## 约化的(3+1)维Hirota方程的呼吸波解、lump解和半有理解

1 集宁师范学院数学与统计学院 内蒙古乌兰察布 012000

2 中国矿业大学数学学院 江苏徐州 221116

## Breather Wave Solutions, Lump Solutions and Semi-Rational Solutions of a Reduced (3+1)Dimensional Hirota Equation

Fang Chunmei,1, Tian Shoufu,2

1 Department of Mathematics and Statistics, Jining Normal University, Inner Mongolia Ulanqab 012000

2 Department of Mathematics, China University of Mining and Technology, Jiangsu Xuzhou 221116

 基金资助: 国家自然科学基金.  11975306内蒙古自治区高等学校科学研究项目.  NJZY20248内蒙古自治区高等学校科学研究项目.  NJZY22307

 Fund supported: the NSFC.  11975306the Higher Educational Scientific Research Projects of Inner Mongolia Autonomous Region.  NJZY20248the Higher Educational Scientific Research Projects of Inner Mongolia Autonomous Region.  NJZY22307

Abstract

In this paper, the long wave limit method is used to study the exact solutions of the (3+1)dimensional Hirota equation under dimensional reduction $z$=$x$. First, the bilinear form is constructed by using Bell polynomials. Based on the bilinear form, the $n$-order breather wave solutions are obtained under some parameter constraints on the $N$-order soliton solution. Secondly, by using the long wave limit method, high order lump wave solutions are obtained. Finally, the combined solutions of the first-order, second-order lump wave solutions and single solitary wave solutions are derived, i.e. semi-rational solutions. All the obtained solutions were analyzed with Maple software for physical characteristics.

Keywords： The (3+1)dimensional Hirota equation ; Bilinear representation ; Breather solutions ; Lump wave solutions ; Semi-rational solutions

Fang Chunmei, Tian Shoufu. Breather Wave Solutions, Lump Solutions and Semi-Rational Solutions of a Reduced (3+1)Dimensional Hirota Equation. Acta Mathematica Scientia[J], 2022, 42(3): 775-783 doi:

## 1 引言

\begin{align} u_{ty}-u_{xxxy}-3(u_{x}u_{y})_{x}-3u_{xx}+3u_{zz}=0, \end{align}

$$$N=2n, \quad\mu_{j}^{*}=\mu_{j+1}, \quad\nu_{j}^{*}=\mu_{j+1}, \quad\eta_{j}^{*}=\eta_{j+1}.$$$

$$$\gamma=\frac{1}{3}, \quad\mu_{1}^{*}=\mu_{2}=-{\rm i}, \quad\nu_{1}^{*}=\mu_{2}=0.5-0.5{\rm i}, \quad\eta_{1}^{0}=\eta_{2}^{0},$$$

$\begin{eqnarray} f_{2}&=&1+2\sinh(3t-0.5y)\cos(-x-0.5y+4t){}\\ &&+2\cosh(0.5y+3t)\cos(-x-0.5y+4t)- 2\sinh(6t+y)+2\cosh(6t+y). \end{eqnarray}$

### 4 Lump波解

$$$u=\gamma y+3(\ln f_{N})_{x},$$$

$\begin{eqnarray} f_{N}&=&\prod\limits_{i=1}^{N}\theta_{i}+\frac{1}{2}\sum\limits_{i, j}^{N}A_{ij}\prod\limits_{\rho\ne i, j}^{N}\theta_{\rho}+\cdots{}\\ &&+\frac{1}{M!2^{M}}\times\sum\limits_{i, j, \cdots, m, l}^{N} \mathop{ \overbrace{A_{ij}A_{k\rho}\cdots A_{ml}}}\limits^M \times\prod\limits_{q\ne i, j, \cdots, m, l}\theta_{q}+\cdots, \end{eqnarray}$

$$$\theta_{i}=x+\frac{3\gamma}{\nu_{i}}t+\nu_{i}y, \quad A_{ij}=\frac{2\nu_{i}\nu_{j}(\nu_{i}+\nu_{j})}{\gamma(\nu_{i}-\nu_{j})^2}, \quad \gamma\ne0.$$$

$$$f=(\theta_{1}\theta_{2}+A_{12})+(\theta_{1}\theta_{2}+A_{12}+A_{13}\theta_{2}+A_{23}\theta_{1}+A_{12}A_{23})e^{\eta_{3}},$$$

$$$A_{i3}=\frac{2\nu_{i}\nu_{3}\mu_{3}(\nu_{i}+\nu_{3})}{-\nu_{i}\mu_{3}^4+\gamma(\nu_{i}-\nu_{3})^2}, \quad i=1, 2,$$$

$\eta_{3}$以及$\theta_{1}, \theta_{2}, A_{12}$分别由(3.3)式, (4.5)式给出. 令

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Hirota R . The Direct Method in Soliton Theory. Cambridge: Cambridge University Press, 2004

Lou S Y , Tang X Y . Nonlinear Mathematical Physics Methods. Beijing: Science Press, 2006

Hao X H , Cheng Z L .

The integrability of the KdV-shallow water wave equation

Acta Math Sci, 2019, 39A (3): 451- 460

Wazwaz A M .

Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities

Nonlinear Dyn, 2016, 83, 591- 596

Ginzburg N S , Rozental R M , Sergeev A S , et al.

Generation of rogue waves in gyrotrons operating in the regime of developed turbulence

Phys Rev Lett, 2017, 119 (3): 034801

Onorato M , Residori S , Bortolozzo U , et al.

Rogue waves and their generating mechanisms in different physical contexts

Phys Rep, 2013, 528 (2): 47- 89

Rao J G , Zhang Y S , Athanassios S F , He J S .

Rogue waves of the nonlocal Davey-Stewartson I equation

Nonlinearity, 2018, 31, 4090- 4107

Ma W X , Yong X L , X .

Soliton solutions to the B-type Kadomtsev-Petviashvili equation under general dispersion relations

Wave Motion, 2021, 103, 102719

Ma W X .

N-soliton solution of a combined pKP-BKP equation

J Geom Phys, 2021, 165, 104191

Ma W X .

N-soliton solutions and the Hirota conditionsin (1+1)-dimensions

Int J Nonlinear Sci Numer Simul, 2021,

Ma W X .

N-soliton solutions and the Hirota conditions in (2+1)-dimensions

Opt Quantum Electron, 2020, 52, 511

Ma W X .

N-soliton solution and the Hirota condition of a (2+1)-dimensional combined equation

Math Comput Simulat, 2021, 190 (C): 270- 279

Ma W X .

Lump solutions to the Kadomtsev-Petviashvili equation

Phys Lett A, 2015, 379 (36): 1975- 1978

Ma W X , Qin Z Y , X .

Lump solutions to dimensionally reduced p-gKP and p-gBKP equations

Nonlinear Dyn, 2016, 84, 923- 931

Tian S F .

Asymptotic behavior of a weakly dissipative modified two-component Dullin-Gottwald-Holm system

Appl Math Lett, 2018, 83, 65- 72

Ablowitz M J , Satsuma J .

Solitons and rational solutions of nonlinear evolution equations

J Math Phys, 1978, 19 (10): 2180- 2186

Satsuma J , Ablowitz M J .

Two-dimensional lumps in nonlinear dispersive systems

J Math Phys, 1979, 20, 1496- 1503

Huang L L , Chen Y .

Lump solutions and interaction phenomenon for (2+1)-dimensional Sawada-Kotera equation

Commun Theor Phys, 2017, 67, 473- 478

Tian S F , Zhang T T .

Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrodinger equation with time-periodic boundary condition

Proc Am Math Soc, 2018, 146 (4): 1713- 1729

Qian C , Rao J G , Liu Y B , He J S .

Rogue waves in the three-dimensional Kadomtsev-Petviashvili equation

Chin Phys Lett, 2016, 33 (11): 110201

Peng W Q , Tian S F , Zhang T T .

Analysis on lump, lumpoff and rogue waves with predictability to the (2+1)-dimensional B-type Kadomtsev-Petviashvili equation

Phys Lett A, 2018, 382, 2701- 2708

Liu Y K , Li B .

Dynamics of rogue waves on multi-soliton background in the Benjamin Ono equation

Pramana, 2017, 88 (4): 57

Rao J G , Fokas A S , He J S .

Doubly localized two-dimensional rogue waves in the Davey-Stewartson I equation

J Nonlinear Sci, 2021, 31, 67

Rao J G , Chow K W , Mihalache D , He J S .

Completely resonant collision of lumps and line solitons in the Kadomtsev-Petviashvili I equation

Stud Appl Math, 2021, 147 (3): 1007- 1035

Rao J G , He J S , Mihalache D .

Doubly localized rogue waves on a background of dark solitons for the Fokas system

Appl Math Lett, 2021, 121, 107435

Rao J G , Cheng Y , He J S .

Rational and semi-rational solutions of the nonlocal Davey-Stewartson equations

Stud Appl Math, 2017, 139 (4): 568- 598

Gao L N , Zhao X Y , Zi Y Y .

Resonant behavior of multiple wave solutions to a Hirota bilinear equation

Comput Math Appl, 2016, 72 (5): 1225- 1229

Dong M J , Tian S F , Yan X W , Zou L .

Solitary waves, homoclinic breather waves and rogue waves of the (3+1)-dimensional Hirota bilinear equation

Comput Math Appl, 2018, 75 (3): 957- 964

X , Ma W X .

Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation

Nonlinear Dyn, 2016, 85 (2): 1217- 1222

Wang C .

Lump solution and integrability for the associated Hirota bilinear equation

Nonlinear Dyn, 2017, 87 (4): 2635- 2642

/

 〈 〉