Breather Solutions and Their Transformation Mechanisms of the (2 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation

Xingqi Xiao

College of Science, University of Shanghai for Science and Technology, Shanghai, China.

DOI: 10.4236/jamp.2025.133045                

Abstract

In this paper, we investigate the transformation phenomenon of the first-order breather and the second-order breather. First, we obtain the 2-solitary wave and 4-solitary wave solutions of the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation by using the Hirota bilinear method. Then by analyzing and controlling characteristic lines, we derive several types of nonlinear transformantion waves and give the corresponding figures.

Keywords

Hirota Bilinear Method, The (2 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation, Characteristic Line, Breather

1. Introduction

With the development of technology, linear equations are often insufficient in fully capturing the complexity and intricacies of natural phenomena. As a result, nonlinear partial differential equations (NLPDEs) have emerged as essential models for describing intricate behaviors in nature. The study of soliton equation is an important direction dedicated to uncovering the nonlinear phenomena, which originates from many fields such as applied physics, life sciences, oceanography and other fields [1] [2] [3] [4].

Recently, the study of solitary wave solutions of the soliton equation has been a focus and many methods have been developed, such as the Darboux transform method [5] [6], the inverse scattering method [7] [8], the Hirota bilinear method [9] [10] [11] [12], tanh-function method [13], ansatz method [14] and other approaches [15]. The Hirota method used in this paper is a powerful and effective tool to investigate the exact solutions of NLPDEs. The (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation was initially derived through the study of the weak Lax pair associated with the KdV equation [16]. Originally, this equation was introduced to describe nonlinear wave phenomena and shock wave propagation in incompressible fluids. Beyond fluid mechanics, the BLMP equation has also been widely applied in wave analysis within optical fiber communications, plasma physics, and solid mechanics. Many studies have been carried out on solving the exact solutions of the (2 + 1) and (3 + 1)-dimensional BLMP equations and structures of their solutions [17] [18] [19] [20] [21] by using different methods.

In this paper, we study the (2 + 1)-dimensional BLMP equation

$u_{xxxy}+u_{yt}-3u_{xx}{u}_y-3u_x{u}_{xy}=0,$ (1.1)

where $u=u\left(x,y,t\right)$ is a function of $x$ , $y$ and $t$ . Through the transformation [22] with a parameter $u_0$

$u=-2{\left[\ln f\right]}_x+u_{0}y,$ (1.2)

the (2 + 1)-dimensional BLMP equation can be rewritten as the following Hirota bilinear form

$\left(D_y{D}_t+D_y{D}_x^3-3u_0{D}_x^2\right)f\cdot f=0,$ (1.3)

where

$\begin{array}{l}{D}_x^m{D}_y^n{D}_t^{p}f\cdot g\\ ={\left({\partial }_x-{\partial }_{x^{\prime }}\right)}^m{\left({\partial }_y-{\partial }_{y^{\prime }}\right)}^n{\left({\partial }_t-{\partial }_{t^{\prime }}\right)}^{p}f\left(x,y,t\right)g{\left(x^{\prime },y^{\prime },t^{\prime }\right)|}_{x^{\prime }=x,y^{\prime }=y,t^{\prime }=t}.\end{array}$ (1.4)

For soliton equations, exact solutions include solitary wave solutions, breather solutions, lump solutions [23] [24], etc. A solitary wave that exhibits periodic oscillations within a certain region, resembling a breathing motion, is defined as a breather [25]. Some researchers have discovered that transformations can occur between different types of nonlinear waves under certain conditions. Yin in [26] investigated the transitions and mechanisms of nonlinear waves in the (2 + 1)-dimensional Sawada-Kotera equation using characteristic lines and phase shift analysis, focusing on the dynamics of transformed waves. Chowdury [27] analyzed a quintic integrable equation of the nonlinear Schrödinger hierarchy, demonstrating how a breather solution can transform into a non-pulsating soliton solution.

In this paper, starting with an examination of relative positions of the characteristic lines, we consider the conditions under which the first-order and second-order breather solutions of the BLMP Equation (1.1) can transformed into other nonlinear waves and explore their localization and oscillatory behavior. According to the Hirota bilinear method, we obtain the 2-solitary wave and the 4-solitary wave solution [28]

$u_2=-2{\left[\ln f_2\right]}_x+u_{0}y,f_2=1+{\text{e}}^{{\eta }_1}+{\text{e}}^{{\eta }_2}+A_{12}{\text{e}}^{{\eta }_1+{\eta }_2},$ (1.5)

$u_4=-2{\left[\ln f_4\right]}_x+u_{0}y,f_4={\displaystyle \sum }_{\kappa =0,1}\exp \left({\displaystyle \sum }_{i=1}^4{\kappa }_i{\eta }_i+{\displaystyle \sum }_{ij}^4{\kappa }_i{\kappa }_j{A}_{ij}\right),$ (1.6)

where

${\eta }_i=k_{i}x+p_{i}y+{\omega }_{i}t+{\varphi }_i,{\omega }_i=\frac{3u_0{k}_i^2}{p_i}-k_i^3,i=1,2,3,4,$ (1.7)

${\text{e}}^{A_{ij}}=\frac{u_0{\left(k_i{p}_j-k_j{p}_i\right)}^2+k_i{k}_j{p}_i{p}_j\left(k_i-k_j\right)\left(p_i-p_j\right)}{u_0{\left(k_i{p}_j-k_j{p}_i\right)}^2+k_i{k}_j{p}_i{p}_j\left(k_i+k_j\right)\left(p_i+p_j\right)},1\le i<j\le 4,$ (1.8)

and $k_i,p_i,{\varphi }_i$ are arbitrary constants, $u_0$ is a nonzero constant, ${{\displaystyle \sum }}_{\kappa =0,1}$ is the summation with all possible combinations of ${\kappa }_i$ , ${\kappa }_j=0,1$ .

The structure of this paper is as follows. In Section 2, we first provide a detailed expression for the first-order breather and obtain the corresponding two characteristic lines. Various nonlinear transformed waves are obtained by controlling the relationships between the characteristic lines and adjusting the parameters. In Section 3, three types of transformation phenomena including non-transformed, semi-transformed and full-transformed are discussed. In Section 4, we present some conclusions.

2. State Transformation of the First-Order Breather Solution

In this section, starting with the 2-solitary wave solution and by taking the complex conjugate of the parameters, we derive the first-order breather solution of the (2 + 1)-dimensional BLMP Equation (1.1) and analyze its transformation mechanism. Firstly, taking

$k_1={\alpha }_1+\text{i}{\beta }_1=k_2^{\text{*}},p_1={\nu }_1+\text{i}{\sigma }_1=p_2^{\text{*}},{\varphi }_1={\xi }_1+\text{i}{\mu }_1={\varphi }_2^{\text{*}},$ (2.1)

in $f_2$ defined by (1.5), where $\ast$ denotes the complex conjugate, ${\alpha }_1$ , ${\beta }_1$ , ${\nu }_1$ , ${\sigma }_1$ , ${\xi }_1$ and ${\mu }_1$ are arbitrary real constants. Then $f_2$ can be rewritten as

$f_2=2{\text{e}}^{{\Lambda }_1}\left[\sqrt{M_1}\cosh \left({\Lambda }_1+\ln \sqrt{M_1}\right)+\cos {\Gamma }_1\right],$ (2.2)

where

${\Lambda }_1={\alpha }_{1}x+{\nu }_{1}y+{\omega }_{1R}t+{\xi }_1,{\Gamma }_1={\beta }_{1}x+{\sigma }_{1}y+{\omega }_{1I}t+{\mu }_1,$ (2.3)

$M_1=\frac{u_0{\left({\beta }_1{\nu }_1-{\alpha }_1{\sigma }_1\right)}^2+{\beta }_1{\sigma }_1\left({\alpha }_1^2+{\beta }_1^2\right)\left({\sigma }_1^2+{\nu }_1^2\right)}{u_0{\left({\beta }_1{\nu }_1-{\alpha }_1{\sigma }_1\right)}^2-{\alpha }_1{\nu }_1\left({\alpha }_1^2+{\beta }_1^2\right)\left({\sigma }_1^2+{\nu }_1^2\right)},$ (2.4)

${\omega }_{1R}=\frac{6{\alpha }_1{\beta }_1{\sigma }_1{u}_0+3{\alpha }_1^2{u}_0{\nu }_1-3{\beta }_1^2{u}_0{\nu }_1+3{\alpha }_1{\beta }_1^2{\nu }_1^2+3{\alpha }_1{\beta }_1^2{\sigma }_1^2-{\alpha }_1^3{\nu }_1^2-{\alpha }_1^3{\sigma }_1^2}{{\sigma }_1^2+{\nu }_1^2},$ (2.5)

${\omega }_{1I}=\frac{6{\alpha }_1{\beta }_1{\nu }_1{u}_0-3{\alpha }_1^2{u}_0{\sigma }_1+3{\beta }_1^2{u}_0{\sigma }_1-3{\alpha }_1^2{\beta }_1{\nu }_1^2-3{\alpha }_1^2{\beta }_1{\sigma }_1^2+{\beta }_1^3{\nu }_1^2+{\beta }_1^3{\sigma }_1^2}{{\sigma }_1^2+{\nu }_1^2}.$ (2.6)

The ${\omega }_{1R}$ and ${\omega }_{1I}$ denote the real and imaginary parts of ${\omega }_1$ respectively. By substituting (2.2) into $u_2$ , which is defined by (1.5), the first-order breather solution takes the form

$u_2=\frac{-2{\alpha }_1\sqrt{M_1}\sinh \left({\Lambda }_1+\ln \sqrt{M_1}\right)+2{\beta }_1\sin {\Gamma }_1}{\sqrt{M_1}\cosh \left({\Lambda }_1+\ln \sqrt{M_1}\right)+\cos {\Gamma }_1}-2{\alpha }_1+u_{0}y.$ (2.7)

From the above equation, a singularity arises in the solution when the denominator equals zero. Therefore, we require that $M_1>1$ in all cases. Furthermore, by the properties of hyperbolic functions, the extreme values of the equation move along the line ${\Lambda }_1+\ln \sqrt{M_1}=0$ .

For the first-order breather solution 2.7), we take the parameters

${\alpha }_1=3,{\beta }_1=2,{\nu }_1=1,{\sigma }_1=-2,{\xi }_1=1,{\mu }_1=1,u_0=1.$ (2.8)

The graph shows that the distance between adjacent peaks is uniform. In each periodic module, the single breather waveform has a main peak and two valleys, resembling a single lump. Therefore, it can also be viewed as an arrangement of multiple lump waves. In addition, the movement of the breather wave along the

line $3x+y-\frac{12}{5}t+1+\frac{1}{2}\text{ln}\frac{196}{131}=0$ is periodic in the propagation direction and decays exponentially in the perpendicular direction.

By analyzing the solution, we can draw the following conclusions about the first-order breather: the first-order breather solution (15) consists of hyperbolic and trigonometric functions from which some properties of the solution can be determined. The hyperbolic functions $\sinh \left({\Lambda }_1+\ln \sqrt{M_1}\right)$ and $\cosh \left({\Lambda }_1+\ln \sqrt{M_1}\right)$ determine the local properties of the breather, while its periodicity is determined by the trigonometric function $\sin {\Gamma }_1$ and $\cos {\Gamma }_1$ . Consequently, the nonlinear wave structure can be viewed as a superposition of two wave solutions: a solitary wave governed by ${\text{Λ}}_1$ and a periodic wave governed by ${\text{Γ}}_1$ .

From the first-order breather solution (2.7), we notice that it has two key characteristic lines $L_1$ and $L_2$ :

$L_1:{\alpha }_{1}x+{\nu }_{1}y+{\omega }_{1R}t+\ln \sqrt{M_1}+{\xi }_1=0,$ (2.9)

$L_2:{\beta }_{1}x+{\sigma }_{1}y+{\omega }_{1I}t+{\mu }_1=0,$ (2.10)

where the characteristic line $L_1$ determines the direction of movement. Then, by adjusting the parameter ${\alpha }_1$ , ${\beta }_1$ , ${\nu }_1$ , ${\sigma }_1$ , ${\xi }_1$ and ${\mu }_1$ , we consider the phenomena arising from the transformation of the first-order breather state when the two characteristic lines are parallel or non-parallel in the $xOt$ plane.

The characteristic lines $L_1$ and $L_2$ are parallel in the $xOt$ plane if and only if

$\frac{{\alpha }_1}{{\beta }_1}=\frac{{\omega }_{1R}}{{\omega }_{1I}}\iff \left|\begin{array}{cc}{\alpha }_1& {\omega }_{1R}\\ {\beta }_1& {\omega }_{1I}\end{array}\right|=0.$ (2.11)

In the following, we will explore the transformation mechanism of the first-order breather based on the above conditions. It is important to emphasize that ensuring the non-singularity of the solution is a critical requirement, and it is necessary to require $M_1>1$ in all cases.

(i) If the relation between $L_1$ and $L_2$ satisfies $\left|\begin{array}{cc}{\alpha }_1& {\omega }_{1R}\\ {\beta }_1& {\omega }_{1I}\end{array}\right|\ne 0$ , i.e., $L_1$ and $L_2$ are not parallel in $xOt$ plane, we can obtain the first-order breather where no transformation occurs in Figure 1.

Figure 1. The first-order breather (2.7) with (2.8) at $y=0$ .

(ii) If the relation between $L_1$ and $L_2$ satisfies $\left|\begin{array}{cc}{\alpha }_1& {\omega }_{1R}\\ {\beta }_1& {\omega }_{1I}\end{array}\right|=0$ , i.e., $L_1$ and $L_2$ are parallel in $xOt$ plane. In this situation, we have

${\beta }_1=\frac{3u_0{\alpha }_1{\sigma }_1}{3u_0{\nu }_1-2{\nu }_1^2{\alpha }_1-2{\alpha }_1{\sigma }_1^2},$ (2.12)

and the first-order breather can be transformed into various nonlinear waves by choosing different parameter relations.

Taking parameters with

$u_0=1,{\alpha }_1=1,{\nu }_1=1,{\beta }_1=\frac{9}{17},{\sigma }_1=-3,{\xi }_1=1,{\mu }_1=1,$ (2.13)

in (2.7) as shown in Figure 2(a) and Figure 2(b) where $\left|\frac{{\beta }_1}{{\alpha }_1}\right|=\left|\frac{{\omega }_{1I}}{{\omega }_{1R}}\right|=\frac{9}{17}$ . The figure illustrates a kink solitary wave converted from the first-order breather. In this case, the solution maintains the characteristics of a solitary wave, without exhibiting oscillatory or periodic behavior.

Taking parameters with

$u_0=1,{\alpha }_1=\frac{7}{10},{\nu }_1=2,{\beta }_1=\frac{21}{26},{\sigma }_1=-2,{\xi }_1=1,{\mu }_1=1,$ (2.14)

in the (2.7) as shown in Figure 2(c) and Figure 2(d) where $\left|\frac{{\beta }_1}{{\alpha }_1}\right|=\left|\frac{{\omega }_{1I}}{{\omega }_{1R}}\right|=\frac{15}{13}$ . We

Figure 2. Transformation waves of the first-order breather (2.7) at $y=0$ .

discover that the first-order breather can be transformed into m-shaped kink solitary wave in Figure 2(c), which possesses two different wave peaks. In this paper, this structure is called m-kink type solitary wave.

Taking parameters with

$u_0=1,{\alpha }_1=\frac{1}{2},{\nu }_1=2,{\beta }_1=\frac{3}{2},{\sigma }_1=-2,{\xi }_1=1,{\mu }_1=1,$ (2.15)

in the (2.7) as shown in Figure 2(e) and Figure 2(f) where $\left|\frac{{\beta }_1}{{\alpha }_1}\right|=\left|\frac{{\omega }_{1I}}{{\omega }_{1R}}\right|=3$ . The figure shows a multi-kink type solitary wave phenomenon with multiple peaks, where the periodicity becomes more apparent and the solitary wave characteristics weaken.

Taking parameters with

$u_0=\frac{4}{63},{\alpha }_1=\frac{1}{20},{\nu }_1=1,{\beta }_1=1,{\sigma }_1=-1,{\xi }_1=1,{\mu }_1=1,$ (2.16)

in the (2.7) as shown in Figure 2(g) and Figure 2(h) where $\left|\frac{{\beta }_1}{{\alpha }_1}\right|=\left|\frac{{\omega }_{1I}}{{\omega }_{1R}}\right|=20$ . From the figure, it can be seen that the oscillation of wave is more apparent, and the peak value decreases. The first-order breather will be transformed into a periodic wave. As the ratio of $\left|\frac{{\beta }_1}{{\alpha }_1}\right|$ increases gradually, the periodicity of the breather becomes more and more obvious, and the solitary wave property almost disappears.

From the analysis above, we can observe that the solution (2.7) exhibits either dominant solitary wave characteristics with weak periodicity or dominant periodicity with weak solitary wave characteristics. Therefore, the first-order breather can be interpreted as a nonlinear superposition of solitary and periodic waves. When the velocities of the two components are identical, the breather may transition into different types of nonlinear waves.

3. State Transformation of the Second-Order Breather Solution

In this section, we investigate the conversion mechanism of the second-order breather derived from $f_4$ defined by (1.6). Similarly, we select conjugate parameters to derive the second-order breather, and under specific parameter constraints, the breathers undergo partial or complete transformation.

The parameters $k_1$ , $k_2$ , $p_1$ , $p_2$ , ${\varphi }_1$ and ${\varphi }_2$ defined by (2.1) and we take

$k_3={\alpha }_2+\text{i}{\beta }_2=k_4^{\text{*}},p_3={\nu }_2+\text{i}{\sigma }_2=p_4^{\text{*}},{\varphi }_3={\xi }_2+\text{i}{\mu }_2={\varphi }_4^{\text{*}},$ (3.1)

in $f_4$ defined by (1.6), where ${\alpha }_2$ , ${\beta }_2$ , ${\nu }_2$ , ${\sigma }_2$ , ${\xi }_2$ and ${\mu }_2$ are arbitrary real constants.

Then $f_4$ can be rewritten as

$\begin{array}{c}{f}_4=1+2{\text{e}}^{{\text{Λ}}_1}\cos {\Gamma }_1+2{\text{e}}^{{\text{Λ}}_2}\cos {\Gamma }_2+M_1{\text{e}}^{2{\text{Λ}}_1}+M_2{\text{e}}^{2{\text{Λ}}_2}+M_1{M}_2{\text{e}}^{2\left({\text{Λ}}_1+{\text{Λ}}_2+b_1+b_2\right)}\\ +2{\text{e}}^{{\text{Λ}}_1+{\text{Λ}}_2+b_1}\cos \left({\text{Γ}}_1+{\text{Γ}}_2+a_1\right)+2{\text{e}}^{{\text{Λ}}_1+{\text{Λ}}_2+b_2}\cos \left({\text{Γ}}_1-{\text{Γ}}_2+a_2\right)\\ +2M_1{\text{e}}^{2{\text{Λ}}_1+{\text{Λ}}_2+b_1+b_2}\cos \left({\text{Γ}}_2+a_1-a_2\right)+2M_2{\text{e}}^{{\text{Λ}}_1+2{\text{Λ}}_2+b_1+b_2}\cos \left({\text{Γ}}_1+a_1+a_2\right),\end{array}$ (3.2)

where ${\text{Λ}}_1$ , ${\text{Γ}}_1$ and $M_1$ are defined by (2.3)-(3.6), in addition

$b_1=\ln \sqrt{{\left(\mathrm{Re}{A}_{13}\right)}^2+{\left(\mathrm{Im}{A}_{13}\right)}^2},a_1=\arctan \frac{\mathrm{Im}{A}_{13}}{\mathrm{Re}{A}_{13}},$ (3.3)

$b_2=\ln \sqrt{{\left(\mathrm{Re}{A}_{14}\right)}^2+{\left(\mathrm{Im}{A}_{14}\right)}^2},a_2=\arctan \frac{\mathrm{Im}{A}_{14}}{\mathrm{Re}{A}_{14}},$ (3.4)

$M_2=\frac{u_0{\left({\beta }_2{\nu }_2-{\alpha }_2{\sigma }_2\right)}^2+{\beta }_2{\sigma }_2\left({\alpha }_2^2+{\beta }_2^2\right)\left({\sigma }_2^2+{\nu }_2^2\right)}{u_0{\left({\beta }_2{\nu }_2-{\alpha }_2{\sigma }_2\right)}^2-{\alpha }_2{\nu }_2\left({\alpha }_2^2+{\beta }_2^2\right)\left({\sigma }_2^2+{\nu }_2^2\right)},$ (3.5)

${\text{Λ}}_2={\alpha }_{2}x+{\nu }_{2}y+{\omega }_{2R}t+{\xi }_2,{\text{Γ}}_2={\beta }_{2}x+{\sigma }_{2}y+{\omega }_{2I}t+{\mu }_2,$ (3.6)

${\omega }_{2R}=\frac{6{\alpha }_2{\beta }_2{\sigma }_2{u}_0+3{\alpha }_2^2{u}_0{\nu }_2-3{\beta }_2^2{u}_0{\nu }_2+3{\alpha }_2{\beta }_2^2{\nu }_2^2+3{\alpha }_2{\beta }_2^2{\sigma }_2^2-{\alpha }_2^3{\nu }_2^2-{\alpha }_2^3{\sigma }_2^2}{{\sigma }_2^2+{\nu }_2^2},$ (3.7)

${\omega }_{2I}=\frac{6{\alpha }_2{\beta }_2{\nu }_2{u}_0-3{\alpha }_2^2{u}_0{\sigma }_2+3{\beta }_2^2{u}_0{\sigma }_2-3{\alpha }_2^2{\beta }_2{\nu }_2^2-3{\alpha }_2^2{\beta }_2{\sigma }_2^2+{\beta }_2^3{\nu }_2^2+{\beta }_2^3{\sigma }_2^2}{{\sigma }_2^2+{\nu }_2^2}.$ (3.8)

We can obtain the second-order breather solution by combining the transformations (1.2). Then we analyze its asymptotic properties in $xOt$ plane to understand the nature of this solution. Without loss of generality, we assume

${\alpha }_1>0,{\alpha }_2>0,\frac{{\omega }_{1R}}{{\alpha }_1}>\frac{{\omega }_{2R}}{{\alpha }_2}.$ (3.9)

Before collision ($t\to -\infty$ ):

(a) Fixing ${\text{Λ}}_1$ , one can obtain ${\text{Λ}}_2\to +\infty$ , then

$f_4=2{\text{e}}^{{\text{Λ}}_1+b_1+b_2}\left[\sqrt{M_1}\cosh \left({\Lambda }_1+b_1+b_2+\ln M_1\right)+\cos \left({\Gamma }_1+a_1+a_2\right)\right],$ (3.10)

and

$\begin{array}{l}u~u_{{\text{Λ}}_1}^{-},u_{{\text{Λ}}_1}^{-}\\ =\frac{-2{\alpha }_1\sqrt{M_1}\sinh \left({\Lambda }_1+b_1+b_2+\ln \sqrt{M_1}\right)+2{\beta }_1\sin \left({\Gamma }_1+a_1+a_2\right)}{\sqrt{M_1}\cosh \left({\Lambda }_1+b_1+b_2+\ln \sqrt{M_1}\right)+\cos \left({\Gamma }_1+a_1+a_2\right)}-2{\alpha }_1+u_{{}_0}y.\end{array}$ (3.11)

(b) Fixing ${\text{Λ}}_2$ , one can obtain ${\text{Λ}}_1\to -\infty$ , then

$f_4=2{\text{e}}^{{\text{Λ}}_2}\left[\sqrt{M_2}\cosh \left({\Lambda }_2+\ln \sqrt{M_2}\right)+\cos {\Gamma }_2\right],$ (3.12)

and

$u~u_{{\Lambda }_2}^{-},u_{{\Lambda }_2}^{-}=\frac{-2{\alpha }_2\sqrt{M_2}\sinh \left({\Lambda }_2+\ln \sqrt{M_2}\right)+2{\beta }_2\sin {\Gamma }_2}{\sqrt{M_2}\cosh \left({\Lambda }_2+\ln \sqrt{M_2}\right)+\cos {\Gamma }_2}-2{\alpha }_2+u_{0}y.$ (3.13)

After collision ($t\to +\infty$ ):

(c) Fixing ${\text{Λ}}_1$ , one can obtain ${\text{Λ}}_2\to -\infty$ , then

$f_4=2{\text{e}}^{{\text{Λ}}_1}\left[\sqrt{M_1}\cosh \left({\Lambda }_1+\ln \sqrt{M_1}\right)+\cos {\Gamma }_1\right],$ (3.14)

and

$u~u_{{\text{Λ}}_1}^{+},u_{{\text{Λ}}_1}^{+}=\frac{-2{\alpha }_1\sqrt{M_1}\sinh \left({\Lambda }_1+\ln \sqrt{M_1}\right)+2{\beta }_1\sin {\Gamma }_1}{\sqrt{M_1}\cosh \left({\Lambda }_1+\ln \sqrt{M_1}\right)+\cos {\Gamma }_1}-2{\alpha }_1+u_{0}y.$ (3.15)

(d) Fixing ${\text{Λ}}_2$ , one can obtain ${\text{Λ}}_1\to +\infty$ , then

$f_4=2{\text{e}}^{{\text{Λ}}_2+b_1+b_2}\left[\sqrt{M_2}\cosh \left({\Lambda }_2+b_1+b_2+\ln M_2\right)+\cos \left({\Gamma }_2+a_1-a_2\right)\right],$ (3.16)

and

$\begin{array}{l}u~u_{{\Lambda }_2}^{+},u_{{\Lambda }_2}^{+}\\ =\frac{-2{\alpha }_2\sqrt{M_2}\sinh \left({\Lambda }_2+b_1+b_2+\ln \sqrt{M_2}\right)+2{\beta }_2\sin \left({\Gamma }_2+a_1-a_2\right)}{\sqrt{M_2}\cosh \left({\Lambda }_2+b_1+b_2+\ln \sqrt{M_2}\right)+\cos \left({\Gamma }_2+a_1-a_2\right)}-2{\alpha }_2+u_{0}y.\end{array}$ (3.17)

The above analysis allows us to obtain an asymptotic expression for the solution defined by (1.2) and (3.2), namely

$u\to \{\begin{array}{l}{u}_{{\Lambda }_1}^{-}+u_{{\Lambda }_2}^{-},t\to -\infty ,\hfill \\ u_{{\Lambda }_1}^{+}+u_{{\Lambda }_2}^{+},t\to +\infty .\hfill \end{array}$ (3.18)

The second-order breather can be obtained in Figure 3 by taking

$\begin{array}{l}{u}_0=1,{\alpha }_1=\frac{1}{2},{\beta }_1=\frac{1}{2},{\nu }_1=\frac{7}{12},{\sigma }_1=\frac{1}{12},{\xi }_1=1,{\mu }_1=0,\\ {\alpha }_2=\frac{1}{3},{\beta }_2=\frac{1}{2},{\nu }_2=\frac{5}{18},{\sigma }_2=\frac{1}{18},{\xi }_2=1,{\mu }_2=0,\end{array}$ (3.19)

in (1.2) and (3.2). As can be seen from the figure, there is no transformation between breathers. According to the first-order transformation mechanism, the breather can be converted by controlling the characteristic lines. In this paper, the transformation of the second-order breather can be classified into non-transformed, semi-transformed and full-transformed according to the type of transformation.

Figure 3. The second-order breather (1.2) and (3.2) with (3.19) at $y=0$ .

In the following, the transformation of the second-order breather solution of the BLMP Equation (1.1) will be investigated based on the relationship between the two sets of characteristic lines in $xOt$ plane. The second-order breather has two groups of characteristic lines $L_1$ and $L_2$ defined by (2.9) and (2.10) as well as $L_3$ and $L_4$ :

$L_3:{\alpha }_{2}x+{\nu }_{2}y+{\omega }_{2R}t+\text{ln}\sqrt{M_2}+{\xi }_2=0,$ (3.20)

$L_4:{\beta }_{2}x+{\sigma }_{2}y+{\omega }_{2I}t+{\mu }_2=0.$ (3.21)

Case I (Non-transformed): We suppose that $L_1$ and $L_2$ are not parallel, and $L_3$ and $L_4$ are not parallel which means that they satisfy

$\left|\begin{array}{cc}{\alpha }_1& {\omega }_{1R}\\ {\beta }_1& {\omega }_{1I}\end{array}\right|\ne 0,\left|\begin{array}{cc}{\alpha }_2& {\omega }_{2R}\\ {\beta }_2& {\omega }_{2I}\end{array}\right|\ne 0.$ (3.22)

At this stage the two breathers collide elastically and neither transforms, as shown in Figure 3.

Case II (Semi-transformed): We suppose that $L_1$ and $L_2$ are not parallel, and $L_3$ and $L_4$ are parallel which means that they satisfy

$\left|\begin{array}{cc}{\alpha }_1& {\omega }_{1R}\\ {\beta }_1& {\omega }_{1I}\end{array}\right|\ne 0,\left|\begin{array}{cc}{\alpha }_2& {\omega }_{2R}\\ {\beta }_2& {\omega }_{2I}\end{array}\right|=0.$ (3.23)

One of the breathesr does not transform and the other one transforms into other nonlinear wave when the condition in the above is satisfied.

Taking parameters with

$\begin{array}{l}{u}_0=1,{\alpha }_1=1,{\beta }_1=1,{\nu }_1=1,{\sigma }_1=-3,{\xi }_1=1,{\mu }_1=0,\\ {\alpha }_2=1,{\beta }_2=\frac{9}{17},{\nu }_2=1,{\sigma }_2=-3,{\xi }_2=1,{\mu }_2=0,\end{array}$ (3.24)

in (1.2) and (3.2), the plots of the solution are shown in Figure 4(a) and Figure 4(b).

Figure 4. Semi-transformation waves of the second-order breather (1.2) with (3.2) at $y=0$ .

The figures show the interaction of a breather and a kink solitary wave, describing the phenomenon that one of two breathers is a non-transformed wave and the other is transformed into a single kink solitary wave.

Taking parameters with

$\begin{array}{l}{u}_0=1,{\alpha }_1=1,{\beta }_1=\frac{9}{7},{\nu }_1=1,{\sigma }_1=-3,{\xi }_1=1,{\mu }_1=0,\\ {\alpha }_2=\frac{7}{10},{\beta }_2=\frac{21}{26},{\nu }_2=2,{\sigma }_2=-2,{\xi }_2=1,{\mu }_2=0,\end{array}$ (3.25)

in (1.2) and (3.2), the plots of the solution are illustrated in Figure 4(c) and Figure 4(d). The figures represent the elastic interaction between the breather and the m-type kink solitary wave, and the phase shift of the two kink solitary waves is relatively obvious.

Taking parameters with

$\begin{array}{l}{u}_0=1,{\alpha }_1=1,{\beta }_1=1,{\nu }_1=1,{\sigma }_1=-3,{\xi }_1=1,{\mu }_1=0,\\ {\alpha }_2=\frac{1}{2},{\beta }_2=\frac{3}{2},{\nu }_2=2,{\sigma }_2=-2,{\xi }_2=1,{\mu }_2=0,\end{array}$ (3.26)

in (1.2) and (3.2), the plots of the solution are displayed in Figure 4(e) and Figure 4(f). The figures demonstrate the interaction of a breather with a multi-kink solitary wave, namely, one breather does not transform and the other transforms into a multi-peak nonlinear wave.

Taking parameters with

$\begin{array}{l}{u}_0=\frac{4}{33},{\alpha }_1=\frac{1}{10},{\beta }_1=1,{\nu }_1=1,{\sigma }_1=-1,{\xi }_1=1,{\mu }_1=0,\\ {\alpha }_2=\frac{1}{10},{\beta }_2=1,{\nu }_2=1,{\sigma }_2=-2,{\xi }_2=1,{\mu }_2=0,\end{array}$ (3.27)

in (1.2) and (3.2), the plots of the solution are shown in Figure 4(g) and Figure 4(h). The figures show the phenomenon of the interaction between a breather and a periodic wave, and we can learn that by controlling the parameters we can make the periodicity of the transformed nonlinear wave more and more apparent.

Case III (Full-transformed): Supposing that $L_1$ and $L_2$ are parallel, as well as $L_3$ and $L_4$ are parallel which means that they satisfy

$\left|\begin{array}{cc}{\alpha }_1& {\omega }_{1R}\\ {\beta }_1& {\omega }_{1I}\end{array}\right|=0,\left|\begin{array}{cc}{\alpha }_2& {\omega }_{2R}\\ {\beta }_2& {\omega }_{2I}\end{array}\right|=0.$ (3.28)

At this stage, both breathers experience transformation. Under certain parameter conditions each breather can be transformed into a different type of nonlinear wave, thus making the phenomenon more abundant.

Taking parameters with

$\begin{array}{l}{u}_0=1,{\alpha }_1=1,{\beta }_1=\frac{9}{17},{\nu }_1=1,{\sigma }_1=-3,{\xi }_1=1,{\mu }_1=0,\\ {\alpha }_2=\frac{7}{10},{\beta }_2=\frac{21}{26},{\nu }_2=2,{\sigma }_2=-2,{\xi }_2=1,{\mu }_2=0,\end{array}$ (3.29)

the plots of transformed nonlinear wave interactions are given by (1.2) with (3.2) in Figure 5(a) and Figure 5(b). The figures describe the transformation of one of two breathers into a single kink solitary wave and the other into an m-type kink solitary wave.

Taking parameters with

$\begin{array}{l}{u}_0=1,{\alpha }_1=1,{\beta }_1=\frac{9}{17},{\nu }_1=1,{\sigma }_1=-3,{\xi }_1=1,{\mu }_1=0,\\ {\alpha }_2=\frac{1}{2},{\beta }_2=\frac{3}{2},{\nu }_2=2,{\sigma }_2=-2,{\xi }_2=1,{\mu }_2=0,\end{array}$ (3.30)

the images of transformed nonlinear wave interactions are presented by (1.2) with (3.2) in Figure 5(c) and Figure 5(d). Similar to the previous case, the figures depict the transformation of one of the two breathers into a single kink solitary wave and the other into a multi-kink solitary wave.

Taking parameters with

$\begin{array}{l}{u}_0=1,{\alpha }_1=\frac{7}{10},{\beta }_1=\frac{21}{26},{\nu }_1=2,{\sigma }_1=-2,{\xi }_1=1,{\mu }_1=0,\\ {\alpha }_2=\frac{1}{2},{\beta }_2=\frac{3}{2},{\nu }_2=2,{\sigma }_2=-2,{\xi }_2=1,{\mu }_2=0,\end{array}$ (3.31)

in Equations (1.2) with (3.2), the plots of the solution are illustrated in Figure 5(e) and Figure 5(f). One breather transforms into an m-type kink solitary wave, while the other transforms into a multi-kink solitary wave.

Figure 5. Full-transformation waves of the second-order breather (1.2) with (3.2).

4. Conclusion

In this paper, by means of the Hirota bilinear method, we obtain the first-order and the second-order breather from the two-solitary wave solution and four-solitary wave solution of the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation, respectively. Based on the solution expressions, the exact solutions consist of hyperbolic and trigonometric functions, where the hyperbolic function represents a solitary wave, and the periodic wave describes a trigonometric function. The transformation mechanism of the first-order and second-order breathers presented in this paper primarily depends on adjusting the parameters and controlling the characteristic lines of the breathers. For different characteristic lines, several types of nonlinear waves in the $xOt$ plane are illustrated, including kink solitary waves, m-kink type solitary waves, multi-kink type solitary waves and periodic waves. The periodic features of the transformed nonlinear waves become evident with the slope of the characteristic lines.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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