Breathers and rogue waves on the double-periodic background for the reverse-space-time derivative nonlinear Schrödinger equation

Nonlinear evolution equations play an important role in integrable systems and due to the applications of their solutions [1, 2], many well-known mathematicians and physicists did some significant work [3, 4, 5, 6, 7]. Breathers and rogue waves are the important solutions of nonlinear evolution equations. There have been a lot of studies about breathers and rogue waves in recent years [8, 9, 10, 11, 12, 13, 14, 15, 16].

The derivative nonlinear Schrödinger equation (DNLS) [20, 17, 18, 19] is given by

where the complex function $q=q(x,t)$ denotes the wave envelope and ^∗ denotes the complex conjugation. Eq. (1.1) arises in the study of circular polarized Alfvén waves in plasma [21], propagating parallel to the magnetic field [22], which is one of the most important integrable systems in mathematics and physics. Recently, the equation is also used to describe large-amplitude magnetohydrodynamic waves [23, 24] of plasmas, nonlinear optics, the sub-picosecond and femtosecond pulses in single-mode optical fiber [25, 26, 27]. The DNLS and nonlocal DNLS equations are reduced from the Kaup-Newell system [28, 29] and are Lax integrable. There generate many new physical phenomena and have important physical significance when nonlocal terms are added to nonlinear equations.

In recent years, many researchers have studied nonlocal DNLS equations from different viewpoints and perspectives. For example, in [30], the global bounded solutions of the nonlocal DNLS equation have been obtained from zero seed solution by Darboux transformation (DT) [31, 32, 33, 34, 35, 36]. Furthermore, solutions and connections of nonlocal DNLS equations have been studied in [37]. In [38], the periodic bounded solutions of the second-type nonlocal DNLS equation from zero seed solutions have been studied. The $PT$-symmetric, reverse-time, and reverse-space-time nonlocal DNLS equations are integrable infinite dimensional Hamiltonian dynamical systems, which were first proposed by Ablowitz and Musslimani [39, 41]. The general N-solitons in these three nonlocal nonlinear Schrödinger equations are presented by Yang in [40]. To investigate the connections between solutions at reverse-space-time points $(x,t)$ and $(-x,-t)$, we need to consider the reverse-space-time reduction. The reverse-space-time DNLS equation is as follows

here the symmetry reductions are nonlocal both in space and time. The reverse-space-time DNLS equation has many physical applications in optics, ocean water waves, quantum entanglement and an unconventional system of magnetics etc [42, 43, 44]. For eq. (1.2), the evolution of the solution at location $(x,t)$, depends both on the local position $(x,t)$ and the distant position $(-x,-t)$. This implies that the states of the solution at distant opposite locations are directly related, reminiscent of quantum entanglement in pairs of particles [40]. The solution of reverse-space-time DNLS equation can extend the solution of the local equation to a more general case and deepen the physical research on the mechanism of ocean rogue waves. These results would also be useful for understanding the corresponding rational soliton phenomena in many fields of nonlocal nonlinear dynamical systems such as nonlinear optics, Bose-Einstein condensates, ocean and other relevant fields [45, 46].

In general, it is extremely nontrivial to construct the rogue waves on a periodic background which is usually associated with complicated Jacobi elliptic functions [47, 48, 49, 50, 51, 52, 53, 54], PT symmetry [55], integrable equations with variable coefficients [56, 57], or vector integrable equations [58]. In [59, 60], the rogue waves on the periodic background have been constructed by using odd-order semi-degenerate DT. In this article, we mainly study the breathers and the rogue waves on the double periodic background by using even-fold DT and even-order semi-degenerate DT respectively. This is an effective new method to construct the solutions on double-periodic background without using Jacobi elliptic functions. It is of great physical significance to study rogue waves on a double-periodic background. For example, the rogue waves on the double-periodic background in the hydrodynamical experiments are possible due to the rogue waves on the continuous wave background observed in laser optics and water tanks [61]. Rogue waves on the double-periodic background could be relevant to diagnostics of rogue waves on the ocean surface and understanding the formation of random waves due to modulation instability [62].

In this work, we construct the breathers and rogue waves on the periodic background by odd-fold DT and odd-order semi-degenerate DT respectively. This is the first time to extend this method to reverse-space-time nonlocal equations. Remarkably, using a plane wave seed solution, the breathers and rogue waves on the double-periodic background are first successfully constructed by even-fold DT and even-order semi-degenerate DT respectively. By taking the dynamics analysis of the first-order rogue waves on double-periodic background, we show two types of structures: the two peaks and four peaks. The interesting thing can also be seen from the dynamic figures of the second-order rogue waves on the double-periodic background. There are two types of structures: one peak and two peaks. We shall also show the transformation process of double-periodic background and plane wave background in this study. These results have not been reported to our best knowledge.

The organizational structure of this paper is as follows. In section 2, the determinant representation of the $n$-fold DT formula is given. In section 3, using a plane wave seed solution, the periodic solution, breathers on the periodic background are given by odd-fold DT. The double-periodic solution, breathers and breathers on the double-periodic background solution are given by even-fold DT. In section 4, we construct higher-order rogue waves on the periodic background and double-periodic background by semi-degenerate DT formula. The final section is devoted to conclusion.

Starting from the seed solutions $q(x,t)=ce^{i(ax+bt)}$ and $q(-x,-t)=ce^{-i(ax+bt)}$, where $b=-ac^{2}+a^{2}$ and $c$ denoting the background height. In this section, the periodic solution, breathers on the periodic background are given by odd-fold DT. In addition, the double-periodic solution, breathers and breathers on the double-periodic background solution are given by even-fold DT.

Solving the Lax pair (2.1) and (2.2), then it gives

where

In order to obtain nontrivial solution of Eq. (1.2), we constructed the new eigenfunctions associated with $\lambda_{k}$ by the linear superposition principle.

Using eigenfunction (3.1) and symmetry condition (2.3), we can get some interesting new solutions.

When $n=1$: For simple, let $\lambda_{1}=i\beta_{1},$ then

where

$\lim\limits_{\begin{subarray}{c}x\rightarrow\infty\\ t\rightarrow\infty\end{subarray}}|q[1]|^{2}=c^{2}-2a$, and the trajectory of the solution (3.2) is $x=(2\beta_{1}^{2}+c^{2}-a)t$. According to these results, we can control the structure of the solution (3.2) by adjusting the parameters.

1. (1)

   When $c^{2}>2a$, Eq. (3.2) can generate soliton solutions. More profound, we find that Eq. (3.2) can generate a dark soliton when $c^{2}>2a>0$, $\frac{c}{2}+\frac{\sqrt{c^{2}-2a}}{2}>\beta_{1}>\frac{c}{2}-\frac{\sqrt{c^{2}-2a}}{2}$ or $c^{2}>0>2a$, $\frac{c}{2}+\frac{\sqrt{c^{2}-2a}}{2}>\beta_{1}>-\frac{c}{2}+\frac{\sqrt{c^{2}-2a}}{2}$ (see Fig. 1(a)). Eq. (3.2) can generate a bright soliton if $c^{2}>2a>0$, $-\frac{c}{2}+\frac{\sqrt{c^{2}-2a}}{2}>\beta_{1}>-\frac{c}{2}-\frac{\sqrt{c^{2}-2a}}{2}$ or $c^{2}>0>2a$, $\frac{c}{2}-\frac{\sqrt{c^{2}-2a}}{2}>\beta_{1}>-\frac{c}{2}-\frac{\sqrt{c^{2}-2a}}{2}$ (see Fig. 1(b)). Eq. (3.2) can generate periodic solutions when $\beta_{1}$ belongs to other intervals.

2. (2)

   $c^{2}\leq 2a$, $\forall\beta_{1}\in R$, Eq. (3.2) also generate periodic solution (see Fig. 1(c)).

For n=2: The two-fold DT formula (2.14) of the reverse-space-time DNLS equation implies a solution

where

We can derive breathers and double-periodic solution according to different reduced methods of spectrum parameter $\lambda_{1}$ and $\lambda_{2}$ as follows.

Case 1: $\lambda_{2}=\pm\lambda_{1}^{*}$, now $q[2]$ is a breathers. For simplicity, we take $\lambda_{2}=-\lambda_{1}^{*}=-\alpha_{1}+i\beta_{1}$ and $\operatorname{Im}\left(-a^{2}-4\lambda_{1}^{4}-4\lambda_{1}^{2}\left(c^{2}-a\right)\right)=0$. Then $\lim\limits_{\begin{subarray}{c}x\rightarrow\infty\\ t\rightarrow\infty\end{subarray}}|q[2]|^{2}=c^{2}$. and the center trajectory equation of solution $q[2]$ can be calculated as $x=4(\beta_{1}^{2}-\alpha_{1}^{2})t$. We can know that the solution evolves periodically along the straight line with a certain angle of $x$ axis and $t$ axis when $\beta_{1}^{2}\neq\alpha_{1}^{2}$ from the above analysis (see Fig. 2(a)). And when $\beta_{1}^{2}=\alpha_{1}^{2}$, the classical Ma breathers (time periodic breather) can be seen in Fig. 2(b) and the Akhmediev breathers (space periodic breather) can be seen in Fig. 2(c).

Next, we construct rogue wave $q_{r}$ by letting the period of the breathers tend to be infinity. Let $c\rightarrow-2\beta_{1}$ in $q[2]$, then

After calculation and analysis, we know that $\lim\limits_{\begin{subarray}{c}x\rightarrow\infty\\ t\rightarrow\infty\end{subarray}}|q_{r}|^{2}=(2\beta_{1})^{2}$. The maximum amplitude of $|q_{r}|^{2}$ equals to $(6\beta_{1})^{2}$ occurs at $x=0$ and $t=0$. This means that the maximum amplitude of the rogue wave $q_{r}$ is $3$ times compared with the asymptotic plane wave at infinity. The min amplitude of $|q_{r}|^{2}$ occurs at ($\sqrt{\frac{27\alpha_{1}^{4}}{16\beta_{1}^{2}(4\alpha_{1}^{2}+\beta_{1}^{2})(\alpha_{1}^{2}+\beta_{1}^{2})^{2}}}$, $\sqrt{\frac{3}{256\beta_{1}^{2}(4\alpha_{1}^{2}+\beta_{1}^{2})(\alpha_{1}^{2}+\beta_{1}^{2})^{2}}}$) and ($-\sqrt{\frac{27\alpha_{1}^{4}}{16\beta_{1}^{2}(4\alpha_{1}^{2}+\beta_{1}^{2})(\alpha_{1}^{2}+\beta_{1}^{2})^{2}}}$, $-\sqrt{\frac{3}{256\beta_{1}^{2}(4\alpha_{1}^{2}+\beta_{1}^{2})(\alpha_{1}^{2}+\beta_{1}^{2})^{2}}}$), which equals to $\frac{108(3\alpha_{1}^{4}-2\alpha_{1}^{2}\beta_{1}^{2}+4\beta_{1}^{4})\alpha_{1}^{4}\beta_{1}^{2}}{169\alpha_{1}^{8}-56\alpha_{1}^{6}\beta_{1}^{2}+6\alpha_{1}^{4}\beta_{1}^{4}-8\alpha_{1}^{2}\beta_{1}^{6}+4\beta_{1}^{8}}$. The rogue wave $|q_{r}|$ with specific parameter $\alpha_{1}=\frac{1}{2}$ and $\beta_{1}=1$ is plotted in Fig. (2(d)). From the graph of the rogue wave $|q_{r}|$ , we can see that the rogue wave $q_{r}$ has a single peak with two caves on both sides of the peak. The optical pulse $q_{r}$ only exists locally with all variables and disappears as time and space go far.

Case 2: $\lambda_{1}=i\beta_{1}$, $\lambda_{2}=i\beta_{2}$ and $\beta_{2}\neq\pm\beta_{1}$, $q[2]$ is represented as a double-periodic wave solution which is similar to the Jacobi elliptic function-type seed solution. From the Fig. 3, we can see clearly that there are two periodic waves with different directions in the double-periodic wave solution, and when the two waves with different directions are superimposed on each other, a higher wave peak can be generated. From a visual perspective, it seems that several parallel breathers are generated under the period background.

We find that the periodic solution can be generated by first-fold DT. The breathers and double-periodic solutions can be constructed respectively according to second-fold DT. Therefore, we consider constructing the breathers on the periodic background by odd-fold DT, and constructing the breathers on the double-periodic background by even-fold DT.

For n=3: Set $\lambda_{2}=-\lambda_{1}^{*}=-\alpha_{1}+i\beta_{1}$, $\lambda_{3}=i\beta_{3}$ and $a=2\alpha_{1}^{2}-2\beta_{1}^{2}+c^{2}$. Parameter values have a great influence on the propagation direction of the breathers. When $\beta_{1}^{2}\neq\alpha_{1}^{2}$, solution $q[3]$ is a general breather solution on periodic background (see Fig. 4(a)). When $\beta_{1}^{2}=\alpha_{1}^{2}$, the classical Ma breathers on the periodic background can be seen in Fig. 4(b) and the Akhmediev breathers on the periodic background can be seen in Fig. 4(c). There are some interesting phenomenons: Under the perturbation of the periodic background, the crest of the Ma breathers is cut and the phase shift occurs at the center of the breathers.

For n=4: Set $\lambda_{2}=-\lambda_{1}^{*}=-\alpha_{1}+i\beta_{1}$, $\lambda_{3}=i\beta_{3}$, $\lambda_{4}=i\beta_{4}$ and $\beta_{4}\neq\pm\beta_{3}$. The breathers on a double-periodic background generated by formula (2.14). Similar to $n=2$, we also let $a=2\alpha_{1}^{2}-2\beta_{1}^{2}+c^{2}$. When $\beta_{1}^{2}\neq\alpha_{1}^{2}$, we can construct the general breathers on the double-periodic (see Fig. 5(a)). Under the disturbance of double-periodic background, the propagation direction of general breathers usually produces shift. When $\beta_{1}^{2}=\alpha_{1}^{2}$, the Ma breathers and Akhmediev breathers can be constructed by adjusting spectrum parameters. As for the Ma breathers on the double-periodic, due to the great influence of the double-periodic background, the image of Ma breathers solution looks like it disappears in the double-periodic background (see Fig. 5(b)). The Akhmediev breathers on the double-periodic background is plotted in Fig. 5(c). Visually, it looks like a breathers with a higher amplitude is generated under the several parallel breathers background.

In addition, if we set $\lambda_{2}=-\lambda_{1}^{*}=-\alpha_{1}+i\beta_{1}$ and $\lambda_{4}=-\lambda_{3}^{*}=-\alpha_{3}+i\beta_{3}$. When $\beta_{3}^{2}-\alpha_{3}^{2}$=$\beta_{1}^{2}-\alpha_{1}^{2}$, we can construct the velocity resonance of two pairs of breathers (see Fig. 6(a) ). Otherwise, we can construct the elastic collision of two pairs of breathers (see Fig. 6(b) ).

Note that the eigenfunction is degenerated when $\lambda=\pm\frac{1}{2}\sqrt{2a-c^{2}}-\frac{1}{2}ic$. In this case, the higher-order rogue waves can be derived. Combined with the methods of constructing the periodic and double-periodic background in the previous section, we will give the solutions of higher-order rogue waves on the periodic and double-periodic background in this section. Since periodic can be derived by odd-fold DT, both breathers and double-periodic solution can be obtained by even-fold DT. We can construct higher-order rogue waves on the periodic by odd-order semi-degenerate DT and higher-order rogue waves on the double-periodic by even-order semi-degenerate DT.