Dynamics of solitons and breathers on a periodic waves background in the nonlocal Mel’nikov equation

Abstract

Dynamics of general line solitons and breathers on a periodic line waves (PLWs) background in the nonlocal Mel’nikov (MK) equation are investigated via the KP hierarchy reduction method. By constraining different parametric conditions for a general type of tau functions of the KP hierarchy, two families of mixed solutions to the nonlocal MK equation are derived. The first family of mixed solutions illustrates general line solitons on a PLWs background. The simplest case of such mixed solutions shows the two-line solitons on a PLWs background, and the two-line solitons possess five different patterns: a mixture of one-dark-soliton and one-antidark-soliton, two-antidark-soliton, two-dark-soliton, degenerated two-dark-soliton, and degenerated two-anti-dark-soliton. The high-order mixed solutions display superposition of several individual simplest solutions. The second family of mixed solutions demonstrates general breathers on a PLWs background or on a nonzero constant background. The breathers are periodic in time and do not move in the (x, y)-plane as time propagates.

Appendix A

In this Appendix, we will present the proof procedure for Theorems 1 and 2. Since the solutions given in Theorems 1 and 2 are under nonzero boundary conditions, we employ the employing the following dependent variable transformations

$$\begin{aligned} \begin{aligned} \Psi =\sqrt{2} \frac{g(x,y,t)}{f(x,y,t)},u= 2( {\log } f(x,y,t))_{xx}, \end{aligned} \end{aligned}$$

(50)

to translate the nonlocal MK equation (14) into the following bilinear equation

$$\begin{aligned}&(D_{x}^{2}-iD_{y})g \cdot f =0,\nonumber \\&(D^{4}_{x}{+}D_xD_t{-}3D^2_{y})f \cdot f =2\kappa [f^2 {-}gg^{*}(-x,y,-t) ].\nonumber \\ \end{aligned}$$

(51)

According to the Sato theory [65,66,67,68,69,70], the following bilinear equations in the single component KP hierarchy

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

(52)

possess the following tau functions expressed via Gramian determinant

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

(53)

where the matrix elements are defined as

$$\begin{aligned} \begin{aligned}&m_{sj}^{(n)}={\widetilde{b}}_{s}\delta _{sj}+\frac{p_s+r_s}{p_s+q_j} \left( -\frac{p_s}{q_j}\right) ^{n}e^{\xi _s+\eta _j},\\&\xi _s= \frac{1}{p_s}x_{-1}+p_sx_1+p_s^2x_2+p_s^3x_3+\xi _{0,s},\\&\eta _j=\frac{1}{q_{j}}x_{-1}+q_jx_1-q_j^2x_2+q_j^3x_3+\eta _{0,j},\\ \end{aligned} \end{aligned}$$

(54)

and $p_s,r_s,q_j,b_{s},\xi _{i0}$ and $\eta _{0,j}$ are arbitrary complex constants.

To construct periodic solutions given in Theorem 1, we first take the variable transformations

$$\begin{aligned} \begin{aligned} x_{-1}=\kappa t, x_1=x, x_{2}=-iy,x_3=-4t, \end{aligned} \end{aligned}$$

(55)

and then the tau function defined in (53) can be rewritten as

$$\begin{aligned} \begin{aligned} \tau _{n}(x,y,t)=\prod _{s=1}^{N}(p_s+r_s)e^{\zeta _{s}}\det \limits _{1\le i,j\le N} ({\widehat{m}}_{ij}^{(n)}), \end{aligned} \end{aligned}$$

(56)

where

$$\begin{aligned} \begin{aligned} {\widehat{m}}_{sj}^{(n)}= {\widetilde{b}}_s\delta _{sj}e^{-\zeta _s}\frac{1}{p_s+r_s} +\frac{1}{p_s+q_j}\left( -\frac{p_s}{q_j}\right) ^n, \end{aligned} \end{aligned}$$

(57)

with

$$\begin{aligned} \begin{aligned} \zeta _s=&\xi _s+\eta _s\\ =&(p_s+q_s)x-i(p_s^2-q_s^2)y\\&+\left( \frac{\kappa }{p_s}+\frac{\kappa }{q_s} -4p_s^3-4q_s^3\right) t+\zeta _{0,s}, \end{aligned} \end{aligned}$$

(58)

In what follows, we consider $(2M+1)\times (2M+1)$ matrix for $\tau _n$ (i.e., $N=2M+1$ in (53)) and take the parameters satisfying the following constraint conditions

$$\begin{aligned} \begin{aligned} r_j&=-p_j+1,p_{M+s}=-p_s,q_{M+s}=-q_s,q_{s}\\&=p_s^*,q_{2M+1}=-p_{2M+1}^*,r_j=-p_j+1,\\ {\widetilde{b}}_{j}&=b_{j},b_{M+s}=-b_s^*,\\&{\widetilde{b}}_{2M+1} =ib_{2M+1},\zeta _{0,M+s}=\zeta _{0,s}, \end{aligned} \end{aligned}$$

(59)

for $j=1,2,\ldots 2M$ and $s=1,2,\ldots M$. Since

$$\begin{aligned} \begin{aligned} \zeta _s=&\xi _s+\eta _s\\ =&(p_s+p^*_s)x-i(p_s^2-p_s^{*2})y\\&+\left( \frac{\kappa }{p_s}+\frac{\kappa }{p^*_s}-4p_s^3-4p_s^{*3}\right) t+\zeta _{0,s}, \\ \zeta _{2M+1}=&\xi _{2M+1}+\eta _{2M+1}\\ =&\left( p_{2M+1}-p_{2M+1}^*\right) x\\&-i\left( p_{2M+1}^2-p_{2M+1}^{*2}\right) y\\&+\left( \frac{\kappa }{p_{2M+1}}-\frac{\kappa }{p^*_{2M+1}}\right. \\&\left. -4p_{2M+1}^3+4p_{2M+1}^{*3}\right) t+\zeta _{0,{2M+1}}, \end{aligned} \end{aligned}$$

(60)

thus

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

(61)

and further obtain

$$\begin{aligned} \begin{aligned} {\widehat{m}}_{M+s,j}^{*(n)}(-x,y,-t)&=-b_{s}^*\delta _{M+s,j} e^{-\zeta _{s}}\\&\quad -\frac{1}{p_{s}^*+p_{M+j}}\left( -\frac{p_{M+j}}{p_{s}^*}\right) ^{-n}\\&=-{\widehat{m}}_{s,M+j}^{(-n)}(x,y,t), \end{aligned} \end{aligned}$$

(62)

similarly, the following conditions can also be derived

$$\begin{aligned} \begin{aligned} {\widehat{m}}_{s,M+j}^{*(n)}(-x,y,-t)=&-{\widehat{m}}_{M+s,j}^{(-n)}(x,y,t),\\ {\widehat{m}}_{M+s,M+j}^{*(n)}(-x,y,-t)=&-{\widehat{m}}_{j,s}^{(-n)}(x,y,t),\\ {\widehat{m}}_{2M+1,2M+1}^{*(n)}(-x,y,-t)=&-{\widehat{m}}_{2M+1,2M+1}^{(-n)}(x,y,t),\\ {\widehat{m}}_{2M+1,j}^{*(n)}(-x,y,-t)=&-{\widehat{m}}_{M+j,2M+1}^{(-n)}(x,y,t),\\ {\widehat{m}}_{2M+1,M+j}^{*(n)}(-x,y,-t)=&-{\widehat{m}}_{j,2M+1}^{(-n)}(x,y,t),\\ {\widehat{m}}_{s,2M+1}^{*(n)}(-x,y,-t)=&-{\widehat{m}}_{2M+1,M+s}^{(-n)}(x,y,t),\\ {\widehat{m}}_{M+s,2M+1}^{*(n)}(-x,y,-t)=&-{\widehat{m}}_{2M+1,s}^{(-n)}(x,y,t), \end{aligned} \end{aligned}$$

(63)

which results in the following nonlocal symmetry condition:

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

(64)

By defining

$$\begin{aligned} \begin{aligned} f(x,y,t)=\tau _0(x,y,t),g(x,y,t)=\tau _{1}(x,y,t), \end{aligned} \end{aligned}$$

(65)

then solutions to the nonlocal MK equation given in Theorem 1 would be obtained that completes the proof for Theorem 1.

Finally, we give the proof for Theorem 2. To derive more general periodic solutions to the nonlocal Mel’nikov equation (14), we choose different variable transformations

$$\begin{aligned} \begin{aligned} x_{-1}=-i\kappa t, x_1=ix, x_{2}=iy,x_3=4it. \end{aligned} \end{aligned}$$

(66)

The tau function $\tau _n$ is rewritten as

$$\begin{aligned} \begin{aligned} \tau _{n}=\prod _{s=1}^{N}(p_s+r_s)e^{\xi _s+\eta _s}\det \limits _{1\le s,j\le N} ({\overline{m}}_{s,j}^{(n)}), \end{aligned} \end{aligned}$$

(67)

where the matrix elements ${\overline{m}}_{s,j}^{(n)}$ given by (67) become the following formula

$$\begin{aligned} \begin{aligned}&{\overline{m}}_{s,j}^{(n)}=\frac{{\widetilde{b}}_{s}\delta _{sj}}{(p_s+r_s)e^{\xi _s+\eta _j}}+\frac{1}{p_s+q_j}\left( -\frac{p_s}{q_j}\right) ^{n} ,\\&\xi _s= ip_sx+ip_s^2y+\left( 4ip_s^3-\frac{i\kappa }{p_s}\right) t+\xi _{0,s},\\&\eta _j=iq_jx-iq_j^2y+\left( 4iq_j^3-\frac{i\kappa }{q_j}\right) t+\eta _{0,j}. \end{aligned} \end{aligned}$$

(68)

Under the parameters satisfying the following constraint conditions

$$\begin{aligned} \begin{aligned}&{\widetilde{b}}_{sj}=1, r_j=q_j, q_j=p_j^*, \end{aligned} \end{aligned}$$

(69)

and $\zeta _{0,s}$ are real for $j=1,2,\ldots N$, then

$$\begin{aligned} \begin{aligned} \zeta _s=&\xi _s+\eta _s\\ =&i(p_s+p_s^*)x+i(p_s^2-p_s^{*2})y\\&+i\left( 4p_s^3+4p_s^{*3} -\frac{\kappa }{p_s}-\frac{\kappa }{p^*_s}\right) t+\zeta _{0,s}, \end{aligned} \end{aligned}$$

(70)

and one can derive the following condition

$$\begin{aligned} \begin{aligned} \zeta _s^*(-x,y,-t)=\zeta _{s}(x,y,t), \end{aligned} \end{aligned}$$

(71)

which can further yield

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

(72)

Again, defining $f=\tau _0,g=\tau _1$ and taking

$$\begin{aligned} \begin{aligned} p_s=\frac{\omega _s}{2}+i\lambda _s,q_s=\frac{\omega _s}{2}-i\lambda _s, \end{aligned} \end{aligned}$$

(73)

the periodic solutions for the nonlocal MK equation given by Theorem 2 are obtained that completes Theorem 2.