General high-order solitons and breathers with a periodic wave background in the nonlocal Hirota–Maccari equation

Abstract

Soliton and breather solutions to the nonlocal Hirota–Maccari equation with a periodic wave background are constructed via the KP hierarchy reduction approach. By constraining tau functions of bilinear equations in the KP hierarchy, we obtain general 2N-line solitons and N-breather solutions with a periodic wave background. What needs to be emphasized are the two-soliton can be divided into non-degenerate and degenerate soliton according to the asymptotic analysis. Meanwhile, one- and two-breather solutions in a periodic wave and constant background are investigated.

Appendix A

In this part, we will prove the Theorem (2.1) and Theorem (3.1). Firstly, we utilize the variable transformations

$$\begin{aligned} u=\frac{g}{f}, \quad u^*(-x,y,-t)=\frac{h}{f},\quad r=-2i(\ln f)_{xy},\nonumber \\ \end{aligned}$$

(25)

to convert the Eq. (2) into the bilinear equation

$$\begin{aligned} \begin{aligned}&(D_t-iD_xD_y+\beta D_x^3+\beta D_x)g\cdot f=0,\\&(3D_x^2+1)f\cdot f=gh. \end{aligned} \end{aligned}$$

(26)

Taking reduction condition

$$\begin{aligned} (\partial _x-\partial _s)f=cf, \end{aligned}$$

(27)

the $(3 + 1)$ dimensional system of Eq. (1) can be obtained

$$\begin{aligned} \begin{aligned}&(D_t-iD_xD_y+\beta D_x^3+\beta D_x)g\cdot f=0,\\&(3D_xD_s+1)f\cdot f=gh, \end{aligned} \end{aligned}$$

(28)

under the nonlocal condition

$$\begin{aligned} f(x,y,t)g^*(-x,y,-t)=f^*(-x,y,-t)h(x,y,t),\nonumber \\ \end{aligned}$$

(29)

where c is constant, g, h are complex-valued functions, and f is a real-valued function.

Based on the Sato theory [31, 32], the bilinear equations in the KP hierarchy

$$\begin{aligned} \begin{aligned}&(D_{x_1}^3+3D_{x_1}D_{x_2}-4D_{x_3})\tau _{n+1}\tau _n=0,\\&(D_{x_1}D_{x_{-1}}-2)\tau _n\tau _n=-2\tau _{n+1}\tau _{n-1}, \end{aligned} \end{aligned}$$

(30)

exist the determinant

$$\begin{aligned} \tau _n=\mathop {det}\limits _{1\le i,j\le N}(m_{ij}^{(n)}), \end{aligned}$$

(31)

where

$$\begin{aligned} \begin{aligned}&m_{ij}^{(n)}=\widetilde{c_i}\delta _{ij}+\frac{p_i+s_i}{p_i+q_j}\left( \frac{-p_i}{q_j}\right) ^ne^{\xi _i+\eta _j},\\&\xi _i=\frac{1}{p_i}x_{-1}+p_ix_1+p_i^2x_2+p_i^3x_3+\xi _{0,i},\\&\eta _j=\frac{1}{q_j}x_{-1}+q_jx_1-q_j^2x_2+q_j^3x_3+\eta _{0,j}. \end{aligned} \end{aligned}$$

(32)

In order to obtain periodic solutions, we employ the independent variables $x_1=\frac{1}{\sqrt{6}}x$, $x_2=\frac{1}{2}i\beta y$, $x_3=-\frac{\sqrt{6}}{9}\beta t-\frac{\sqrt{6}}{9}x$, then the Eq. (31) can rewrite

$$\begin{aligned} \tau _n=\prod _{i=1}^{N}(p_i+s_i)e^{\zeta _i}\mathop {det}\limits _{1\le i,j\le N}(\widehat{m_{ij}^{(n)}}), \end{aligned}$$

(33)

where

$$\begin{aligned} \widehat{m_{ij}^{(n)}}=\widetilde{c_i}\delta _{ij}e^{-\zeta _i}\frac{1}{p_i+s_i}+\frac{1}{p_i+q_j}(-\frac{p_i}{q_j})^n, \end{aligned}$$

(34)

with

$$\begin{aligned} \begin{aligned} \zeta _i&=\xi _i+\eta _i\\&=\frac{1}{\sqrt{6}}(p_i+q_j)x+\frac{i}{2}\beta (p_i^2-q_j^2)y\\ {}&\quad -\frac{\sqrt{6}}{9}(\beta t+x)(p_i^3+q_j^3)+\chi _{0,i}. \end{aligned} \end{aligned}$$

(35)

To yield soliton solutions, we consider $M=2N+1$ in Eq. (31) and restrict the parameters obeying the following conditions

$$\begin{aligned} \begin{aligned}&s_j=-p_j+1,\quad p_{M+i}=-p_i,\quad q_{M+i}=-q_i,\quad q_i=p_i^*,\\&q_{2M+1}\!=\!-p_{2M+1}^*,\quad \widetilde{c_j}\!=\!b_j,\quad c_{M+i}\!=\!-c_i^*,\quad \widetilde{c_{2M+1}}\!=\!ic_{2M+1}, \end{aligned}\nonumber \\ \end{aligned}$$

(36)

then one can obtain

$$\begin{aligned} \begin{aligned} \chi _{2M+1}^*(-x,y,-t)&=\chi _{2M+1}(x,y,t),\\ \chi _{M+i}^*(-x,y,-t)&=\chi _i(x,y,t), \end{aligned} \end{aligned}$$

(37)

and derive

$$\begin{aligned} \begin{aligned} \widehat{m_{M+i,j}^{*(n)}}(-x,y,-t)&=-c_i^*\delta _{M+i,j}e^{-\chi _i}\\&\quad -\frac{1}{p_i^*+p_{M+j}} \left( -\frac{p_{M+j}}{p_i^*}\right) ^{-n}\\&=-\widehat{m_{i,M+j}}^{(-n)}(x,y,t), \end{aligned}\nonumber \\ \end{aligned}$$

(38)

which implies

$$\begin{aligned} \tau _n^*(-x,y,-t)=(-1)^{3N}\tau _{-n}(x,y,t). \end{aligned}$$

(39)

As a result, Eq. (30) can be reduced to the bilinear Eqs. (28) with $f=\tau _0, g=\tau _1, h=\tau _{-1}$.

To generate breathers to Eq. (1), we select several variable transformations

$$\begin{aligned}{} & {} x_{-1}=-\frac{1}{\sqrt{6}}is, \qquad x_1=\frac{1}{\sqrt{6}}ix, \qquad x_2=-\frac{1}{2}i\beta y, \nonumber \\{} & {} \qquad x_3=-i\frac{\sqrt{6}}{9}\beta t-i\frac{\sqrt{6}}{9}x, \end{aligned}$$

(40)

then the tau function becomes

$$\begin{aligned} \tau _n=\prod _{i=1}^{N}(p_i+s_i)e^{\chi _i}\mathop {det}\limits _{1\le i,j\le N}(\overline{m_{i,j}^{(n)}}), \end{aligned}$$

(41)

where $\overline{m_{i,j}^{(n)}}$ is given by

$$\begin{aligned}{} & {} \overline{m_{ij}^{(n)}}=\frac{\widetilde{c_i}\delta _{ij}}{(p_i+s_i)e^{\xi _i+\eta _j}}\nonumber \\{} & {} \quad +\frac{1}{p_i+q_j}\left( -\frac{p_i}{q_j}\right) ^n,\nonumber \\{} & {} \chi _i=\xi _i+\eta _i=\frac{1}{\sqrt{6}}(p_i+p_i^*)ix-\frac{1}{2}i\beta (p_i^2-p_i^{*2})y\nonumber \\{} & {} \quad -\frac{\sqrt{6}}{9}i(p_i^3+p_i^{*3})(\beta t+x)+\chi _i^0. \end{aligned}$$

(42)

When the parameters satisfy $\widetilde{c_{i,j}}=1, s_j=q_j, q_j=p_j^*$, one could conclude that $\chi _i^*(-x,y,-t)=\chi _i(x,y,t)$. In this case, we further derive

$$\begin{aligned} \overline{m_{j,i}^{*(-n)}}(x,y,t)= & {} \overline{m_{i,j}^{*(n)}}(-x,y,-t),\nonumber \\{} & {} \tau _n^*(-x,y,-t)=\tau _{-n}(x,y,t). \end{aligned}$$

(43)

Additionally, determining $f=\tau _0, g=\tau _1, h=\tau _{-1}$ and letting

$$\begin{aligned} p_i=\frac{\omega _i}{2}+i\lambda _i,\qquad q_i=\frac{\omega _i}{2}-i\lambda _i, \end{aligned}$$

(44)

solutions for the breather are discovered.