petviashvili reduction

TL;DR

This work develops general -soliton solutions for the coupled Hirota equation and its Sasa-Satsuma reduction via Kadomtsev-Petviashvili hierarchy reduction, casting solutions in determinant (-function) form. It systematically derives bright-bright, dark-dark, and bright-dark solitons for the Hirota equation and provides corresponding bright and dark solitons for the SS equation, including new determinant representations that extend previous forms. The analysis covers first- and second-order dynamics, revealing a rich taxonomy of soliton shapes (oscillating, single-hump, two-hump, Mexican hat, double-hole, etc.) and collision-induced phase shifts, with additional phenomena such as Y-shaped solitons under degeneracy. The results highlight the power of KP reduction and bilinear methods to generate and classify multi-component, higher-order solitons in integrable systems, with implications for nonlinear optics and related fields. Overall, the paper provides a unified, determinant-based framework for multi-soliton solutions and their dynamic interactions in the cHirota and SS equations.

Abstract

In this paper, we are concerned with various soliton solutions to the coupled Hirota equation, as well as to the Sasa-Satsuma equation which can be viewed as one reduction case of the coupled Hirota equation. First, we derive bright-bright, dark-dark, and bright-dark soliton solutions of the coupled Hirota equation by using the Kadomtsev-Petviashvili reduction method. Then, we present the bright and dark soliton solutions to the Sasa-Satsuma equation which are expressed by determinants of instead of in the literature. The dynamics of first-, second-order solutions are investigated in detail. It is intriguing that, for the SS equation, the bright soliton for is also the soliton to the complex mKdV equation while the amplitude and velocity of dark soliton for are determined by the background plane wave. For , the bright soliton can be classified into three types: oscillating, single-hump, and two-hump ones while the dark soliton can be classified into five types: dark (single-hole), anti-dark, Mexican hat, anti-Mexican hat and double-hole. Moreover, the types of bright solitons for the Sasa-Satsuma equation can be changed due to collision.

Paper Structure

This paper contains 15 sections, 12 theorems, 124 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Bright-bright soliton solution to the coupled Hirota equation
  3. Derivation of bright-bright soliton solution to the coupled Hirota equation
  4. Dynamics of the bright-bright soliton solution to the coupled Hirota equation
  5. Dark-Dark soliton solution to the coupled Hirota equation
  6. Derivation of dark-dark soliton solution to the coupled Hirota equation
  7. Dynamics of dark-dark soliton solution
  8. Bright-dark soliton solution to the coupled Hirota equation
  9. Derivation of dark-dark soliton solution to the coupled Hirota equation
  10. Dynamics of the bright-dark soliton solution
  11. Bright soliton solution to the Sasa-Satsuma equation
  12. Dynamics of the bright soliton solution to the Sasa-Satsuma equation
  13. Dark soliton solution to the Sasa-Satsuma equation
  14. Dynamics of the dark soliton solution to the Sasa-Satsuma equation
  15. Conclusion

Key Result

Theorem 2.1

Equation chirota_1-chirota_2 admits the bright soliton solutions given by $u_1=g_1/f, \ u_2=g_2/f$ with $f,\ g_1, \ g_2$ defined as where $M$ is $N\times N$ matrix, $\Phi$, $\bar{\Psi}$, are $N$-component vectors whose elements are defined respectively as Here, $p_i$, $\xi_{i0}$, $C_i$, $D_i$ are complex parameters.

Figures (13)

  • Figure 1: One-bright soliton solution to the cHirota equation with parameters $p_1=1+\mathrm{i}, C_1=1, D_1=2, \xi_{1,0} = 0, \varepsilon_1 = \varepsilon_2 = -1$.
  • Figure 2: Two-bright soliton solution to the cHirota equation with parameters $p_1=1+\mathrm{i} /\sqrt{5}, p_2 = 1+\mathrm{i} /\sqrt{10}, C_1 = 2, D_1 = 1, C_2 = 1, D_2 = 2, \xi_{1,0} = \xi_{2,0} = 0, \varepsilon_1 = \varepsilon_2 = -1$.
  • Figure 3: Y-shaped bright soliton solution to the cHirota equation with parameters $p_1=1-\mathrm{i} /\sqrt{5}, p_2 = 1+\mathrm{i} /\sqrt{10}, C_1 = 1, D_1 = 0, C_2 = 1, D_2 = 1, \xi_{1,0} = \xi_{2,0} = 0, \varepsilon_1 = \varepsilon_2 = -1$.
  • Figure 4: One-dark soliton solution to the cHirota equation with parameters $p_1= \sqrt{\left(\sqrt{113}+9\right)/2} + \mathrm{i}, d_1 = 1, \alpha_1 = 2, \alpha_2 =1, \rho_1 = 1, \rho_2 = 2, \xi_{1,0} = 0, \varepsilon_1 = \varepsilon_2 = 1$.
  • Figure 5: One-breather solution to the cHirota equation with parameters $p_1= 1 + \mathrm{i}, p_2 = \left(1/4-\mathrm{i}/4\right) \left(-8-\mathrm{i}+\sqrt{-5+36 \mathrm{i}}\right), d_1 = 1, \alpha_1 = 2, \alpha_2 =1, \rho_1 = 1, \rho_2 = 2, \xi_{1,0} = \xi_{2,0} = 0, \varepsilon_1 = \varepsilon_2 = -1$.
  • ...and 8 more figures

Theorems & Definitions (20)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 10 more