1. Introduction
Recently, soliton solutions and hybrid solutions have attracted more and more experts' attention in soliton theory, which can be well described by using nonlinear partial differential equations [1]. And Darboux transformation has been proved to be one of the most useful method to get exact solutions of some soliton equations [2-6]. By Darboux transformation, Xu et al discussed the interaction between asymptotic solitons of several types of nonlinear equations, such as nonlocal nonlinear Schrödinger equation [4, 5] and Hirota equation [6]. In 2013, He et al proposed the generation mechanism of higher-order rouge waves [7] for nonlinear Schrödinger equation by Darboux transformation. They also obtained smooth positons and dynamic properties of this type of positons [8-12]. Similarly, their team applied this mechanism to other integrable equations, such as Fokas-Lenells (FL) equation [9] and modified KdV equation [10-12]. In addition, they also used double degeneracy to generate higher-order rogue waves from multibreathers [13]. Besides, Zhang et al applied Darboux transformation to study some exact solutions of nonlocal FL equation [14].
In previous studies, a common phenomenon is that the soliton solution can only be obtained from a zero seed solution [3, 15], and n-fold Darboux transformation can only obtain n-soliton solutions [12, 14, 15]. Lou used the bilinear method to obtain the solutions with even numbers for the nonlocal Boussinesq-KdV type system [16]. Xu mentioned that, under the background of continuous wave, the iterative solution after n-fold Darboux transformation in general exhibits various elastic interactions among 2n solitons for a nonlocal nonlinear Schrödinger equation. And the two-soliton and four-soliton interactions are discussed in detail in the asymptotic analysis in [4]. This means we can also get 2n-soliton solutions through some methods, such as Darboux transformation.
We consider nonlocal FL equation [14] as follows:
$\begin{eqnarray}{q}_{{xt}}+q-2{\rm{i}}{{qq}}_{x}q(-x,-t)=0.\end{eqnarray}$
Some exact solutions to equation (1 ) have been given in [14]. Combining [9, 14], we can get a new solution q[n] of equation (1 ) through n-fold Darboux transformation with a plane wave solution ${q}^{[0]}=a{{\rm{e}}}^{{\rm{i}}\theta },\theta ={bx}+\tfrac{\left(2\,{a}^{2}b+1\right)t}{b}$:
$\begin{eqnarray}{q}^{[n]}={q}^{[0]}-2\displaystyle \frac{{N}_{2n}}{{D}_{2n}},\end{eqnarray}$
with $\begin{eqnarray*}{D}_{2n}=\left|\begin{array}{ccccc}{\lambda }_{1}^{n}{\phi }_{1} & {\lambda }_{1}^{n-1}{\varphi }_{1} & \cdots & {\lambda }_{1}^{-(n-2)}{\phi }_{1} & {\lambda }_{1}^{-(n-1)}{\varphi }_{1}\\ {\lambda }_{2}^{n}{\phi }_{2} & {\lambda }_{2}^{n-1}{\varphi }_{2} & \cdots & {\lambda }_{2}^{-(n-2)}{\phi }_{2} & {\lambda }_{2}^{-(n-1)}{\varphi }_{2}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {\lambda }_{2n-1}^{n}{\phi }_{2n-1} & {\lambda }_{2n-1}^{n-1}{\varphi }_{2n-1} & \cdots & {\lambda }_{2n-1}^{-(n-2)}{\phi }_{2n-1} & {\lambda }_{2n-1}^{-(n-1)}{\varphi }_{2n-1}\\ {\lambda }_{2n}^{n}{\phi }_{2n} & {\lambda }_{2n}^{n-1}{\varphi }_{2n} & \cdots & {\lambda }_{2n}^{-(n-2)}{\phi }_{2n} & {\lambda }_{2n}^{-(n-1)}{\varphi }_{2n}\end{array}\right|,\end{eqnarray*}$
where $\begin{eqnarray*}{\phi }_{j}=\left(2\,{\text{}}{ab}\,{\lambda }_{j}{{\rm{e}}}^{{\rm{i}}\sigma {\rm{\Omega }}}+\left(-{\lambda }_{j}^{2}+b+2\,\sigma \right){{\rm{e}}}^{-{\rm{i}}\sigma {\rm{\Omega }}}\right){{\rm{e}}}^{\tfrac{{\rm{i}}\theta }{2}},\end{eqnarray*}$
$\begin{eqnarray*}{\varphi }_{j}=\left(\left(-{\lambda }_{j}^{2}+b+2\,\sigma \right){{\rm{e}}}^{{\rm{i}}\sigma {\rm{\Omega }}}+2\,{ab}{\lambda }_{j}{{\rm{e}}}^{-{\rm{i}}\sigma {\rm{\Omega }}}\right){{\rm{e}}}^{-\tfrac{{\rm{i}}\theta }{2}},\end{eqnarray*}$
$\begin{eqnarray*}{\rm{\Omega }}=-\displaystyle \frac{t}{b{\lambda }_{j}^{2}}+x,\sigma =\displaystyle \frac{1}{2}\,\sqrt{{\lambda }_{j}^{4}+\left(-4\,{a}^{2}{b}^{2}-2\,b\right){\lambda }_{j}^{2}+{b}^{2}},\end{eqnarray*}$
and N2n is a determinant given by D2n through replacing its last column with vector ${[{\lambda }_{1}^{-n}{\phi }_{1},{\lambda }_{2}^{-n}{\phi }_{2},\cdots ,{\lambda }_{2n-1}^{-n}{\phi }_{2n}]}^{{\rm{T}}}$.The aim of this paper is to study some exact solutions sitting on a nonzero background via Darboux transformation with a plane wave solution. These solutions includes $2n$-soliton solutions, hybrid solutions consisting of solitons and a kink, hybrid solutions consisting of smooth positons. In section 2 , we present the even soliton solutions and some hybrid solutions. In section 3 , interactions between smooth positons are researched by double degenerate Darboux transformation with a nonzero seed solution. The last section is a short summary and discussion.
2. Even soliton solutions with a nonzero background
On the basis of Darboux transformation, a hybrid solution of $2m$-soliton and $l$th-order breather ${q}_{2m-l-{\rm{hyb}}}$ is given by
$\begin{eqnarray}{q}_{2m-l-{\rm{hyb}}}={q}^{[0]}-\displaystyle \frac{{N}_{2n}}{{D}_{2n}},\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{\lambda }_{1} & \ne & {\lambda }_{2}\ne \cdots \,\ne {\lambda }_{2m},{\lambda }_{2m+2j-1}=-{\lambda }_{2m+2j}^{* },\\ j & = & 1,\cdots ,\,l,\left|\alpha \right|\gt 1,\beta =0,2n=2m+2l,\end{array}\end{eqnarray*}$
and ${D}_{2n},{N}_{2n},{q}^{[0]}$ are given by equation (Looking at figure 1, it is clear that the soliton solutions obtained by Darboux transformation appear in pairs. In other words, after n-fold Darboux transformations, there will be 2n solitons. And $2n$-soliton solutions consist of n ‘bright' solitons and n ‘dark' solitons. In many related studies [7-15], a lot of authors can only obtain n-soliton solutions via n-fold Darboux transformation with a zero seed solution. In fact, if we start with the zero seed solution, n-soliton solution for nonlocal FL equation can only be obtained. It must be mentioned that Lou find $2n$-soliton solutions by bilinear method for the first time [16]. Further, a hybrid solution of solitons and breathers sitting on a nonzero background is researched through Darboux transformation with some parameter constraints.
Figure 1. Solutions obtained by Darboux transformation with $a=2{\rm{i}}$, b = 1: (a) two-soliton solution ${q}_{2-0-{\rm{hyb}}}$ with parameters ${\lambda }_{1}={\rm{i}},{\lambda }_{2}=2{\rm{i}};$ (b) four-soliton solution ${q}_{4-0-{\rm{hyb}}}$ with parameters ${\lambda }_{1}={\rm{i}},{\lambda }_{2}=2{\rm{i}},{\lambda }_{3}=\tfrac{3{\rm{i}}}{2},{\lambda }_{4}=\tfrac{{\rm{i}}}{2};$ (c) a hybrid solution ${q}_{2-1-{\rm{hyb}}}$ consisting of a two-soliton and a breather with ${\lambda }_{1}=1+{\rm{i}},{\lambda }_{2}=-1+{\rm{i}},{\lambda }_{3}={\rm{i}},{\lambda }_{4}=2{\rm{i}}$. |
On the basis of Darboux transformation, a hybrid solution of $2m$-soliton and a kink ${q}_{2m-{\rm{kink}}-{\rm{hyb}}}$ is given by
$\begin{eqnarray}{q}_{2m-{\rm{kink}}-{\rm{hyb}}}={q}^{[0]}-\displaystyle \frac{{N}_{2n}}{{D}_{2n}},\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{\lambda }_{1} & = & {\xi }_{1}+{\rm{i}}{\eta }_{1},{\lambda }_{2}=-{\rm{i}}{\eta }_{1},{\lambda }_{3}\ne {\lambda }_{4}\ne \cdots \ne {\lambda }_{2{m}+2},\left|\alpha \right|\gt 1,\\ \beta & = & 0,2{n}=2{m}+2.\end{array}\end{eqnarray*}$
In proposition 2 , the idea of ${\lambda }_{1}={\xi }_{1}+{\rm{i}}{\eta }_{1},{\lambda }_{2}=-{\rm{i}}{\eta }_{1}$ comes from [14]. We use a specific example described by proposition 2 to observe the hybrid solutions.
Obviously, we can see from figure 2 that for each fold Darboux transformation, two solitons will be added, one ‘bright' and the other ‘dark'. At the same time, we have to admit that it is very difficult to figure out the dynamic property of hybrid solutions described by proposition 2 .
Figure 2. A hybrid solution of solitons and a kink obtained by Darboux transformation with $\alpha =2,b=1$: (a) a hybrid solution ${q}_{2-{\rm{kink}}-{\rm{hyb}}}$ of two-soliton and a kink with parameters ${\lambda }_{1}=1-3{\rm{i}},{\lambda }_{2}=3{\rm{i}},{\lambda }_{3}=2{\rm{i}},{\lambda }_{4}={\rm{i}};$ (b) a hybrid solution ${q}_{4-{\rm{kink}}-{\rm{hyb}}}$ of four-soliton and a kink with parameters ${\lambda }_{1}=1-3{\rm{i}},{\lambda }_{2}=3{\rm{i}},{\lambda }_{3}=2{\rm{i}},{\lambda }_{4}={\rm{i}},{\lambda }_{5}=\tfrac{{\rm{i}}}{2},{\lambda }_{6}=\tfrac{{\rm{i}}}{3}$. |
3. Smooth positons by double degenerate Darboux transformation
Propositions 1 and 2 describe the multisoliton solutions with even number and some hybrid solutions over a nonzero background. In order to further study the generation solution of the multisoliton solution, we introduce the double degenerate Darboux transformation [13] to analyze some new and interesting results.
On the basis of the double degenerate Darboux transformation, an interaction between two $n$ th-order smooth positons ${q}_{n-n-{\rm{pos}}}$ is given by
$\begin{eqnarray}{q}_{n-n-{\rm{pos}}}={q}^{[0]}-\displaystyle \frac{{N}_{2n}^{{\prime} }}{{D}_{2n}^{{\prime} }},\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{rcl}{N}_{2n}^{{\prime} } & = & {\left({\left.\displaystyle \frac{{\partial }^{h(i)}}{\partial {\epsilon }^{h(i)}}\right|}_{\epsilon =0}{\left({N}_{2n}\right)}_{{ij}}({\lambda }_{j}+{\epsilon }^{2})\right)}_{2n\times 2n},\\ {D}_{2n}^{{\prime} } & = & {\left({\left.\displaystyle \frac{{\partial }^{h(i)}}{\partial {\epsilon }^{h(i)}}\right|}_{\epsilon =0}{\left({D}_{2n}\right)}_{{ij}}({\lambda }_{j}+{\epsilon }^{2})\right)}_{2n\times 2n}\\ h(x) & = & \left\{\begin{array}{ll}2(x-1), & x\leqslant n\\ 2(x-1-n), & x\gt n\end{array}\right.\end{array},\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{rcl}{\lambda }_{1} & = & {\lambda }_{2}\,=\,\cdots \,=\,{\lambda }_{n},{\lambda }_{n+1}={\lambda }_{n+2}\,=\,\cdots \,=\,{\lambda }_{2n},\\ {\lambda }_{1} & \ne & {\lambda }_{n+1},\left|\alpha \right|\gt 1,\beta =0.\end{array}\end{eqnarray*}$
Proposition 3 describes a new interaction between smooth positions with a nonzero background. Figure 3 vividly shows the elastic collision between two nth-order smooth positons. The interacting positons can recover their individual shapes upon the interaction and experience only the phase shifts. We can observe that smooth positons described by proposition 3 are composed of ‘bright part' and ‘dark part'. As $\left|t\right|\to \infty $, the ‘bright part' and the ‘dark part' approach different fixed values, respectively. Compared with previous research, proposition 3 has two improvements: one is that the interaction between smooth positions with a nonzero background is found, and the other is that proposition 3 requires fewer eigenvalues than proposition 3 in [17]. When we get a collision between two second-order smooth positons, four eigenvalues are needed instead of eight in [17]. Similarly, according to the result of proposition 2 , we can also get a hybrid solution of positons and a kink. Here we do not describe it in detail.
Figure 3. (a) An interaction ${q}_{2-2-{\rm{pos}}}$ between two second-order smooth positons with parameters ${\lambda }_{1}=2{\rm{i}},{\lambda }_{2}=2{\rm{i}},{\lambda }_{3}={\rm{i}},{\lambda }_{4}={\rm{i}},{a}=2{\rm{i}},{b}=1;$ (b) density plot of (a); (c) an interaction ${q}_{3-3-{\rm{pos}}}$ between two third-order smooth positons with parameters ${\lambda }_{1}=2{\rm{i}},{\lambda }_{2}=2{\rm{i}},{\lambda }_{3}=2{\rm{i}},{\lambda }_{4}={\rm{i}},{\lambda }_{5}={\rm{i}},{\lambda }_{6}={\rm{i}},{a}=2{\rm{i}},{b}=1$. |
In addition to that, we have a new result, which is another kind of smooth positons. In order to facilitate the distinction between these two types of smooth positons, we refer to positons described by Proposition 3 as stable positons and positons described by following propositions as semi-stable positons according to the dynamic properties of these two types of positons.
On the basis of the degenerate Darboux transformation, an interaction between two $n$th-order smooth positons called semi-stable positons ${q}_{n-n-\mathrm{semipos}}$ is given by
$\begin{eqnarray}{q}_{n-n-\mathrm{semipos}}={q}^{[0]}-\displaystyle \frac{{N}_{2n}^{{\prime} }}{{D}_{2n}^{{\prime} }},\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{rcl}{N}_{2n}^{{\prime} } & = & {\left({\left.\displaystyle \frac{{\partial }^{h(i)}}{\partial {\epsilon }^{h(i)}}\right|}_{\epsilon =0}{\left({N}_{2n}\right)}_{{ij}}({\lambda }_{j}+{\epsilon }^{2})\right)}_{2n\times 2n},\\ {D}_{2n}^{{\prime} } & = & {\left({\left.\displaystyle \frac{{\partial }^{h(i)}}{\partial {\epsilon }^{h(i)}}\right|}_{\epsilon =0}{\left({D}_{2n}\right)}_{{ij}}({\lambda }_{j}+{\epsilon }^{2})\right)}_{2n\times 2n},\\ h(x) & = & 2\left[\displaystyle \frac{x+1}{2}\right],\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{rcl}{\lambda }_{2j-1} & = & {\lambda }_{1}\to {ba}-\sqrt{{a}^{2}{b}^{2}+b},\\ {\lambda }_{2j} & = & {\lambda }_{2}\to {ba}+\sqrt{{a}^{2}{b}^{2}+b},j=1,\,\cdots ,\,n,\alpha =0,\beta \ne 0,\end{array}\end{eqnarray*}$
here $\left[x\right]$ denotes the floor function of $x$.In particular, as shown in figure 4(a), we can obtain bright two-soliton sitting on nonzero background via one-fold double degenerate Darboux transformation. However, two-soliton consisting of a dark soliton and a bright soliton can only be generated through proposition 1 . From the results shown in figures 4(a) and (c), solutions described by proposition 4 are somewhat similar to the so-called rational positons in [18]. It can be seen from the figure 4 that the collision between nth-order semi-stable positons maybe an elastic collision.
Figure 4. Interaction between two first-order smooth positons ${q}_{n-n-\mathrm{semipos}}$ with parameters $a=\tfrac{1}{3},b=1$ (a) ${q}_{1-1-\mathrm{semipos}}$ with parameters ${\lambda }_{1}=\tfrac{1}{3}-\tfrac{1}{3}\,\sqrt{10},{\lambda }_{2}=\tfrac{1}{3}+\tfrac{1}{3}\,\sqrt{10};$ (b) density plot of (a); (c) ${q}_{2-2-\mathrm{semipos}}$ with parameters ${\lambda }_{1}=\tfrac{1}{3}-\tfrac{1}{3}\,\sqrt{10},{\lambda }_{2}=\tfrac{1}{3}+\tfrac{1}{3}\,\sqrt{10}$. |
4. Conclusion
In this paper, we mainly study some new solutions sitting on a nonzero background of nonlocal FL equation via Darboux transformation with a plane wave solution. These solutions includes $2n$-soliton solutions, hybrid solutions consisting of solitons and breathers, interaction solutions between smooth positons. We propose four propositions to generate these exact solutions mentioned above. Overall, this paper has two major improvements: one is that $2n$-soliton solutions can be obtained by n-fold Darboux transformation, however many experts get only n-soliton solutions [7-11]; the other is that a new interaction between smooth positons sitting on a nonzero background is derived. In addition, the solitons obtained by Darboux transformation appear in pairs. And pairs of solitons are always one bright and one dark. This is the difference between this paper and other literatures [7-15], and also the two highlights of this research. In the following work, we will do further analysis for the results in this paper: What are the dynamic properties of hybrid solutions described by proposition 4 ? Meanwhile, we also hope that our results will provide some valuable information in the field of nonlinear science.