1. Introduction
Nonlinear partial differential equations are often used to model various phenomena in fields such as physics, chemistry, biology and even social sciences [1–11]. With the help of exact solutions, various nonlinear phenomena can be better explained [1–7]. The construction of exact solutions of nonlinear equations, especially soliton solutions [12–14], is one of the most important and essential tasks in nonlinear science.
It is well known that the famous focusing nonlinear Schrödinger equation (FNLS equation for short)
$\begin{eqnarray}{\rm{i}}{u}_{t}+{u}_{{xx}}+| u{| }^{2}u=0,\end{eqnarray}$
can be used to describe optical solitons, self-trapping phenomena of nonlinear optics, propagation of thermal pulses in solids, Langnui waves in plasma, the motion of superconducting electrons in electromagnetic fields, Bose–Einstein condensation effect of atoms in lasers, and so on [15–17].There have been hundreds of studies and monographs on how to find solutions to equation (1 ) [18–34]. N-soliton solutions, breather solutions, rogue wave solutions, and multiple-pole solutions of the FNLS equation have been derived by Darboux transformation, inverse scattering and other approaches [23–30]. In particular, a very mature mechanism has been developed for obtaining multiple-pole solutions, higher-order rogue wave solutions and rogue waves with a double-periodic background using the Darboux transformation [25–30].
The multiple-pole solution is actually a weakly bound state of solitons, where the velocities and amplitudes of the solitons are almost the same. In this weakly bound state, the soliton travels along a curve, and the velocity and amplitude tend to be fixed values when the time is large enough. As with the multiple-pole solutions, similar asymptotic behavior exists for degeneration of breather solutions. As early as last century, it was reported that multiple-pole solutions of the FNLS equation were obtained by the inverse scattering method [31, 32]. In 2017, Schiebold [23] produced an important work that gave a rigorous and complete asymptotic description of the multiple-pole solutions for the FNLS equation. In the same year, Wang et al [29] first discovered the degeneration of N-breather solutions using the generalized Darboux transformation and creatively pointed out the connection between it and higher-order rogue waves. Subsequently, some scholars linked the inverse scattering method [24] or the Darboux transformation method [33] to the Riemann–Hilbert problem to explore the higher-order multiple-pole solutions.
However, there is no report on whether the Hirota’s method [14, 35–40], as a direct method to obtain the exact solution of the integrable system, can derive the multiple-pole solution of FNLS equation. According to the bilinear method, the N-soliton solution of the FNLS equation takes the following form [19, 20, 41]:
$\begin{eqnarray}u=\displaystyle \frac{{g}_{N}}{{f}_{N}},\end{eqnarray}$
with $\begin{eqnarray*}{f}_{N}=\sum _{\mu =0,1}{A}_{1}\left(\mu \right)\exp \left(\sum _{j=1}^{2N}{\mu }_{j}{\xi }_{j}+\sum _{1\leqslant j\lt l}^{2N}{\mu }_{j}{\mu }_{l}{\theta }_{{jl}}\right),\end{eqnarray*}$
$\begin{eqnarray*}{g}_{N}=\sum _{\mu =0,1}{A}_{2}\left(\mu \right)\exp \left(\sum _{j=1}^{2N}{\mu }_{j}{\xi }_{j}+\sum _{1\leqslant j\lt l}^{2N}{\mu }_{j}{\mu }_{l}{\theta }_{{jl}}\right),\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{l}{\xi }_{j}={\rm{i}}{k}_{j}^{2}t+{k}_{j}x+{\xi }_{j}^{(0)},{\xi }_{N+j}={\xi }_{j}^{* },\\ \left(j=1,2,3\cdots ,N\right),\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\exp \left({\theta }_{{jl}}\right)=\left\{\begin{array}{ll}2{\left({k}_{j}-{k}_{l}\right)}^{2} & j\lt l\leqslant N,\\ \displaystyle \frac{1}{2{\left({k}_{j}+{k}_{l-N}^{* }\right)}^{2}} & j\leqslant N,l\gt N,\\ 2{\left({k}_{j-N}^{* }-{k}_{l-N}^{* }\right)}^{2} & N\lt j\lt l\leqslant 2N.\end{array}\right.\end{eqnarray*}$
Here ${A}_{1}\left(\mu \right)$ and ${A}_{2}\left(\mu \right)$ are the summation over all possible combinations of μ1 = {0, 1}, μ2 = {0, 1}, ⋯, μ2N = {0, 1}. Moreover, ${A}_{1}\left(\mu \right)$ and ${A}_{2}\left(\mu \right)$ also satisfy conditions ${\sum }_{j=1}^{N}{\mu }_{j}={\sum }_{j=1}^{N}{\mu }_{N+j}$ and ${\sum }_{j=1}^{N}{\mu }_{j}={\sum }_{j=1}^{N}{\mu }_{N+j}+1$, respectively. Note that ${\xi }_{j}^{(0)}$ is a real parameter that describes the initial position of the soliton.The latest research [42] mentions a way to derive multiple-pole solutions directly from N-soliton solutions, which provides a possibility to concisely derive degenerate solutions from equation (2 ). After tedious verification, the idea mentioned in [42] is feasible for the FNLS equation. In other words, the multiple-pole solutions of equation (1 ) can be obtained by the Hirota’s bilinear method. Similarly, degenerate solutions can be derived directly from N-breather solutions without too much esoteric mathematics using these skillful limit tricks.
2. Double-pole solutions
By setting
$\begin{eqnarray}\begin{array}{l}N=2,{k}_{2}={k}_{1}+\epsilon ,{\xi }_{1}^{(0)}={\eta }_{1}^{(0)}+\mathrm{ln}\left(-\displaystyle \frac{\beta }{\epsilon }\right),{\xi }_{2}^{(0)}={\eta }_{1}^{(0)}+\mathrm{ln}\left(\displaystyle \frac{\beta }{\epsilon }\right),\end{array}\end{eqnarray}$
in equation ( $\begin{eqnarray}{u}_{2-p}=\displaystyle \frac{g}{f},\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{l}g=\beta {\partial }_{{k}_{1}}\exp \left({\eta }_{1}\right)-\displaystyle \frac{{\beta }^{3}{\partial }_{{k}_{1}^{* }}\exp \left(2{\eta }_{1}+{\eta }_{1}^{* }\right)}{2{B}^{4}}\\ +\displaystyle \frac{2{\beta }^{3}\exp \left(2{\eta }_{1}+{\eta }_{1}^{* }\right)}{{B}^{5}},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}f=1-\displaystyle \frac{{\beta }^{2}{\partial }_{{k}_{1}}\exp \left({\eta }_{1}+{\eta }_{1}^{* }\right)}{{B}^{3}}-\displaystyle \frac{{\beta }^{2}{\partial }_{{k}_{1}^{* }}\exp \left({\eta }_{1}+{\eta }_{1}^{* }\right)}{{B}^{3}}\\ +\displaystyle \frac{{\beta }^{2}{\partial }_{{k}_{1}{k}_{1}^{* }}^{2}\exp \left({\eta }_{1}+{\eta }_{1}^{* }\right)}{2{B}^{2}}\\ +\displaystyle \frac{3{\beta }^{2}\exp \left({\eta }_{1}+{\eta }_{1}^{* }\right)}{{B}^{4}}+\displaystyle \frac{{\beta }^{4}\exp \left(2{\eta }_{1}+2{\eta }_{1}^{* }\right)}{4{B}^{8}},\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}{\eta }_{j}={k}_{j}x+{{\rm{i}}{k}}_{j}^{2}t+{\eta }_{j}^{(0)},\quad B=2{\rm{Re}}\left({k}_{1}\right).\end{eqnarray*}$
In proposition 2.1, both β and ε are real parameters. In this study, we define ${\partial }_{{k}_{1}^{* }}\exp \left({\eta }_{1}^{* }\right)$ as follows:
$\begin{eqnarray*}{\partial }_{{k}_{1}^{* }}\exp \left({\eta }_{1}^{* }\right)={\partial }_{\omega }{\left.\exp \left(\omega x-{\rm{i}}{\omega }^{2}t+{\eta }_{j}^{(0)* }\right)\right|}_{\omega ={k}_{1}^{* }}.\end{eqnarray*}$
According to the definition of the above equation, we know that ${\partial }_{{k}_{1}^{* }}\exp \left({\eta }_{1}^{}\right)=0$. When N = 2, g2 and f2 in equation (
$\begin{eqnarray}\begin{array}{l}{g}_{2}=\exp \left({\xi }_{1}\right)+\exp \left({\xi }_{2}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{3}+{\theta }_{12}+{\theta }_{13}+{\theta }_{23}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{4}+{\theta }_{12}+{\theta }_{14}+{\theta }_{24}\right),\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}{f}_{2}=1+\exp \left({\xi }_{1}+{\xi }_{3}+{\theta }_{13}\right)+\exp \left({\xi }_{1}+{\xi }_{4}+{\theta }_{14}\right)\\ +\,\exp \left({\xi }_{2}+{\xi }_{3}+{\theta }_{23}\right)+\exp \left({\xi }_{2}+{\xi }_{4}+{\theta }_{24}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{3}+{\xi }_{4}+{\theta }_{12}+{\theta }_{13}\right.\\ \left.+\,{\theta }_{14}+{\theta }_{23}+{\theta }_{24}+{\theta }_{34}\right),\end{array}\end{eqnarray}$
respectively. Substituting the constraints mentioned in proposition 2.1 into equation (Note that these calculations are so boring and tedious that they cannot be obtained correctly without the use of a computer.□
Compared with the tedious Darboux transformation [33, 34], the method of obtaining multiple-pole solutions presented by proposition 2.1 is very fast and simple. In contrast to the mathematically demanding inverse scattering method [23, 31, 32], this method does not require the knowledge of complex analysis to achieve the same results. Figure 1 graphically illustrates that the results obtained in this paper are in high agreement with those obtained by inverse scattering [23, 31, 32] and Darboux transformation [33, 34].
Figure 1. (a) $\left|{u}_{2-p}\right|$ described by equation ( |
In equation (4 ), ${\eta }_{1}^{(0)}$ and β jointly determine the position of the double-pole solution at time t = 0. When $\left\{{k}_{1}=a+{\rm{i}}{b},\beta =\pm 2a,{\eta }_{1}^{(0)}=0\right\}$ (a and b are real parameters), the modulus square of a double-pole solution equation (4 ) can be approximated as two modulus squares of a soliton with varying velocity as ∣t∣ goes to infinity:
$\begin{eqnarray}\begin{array}{l}{\left|{u}_{2-p}\right|}^{2}\approx 2{a}^{2}{{\rm{sech}} }^{2} \left(2{abt}-{ax}+\displaystyle \frac{\mathrm{ln}\left(16{a}^{4}{t}^{2}\right)}{2}+\displaystyle \frac{\mathrm{ln}\left(8{a}^{2}\right)}{2}\right)\\ +\,2{a}^{2}{{\rm{sech}} }^{2}\left(2{abt}-{ax}\right. \left.-\,\displaystyle \frac{\mathrm{ln}\left(16{a}^{4}{t}^{2}\right)}{2}+\displaystyle \frac{\mathrm{ln}\left(8{a}^{2}\right)}{2}\right).\end{array}\end{eqnarray}$
The proof process of equation (
By setting
$\begin{eqnarray}\begin{array}{rcl}N & = & 4,{k}_{2}={k}_{1}+\epsilon ,{\xi }_{1}^{(0)}={\eta }_{1}^{(0)}\\ & & +\mathrm{ln}\left(-\displaystyle \frac{\beta }{\epsilon }\right),{\xi }_{2}^{(0)}={\eta }_{1}^{(0)}+\mathrm{ln}\left(\displaystyle \frac{\beta }{\epsilon }\right),\\ {k}_{4} & = & {k}_{3}+\epsilon ,{\xi }_{3}^{(0)}={\eta }_{3}^{(0)}\\ & & +\mathrm{ln}\left(-\displaystyle \frac{\beta }{\epsilon }\right),{\xi }_{4}^{(0)}={\eta }_{3}^{(0)}+\mathrm{ln}\left(\displaystyle \frac{\beta }{\epsilon }\right),\end{array}\end{eqnarray}$
in equation (It can be observed from equation (7 ) that the position of the crest of $\left|{u}_{2-p}\right|$ is jointly determined by the linear term $2{bt}+\tfrac{\mathrm{ln}\left(8{a}^{2}\right)}{2a}$ and the nonlinear term $\pm \tfrac{\mathrm{ln}\left(16{a}^{4}{t}^{2}\right)}{2a}$, among which the linear term plays a major role. Thus, two double-pole solutions will undergo a brief collision shown in figure 2(a) when ${\rm{Im}}\left({k}_{1}\right)\ne {\rm{Im}}\left({k}_{3}\right)$. It is logical that if ${\rm{Im}}\left({k}_{1}\right)={\rm{Im}}\left({k}_{3}\right)$ and the positions of the two double-pole solutions are close enough to each other at the initial moment, then the strong coupling effect is shown in figure 2(b) will occur. In [23], Schiebold also discovered these phenomena in figure 2 through the esoteric and cumbersome operator theoretic approach.
Figure 2. Two different types of interactions between double-pole solutions: (a) a brief collision described by proposition 2.2 with parameters $\left\{{k}_{1}=1+\tfrac{{\rm{i}}}{2},{k}_{3}=1-\tfrac{{\rm{i}}}{2},\beta =2,{\eta }_{1}^{(0)}=0,{\eta }_{3}^{(0)}=0\right\};$ (b) a strong coupling described by proposition 2.2 with parameters $\left\{{k}_{1}=1,{k}_{3}=\tfrac{9}{10},\beta =2,{\eta }_{1}^{(0)}=0,{\eta }_{3}^{(0)}=\tfrac{1}{2}\right\}$. |
Unfortunately, continuing to introduce the module resonance condition ${k}_{1}=\pm {k}_{3}^{* }$ would lead to a denominator of 0 for $\exp \left({\theta }_{{jl}}\right)$ in equation (2 ). This means that in the FNLS system, it is impossible to use two double-pole solutions to generate a degenerate solution of a second-order breather wave through module resonance like the mKdV [42] and complex mKdV [45] equations.
More generally, by setting2 ), then a 2M-soliton solution will be reduced to an interaction between M double-pole solutions when ε → 0.
$\begin{eqnarray}\begin{array}{l}N=2M,{k}_{2}={k}_{1}+\epsilon ,{k}_{4}={k}_{3}+\epsilon ,{k}_{6}\\ ={k}_{5}+\epsilon ,\cdots ,{k}_{2M}={k}_{2M-1}+\epsilon ,\\ {\xi }_{1}^{(0)}={\eta }_{1}^{(0)}+\mathrm{ln}\left(-\displaystyle \frac{\beta }{\epsilon }\right),\\ {\xi }_{2}^{(0)}={\eta }_{1}^{(0)}\,+\,\mathrm{ln}\left(\displaystyle \frac{\beta }{\epsilon }\right),\cdots ,\\ {\xi }_{2M-1}^{(0)}={\eta }_{2M-1}^{(0)}\,+\,\mathrm{ln}\left(-\displaystyle \frac{\beta }{\epsilon }\right),\\ {\xi }_{2M}^{(0)}={\eta }_{2M-1}^{(0)}+\mathrm{ln}\left(\displaystyle \frac{\beta }{\epsilon }\right),\end{array}\end{eqnarray}$
in equation (Regretfully, the computational difficulty of the N-soliton solution of the FNLS equation is equivalent to that of the 2N-soliton solution of the mKdV equation [42]. Therefore, it is difficult to derive general expressions for the interaction of M double-pole solutions. However, the Darboux transformation method which requires a higher mathematical foundation does not have this defect [43].
3. Multiple-pole solutions
By setting
$\begin{eqnarray}\begin{array}{l}N=3,{k}_{2}={k}_{1}+\epsilon ,{k}_{3}={k}_{1}+2\epsilon ,\\ {\xi }_{1}^{(0)}={\eta }_{1}^{(0)}\,+\,\mathrm{ln}\left(\displaystyle \frac{\beta }{{\epsilon }^{2}}\right),\\ {\xi }_{2}^{(0)}={\eta }_{1}^{(0)}\,+\,\mathrm{ln}\left(\displaystyle \frac{-2\beta }{{\epsilon }^{2}}\right),\\ {\xi }_{3}^{(0)}={\eta }_{1}^{(0)}+\mathrm{ln}\left(\displaystyle \frac{\beta }{{\epsilon }^{2}}\right),\end{array}\end{eqnarray}$
in equation ( $\begin{eqnarray}{u}_{3-p}=\displaystyle \frac{g}{f},\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{l}g={w}_{1}\exp \left({\eta }_{1}\right)+{w}_{2}\exp \left(2{\eta }_{1}+{\eta }_{1}^{* }\right)\\ +{w}_{3}\exp \left(3{\eta }_{1}+2{\eta }_{1}^{* }\right),\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}f=1+{s}_{1}\exp \left({\eta }_{1}+{\eta }_{1}^{* }\right)+{s}_{2}\exp \left(2{\eta }_{2}+2{\eta }_{2}^{* }\right)\\ +\,\displaystyle \frac{8{\beta }^{6}}{{B}^{18}}\exp \left(3{\eta }_{3}+3{\eta }_{3}^{* }\right),\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}{w}_{1}=\beta {\eta }_{1,{k}_{1}{k}_{1}}+\beta {\eta }_{1,{k}_{1}}^{2},\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{w}_{2}=\displaystyle \frac{1}{{B}^{8}}\left[-{B}^{4}{\beta }^{3}{\left({\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }\right)}^{2}{\eta }_{1,{k}_{1}}^{2}-{B}^{4}{\beta }^{3}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }{\left({\eta }_{1,{k}_{1}}\right)}^{2}\right.\\ +\,{B}^{4}{\beta }^{3}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}{\eta }_{1,{k}_{1}{k}_{1}}+\,{B}^{4}{\beta }^{3}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }{\eta }_{1,{k}_{1}{k}_{1}}\\ +\,8{B}^{3}{\beta }^{3}{\eta }_{1,{k}_{1}^{* }}^{* }{\left({\eta }_{1,{k}_{1}}\right)}^{2}+\,4{B}^{3}{\beta }^{3}{\eta }_{1,{k}_{1}}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}\\ +\,4{B}^{3}{\beta }^{3}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }{\eta }_{1,{k}_{1}}-8{B}^{3}{\beta }^{3}{\eta }_{1,{k}_{1}^{* }}^{* }{\eta }_{1,{k}_{1}{k}_{1}}\\ -\,20{B}^{2}{\beta }^{3}{\left({\eta }_{1,{k}_{1}}\right)}^{2}-40{B}^{2}{\beta }^{3}{\eta }_{1,{k}_{1}}{\eta }_{1,{k}_{1}^{* }}^{* }-2{B}^{2}{\beta }^{3}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}\\ +\,20{B}^{2}{\beta }^{3}{\eta }_{1,{k}_{1}{k}_{1}}-2{B}^{2}{\beta }^{3}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }\\ \left.120B{\beta }^{3}{\eta }_{1,{k}_{1}}+24B{\beta }^{3}{\eta }_{1,{k}_{1}^{* }}^{* }-84{\beta }^{3}\right].\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{w}_{3}=\displaystyle \frac{1}{{B}^{14}}\left[4{B}^{2}{\beta }^{5}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}-4{B}^{2}{\beta }^{5}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }\right.\\ \left.-48B{\beta }^{5}{\eta }_{1,{k}_{1}^{* }}^{* }+120{\beta }^{5}\right],\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{s}_{1}=\displaystyle \frac{1}{2{B}^{6}}\left[{B}^{4}{\beta }^{2}{\left({\eta }_{1,{k}_{1}}\right)}^{2}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}+{B}^{4}{\beta }^{2}{\left({\eta }_{1,{k}_{1}}\right)}^{2}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }\right.\\ +\,{B}^{4}{\beta }^{2}{\eta }_{1,{k}_{1}{k}_{1}}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}+{B}^{4}{\beta }^{2}{\eta }_{1,{k}_{1}{k}_{1}}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }\\ -\,4{B}^{3}{\beta }^{2}{\eta }_{1,{k}_{1}^{* }}^{* }{\left({\eta }_{1,{k}_{1}}\right)}^{2}-4{B}^{3}{\beta }^{2}{\eta }_{1,{k}_{1}}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}\\ -\,4{B}^{3}{\beta }^{2}{\eta }_{1,{k}_{1}}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }-4{B}^{3}{\beta }^{2}{\eta }_{1,{k}_{1}^{* }}^{* }{\eta }_{1,{k}_{1}{k}_{1}}\\ +\,6{B}^{2}{\beta }^{2}{\left({\eta }_{1,{k}_{1}}\right)}^{2}+24{B}^{2}{\beta }^{2}{\eta }_{1,{k}_{1}}{\eta }_{1,{k}_{1}^{* }}^{* }+6{B}^{2}{\beta }^{2}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}\\ +\,6{B}^{2}{\beta }^{2}{\eta }_{1,{k}_{1}{k}_{1}}+6{B}^{2}{\beta }^{2}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }\\ \left.-48B{\eta }_{1,{k}_{1}}{\beta }^{2}-48B{\eta }_{1,{k}_{1}^{* }}^{* }{\beta }^{2}+120{\beta }^{2}\right],\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{s}_{2}=\displaystyle \frac{1}{{B}^{12}}\left[{B}^{4}{\beta }^{4}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}{\left({\eta }_{1,{k}_{1}}\right)}^{2}-{B}^{4}{\beta }^{4}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }{\left({\eta }_{1,{k}_{1}}\right)}^{2}\right.\\ -\,{B}^{4}{\beta }^{4}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}{\eta }_{1,{k}_{1}{k}_{1}}+{B}^{4}{\beta }^{4}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }{\eta }_{1,{k}_{1}{k}_{1}}\\ -\,8{B}^{3}{\beta }^{4}{\eta }_{1,{k}_{1}^{* }}^{* }{\left({\eta }_{1,{k}_{1}}\right)}^{2}-8{B}^{3}{\beta }^{4}{\eta }_{1,{k}_{1}}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}\\ +\,8{B}^{3}{\beta }^{4}{\eta }_{1,{k}_{1}}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }+8{B}^{3}{\beta }^{4}{\eta }_{1,{k}_{1}^{* }}^{* }{\eta }_{1,{k}_{1}{k}_{1}}\\ +\,12{B}^{2}{\beta }^{4}{\left({\eta }_{1,{k}_{1}}\right)}^{2}+72{B}^{2}{\beta }^{4}{\eta }_{1,{k}_{1}^{* }}^{* }{\eta }_{1,{k}_{1}}+12{B}^{2}{\beta }^{4}{\left({\eta }_{1,{k}_{1}^{* }}^{* }\right)}^{2}\\ -\,12{B}^{2}{\beta }^{4}{\eta }_{1,{k}_{1}{k}_{1}}-12{B}^{2}{\beta }^{4}{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }\\ \left.-\,120B{\beta }^{4}{\eta }_{1,{k}_{1}}-120B{\beta }^{4}{\eta }_{1,{k}_{1}^{* }}^{* }+228{\beta }^{4}\right].\end{array}\end{eqnarray*}$
In proposition 3.1, the notations ${\eta }_{1,{k}_{1}}$, ${\eta }_{1,{k}_{1}{k}_{1}}$, ${\eta }_{1,{k}_{1}^{* }}^{* }$ and ${\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }$ are defined as follows
$\begin{eqnarray*}\begin{array}{l}{\eta }_{1,{k}_{1}}={\partial }_{{k}_{1}}{\eta }_{1},{\eta }_{1,{k}_{1}{k}_{1}}={\partial }_{{k}_{1}{k}_{1}}^{2}{\eta }_{1},{\eta }_{1,{k}_{1}^{* }}^{* }\\ ={\partial }_{{k}_{1}^{* }}{\eta }_{1}^{* },{\eta }_{1,{k}_{1}^{* }{k}_{1}^{* }}^{* }={\partial }_{{k}_{1}^{* }{k}_{1}^{* }}^{2}{\eta }_{1}^{* }.\end{array}\end{eqnarray*}$
When N = 3, g3 and f3 in equation (
$\begin{eqnarray}\begin{array}{l}{g}_{3}=\exp \left({\xi }_{1}\right)+\exp \left({\xi }_{2}\right)\\ +\,\exp \left({\xi }_{3}\right)+\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{4}+{\theta }_{12}+{\theta }_{14}+{\theta }_{24}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{5}+{\theta }_{12}+{\theta }_{15}+{\theta }_{25}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{6}+{\theta }_{12}+{\theta }_{16}+{\theta }_{26}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{3}+{\xi }_{4}+{\theta }_{13}+{\theta }_{14}+{\theta }_{34}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{3}+{\xi }_{5}+{\theta }_{13}+{\theta }_{15}+{\theta }_{35}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{3}+{\xi }_{6}+{\theta }_{13}+{\theta }_{16}+{\theta }_{36}\right)\\ +\,\exp \left({\xi }_{2}+{\xi }_{3}+{\xi }_{4}+{\theta }_{23}+{\theta }_{24}+{\theta }_{34}\right)\\ +\,\exp \left({\xi }_{2}+{\xi }_{3}+{\xi }_{5}+{\theta }_{23}+{\theta }_{25}+{\theta }_{35}\right)\\ +\,\exp \left({\xi }_{2}+{\xi }_{3}+{\xi }_{6}+{\theta }_{23}+{\theta }_{26}+{\theta }_{36}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{3}+{\xi }_{4}+{\xi }_{5}+{\theta }_{12}+{\theta }_{13}+{\theta }_{14}+{\theta }_{15}\right.\\ \left.+{\theta }_{23}+{\theta }_{24}+{\theta }_{25}+{\theta }_{34}+{\theta }_{35}+{\theta }_{45}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{3}+{\xi }_{4}+{\xi }_{6}+{\theta }_{12}+{\theta }_{13}+{\theta }_{14}+{\theta }_{16}\right.\\ \left.+{\theta }_{23}+{\theta }_{24}+{\theta }_{26}+{\theta }_{34}+{\theta }_{36}+{\theta }_{46}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{3}+{\xi }_{5}+{\xi }_{6}+{\theta }_{12}+{\theta }_{13}+{\theta }_{15}+{\theta }_{16}\right.\\ \left.+{\theta }_{23}+{\theta }_{25}+{\theta }_{26}+{\theta }_{35}+{\theta }_{36}+{\theta }_{56}\right),\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}{f}_{3}=1+\,\exp \left({\xi }_{1}+{\xi }_{4}+{\theta }_{14}\right)+\,\exp \left({\xi }_{1}+{\xi }_{5}+{\theta }_{15}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{6}+{\theta }_{16}\right)+\exp \left({\xi }_{2}+{\xi }_{4}+{\theta }_{24}\right)\\ +\,\exp \left({\xi }_{2}+{\xi }_{5}+{\theta }_{25}\right)+\exp \left({\xi }_{2}+{\xi }_{6}+{\theta }_{26}\right)\\ +\exp \left({\xi }_{3}+{\xi }_{4}+{\theta }_{34}\right)+\exp \left({\xi }_{3}+{\xi }_{5}+{\theta }_{35}\right)\\ +\exp \left({\xi }_{3}+{\xi }_{6}+{\theta }_{36}\right)+\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{4}+{\xi }_{6}\right.\\ \left.+{\theta }_{12}+{\theta }_{14}+{\theta }_{16}+{\theta }_{24}+{\theta }_{26}+{\theta }_{46}\right)\\ +\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{4}+{\xi }_{5}+{\theta }_{12}+{\theta }_{14}+{\theta }_{15}\right.\\ \left.+{\theta }_{24}+{\theta }_{25}+{\theta }_{45}\right)\\ +\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{5}+{\xi }_{6}+{\theta }_{12}+{\theta }_{15}\right.\\ \left.+{\theta }_{16}+{\theta }_{25}+{\theta }_{26}+{\theta }_{56}\right)\\ +\exp \left({\xi }_{1}+{\xi }_{3}+{\xi }_{4}+{\xi }_{5}+{\theta }_{13}+{\theta }_{14}\right.\\ \left.+{\theta }_{15}+{\theta }_{34}+{\theta }_{35}+{\theta }_{45}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{3}+{\xi }_{4}+{\xi }_{6}+{\theta }_{13}+{\theta }_{14}\right.\\ \left.+{\theta }_{16}+{\theta }_{34}+{\theta }_{36}+{\theta }_{46}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{3}+{\xi }_{5}+{\xi }_{6}+{\theta }_{13}+{\theta }_{15}\right.\\ \left.+{\theta }_{16}+{\theta }_{35}+{\theta }_{36}+{\theta }_{56}\right)\\ +\,\exp \left({\xi }_{2}+{\xi }_{3}+{\xi }_{4}+{\xi }_{5}+{\theta }_{23}+{\theta }_{24}\right.\\ \left.+{\theta }_{25}+{\theta }_{34}+{\theta }_{35}+{\theta }_{45}\right)\\ +\,\exp \left({\xi }_{2}+{\xi }_{3}+{\xi }_{4}+{\xi }_{6}+{\theta }_{23}+{\theta }_{24}\right.\\ \left.+{\theta }_{26}+{\theta }_{34}+{\theta }_{36}+{\theta }_{46}\right)\\ +\,\exp \left({\xi }_{2}+{\xi }_{3}+{\xi }_{5}+{\xi }_{6}+{\theta }_{23}+{\theta }_{25}\right.\\ \left.+{\theta }_{26}+{\theta }_{35}+{\theta }_{36}+{\theta }_{56}\right)\\ +\,\exp \left({\xi }_{1}+{\xi }_{2}+{\xi }_{3}+{\xi }_{4}+{\xi }_{5}+{\xi }_{6}+{\theta }_{12}\right.\\ +{\theta }_{13}+{\theta }_{14}+{\theta }_{15}+{\theta }_{16}\\ +{\theta }_{23}+{\theta }_{24}+{\theta }_{25}+{\theta }_{26}+{\theta }_{34}\\ +{\theta }_{35}+{\theta }_{36}\\ \left.+{\theta }_{45}+{\theta }_{46}+{\theta }_{56}\right),\end{array}\end{eqnarray}$
respectively. Substituting the constraints equation (Figure 3 clearly depicts the space-time structure and asymptotic behavior of a triple-pole solution. When $\left\{{k}_{1}=a+{\rm{i}}{b},\beta =\pm 2{a}^{2},{\eta }_{1}^{(0)}=0\right\}$ (a and b are real parameters), the modulus square of a triple-pole solution equation (11 ) can be approximated as14 ) implies that the extreme value $\sqrt{2}| a| $ of ∣u2−p∣ can be obtained along the trajectories $2{abt}-{ax}\pm \mathrm{ln}\left(8{a}^{4}{t}^{2}\right)+\tfrac{\mathrm{ln}\left(8{a}^{2}\right)}{2}=0$ and $2{abt}-{ax}+\tfrac{\mathrm{ln}\left(8{a}^{2}\right)}{2}=0$. The cyan curve in figure 3(b) visually verifies the correctness of equation (14 ).
$\begin{eqnarray}\begin{array}{l}{\left|{u}_{3-p}\right|}^{2}\approx 2{a}^{2}{{\rm{sech}} }^{2}\left(2{abt}-{ax}+\mathrm{ln}\left(8{a}^{4}{t}^{2}\right)+\displaystyle \frac{\mathrm{ln}\left(8{a}^{2}\right)}{2}\right)\\ +2{a}^{2}{{\rm{sech}} }^{2}\left(2{abt}-{ax}+\displaystyle \frac{\mathrm{ln}\left(8{a}^{2}\right)}{2}\right)\\ +2{a}^{2}{{\rm{sech}} }^{2}\left(2{abt}-{ax}-\mathrm{ln}\left(8{a}^{4}{t}^{2}\right)+\displaystyle \frac{\mathrm{ln}\left(8{a}^{2}\right)}{2}\right),\end{array}\end{eqnarray}$
when ∣t∣ goes to infinity. Equation (
Figure 3. (a) $\left|{u}_{3-p}\right|$ described by equation ( |
More generally, by setting
$\begin{eqnarray}\begin{array}{l}{k}_{2}={k}_{1}+\epsilon ,{k}_{3}={k}_{1}+2\epsilon ,{k}_{4}={k}_{1}\\ +3\epsilon ,\cdots ,{k}_{N}={k}_{1}+\left(N-1\right)\epsilon ,\\ {\xi }_{1}^{\left(0\right)}={\eta }_{1}^{\left(0\right)}+\mathrm{ln}\displaystyle \frac{{\left(-1\right)}^{N+1}{C}_{N-1}^{0}}{{\epsilon }^{N-1}},\\ {\xi }_{2}^{\left(0\right)}={\eta }_{1}^{\left(0\right)}+\mathrm{ln}\displaystyle \frac{{\left(-1\right)}^{N+2}{C}_{N-1}^{1}}{{\epsilon }^{N-1}},\\ {\xi }_{3}^{\left(0\right)}={\eta }_{1}^{\left(0\right)}+\mathrm{ln}\displaystyle \frac{{\left(-1\right)}^{N+3}{C}_{N-1}^{2}}{{\epsilon }^{N-1}},\cdots ,\\ {\xi }_{N}^{\left(0\right)}={\eta }_{1}^{\left(0\right)}+\mathrm{ln}\displaystyle \frac{{\left(-1\right)}^{2N}{C}_{N-1}^{N-1}}{{\epsilon }^{N-1}},\end{array}\end{eqnarray}$
in equation (Here ${C}_{n}^{r},\left(0\leqslant r\leqslant n,r,n\in {\mathbb{N}}\right)$ implies binomial coefficients, and ${C}_{n}^{r}=\tfrac{n!}{r!(n-r)!}$.
If N = 4, then proposition 3.2 will represent a quadruple-pole solution as shown in figure 4. When ∣t∣ goes to infinity, the quadruple-pole depicted in figure 4 has the following asymptotic behavior:
$\begin{eqnarray}\begin{array}{l}{\left|{u}_{4-p}\right|}^{2}\approx 2{{\rm{sech}} }^{2}\left(-x+\displaystyle \frac{\mathrm{ln}\left(4{t}^{2}\right)}{2}+\displaystyle \frac{3\mathrm{ln}\left(2\right)}{2}\right)\\ +2{{\rm{sech}} }^{2}\left(-x-\displaystyle \frac{\mathrm{ln}\left(4{t}^{2}\right)}{2}+\displaystyle \frac{3\mathrm{ln}\left(2\right)}{2}\right)\\ +2{{\rm{sech}} }^{2}\left(-x+\displaystyle \frac{3}{2}\mathrm{ln}\left(\displaystyle \frac{8}{3}\sqrt[3]{6}{t}^{2}\right)+\displaystyle \frac{3\mathrm{ln}\left(2\right)}{2}\right)\\ +2{{\rm{sech}} }^{2}\left(-x-\displaystyle \frac{3}{2}\mathrm{ln}\left(\displaystyle \frac{8}{3}\sqrt[3]{6}{t}^{2}\right)+\displaystyle \frac{3\mathrm{ln}\left(2\right)}{2}\right).\end{array}\end{eqnarray}$
The cyan curves in figure 4(b) represent the four trajectories $\{-x\pm \tfrac{\mathrm{ln}\left(4{t}^{2}\right)}{2}+\tfrac{3\mathrm{ln}\left(2\right)}{2}=0$, $-x\pm \tfrac{3}{2}\mathrm{ln}\left(\tfrac{8}{3}\sqrt[3]{6}{t}^{2}\right)+\tfrac{3\mathrm{ln}\left(2\right)}{2}=0\}$ along which the extreme points of ∣${u}_{4-p}$∣ can be reached. Unfortunately, the general expression for the quadruple-pole solution is so lengthy that its asymptotic behavior is difficult to find.
Figure 4. (a) $\left|{u}_{4-p}\right|$ described by proposition 3.2 with parameters $\left\{N=4,{k}_{1}=1,\beta =\tfrac{4}{3},{\eta }_{1}^{(0)}=0\right\}$; (b) Contour plot of (a), and the cyan curve represents the trajectory predicted by equation ( |
4. Degeneration of N-breather solutions
The method of deriving multiple-pole solutions directly from N-soliton solutions described in the previous two sections is also applicable to finding degenerate solutions of higher-order breather waves. With reference to [21], the N-breather solution of the FNLS equation takes the following form:17 ), k, ρ0 are real parameters and ${\phi }_{j},{{\rm{\Omega }}}_{j}^{(0)}$ are complex parameters.
$\begin{eqnarray}u={\rho }_{0}\exp \left({\rm{i}}\theta \right)\displaystyle \frac{{g}_{N}}{{f}_{N}},\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{l}{f}_{N}=\sum _{\mu =0,1}\exp \left(\sum _{j=1}^{2N}{\mu }_{j}{{\rm{\Gamma }}}_{j}+\sum _{1\leqslant j\lt l}^{2N}{\mu }_{j}{\mu }_{l}{A}_{{jl}}\right),\\ \quad {g}_{N}=\sum _{\mu =0,1}\exp \left(\sum _{j=1}^{2N}{\mu }_{j}{{\rm{\Omega }}}_{j}+\sum _{1\leqslant j\lt l}^{2N}{\mu }_{j}{\mu }_{l}{A}_{{jl}}\right),\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{l}\theta ={kx}-\left({k}^{2}-{\rho }_{0}^{2}\right)t,\quad {{\rm{\Gamma }}}_{j}={{\rm{\Omega }}}_{j}-2{\rm{i}}{\phi }_{j},\\ \exp \left({A}_{{jl}}\right)={\left[\displaystyle \frac{\sin \tfrac{1}{2}\left({\phi }_{j}-{\phi }_{l}\right)}{\sin \tfrac{1}{2}\left({\phi }_{j}+{\phi }_{l}\right)}\right]}^{2},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{{\rm{\Omega }}}_{j}={\rm{i}}\sqrt{2}{\rho }_{0}x\sin {\phi }_{j}\\ -\left(2{\rm{i}}k\sqrt{2}{\rho }_{0}\sin {\phi }_{j}+2{{\rho }_{0}}^{2}{\sin }^{2}{\phi }_{j}\cot {\phi }_{j}\right)t\\ +2\ {\rm{i}}{\phi }_{j}+{{\rm{\Omega }}}_{j}^{(0)},j=1,2,\cdots ,2N,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{\phi }_{n}={\phi }_{n+N}^{* }+\pi ,\quad {{\rm{\Omega }}}_{n}^{(0)}={{\rm{\Omega }}}_{n+N}^{(0)* },\\ n=1,2,\cdots ,N.\end{array}\end{eqnarray*}$
Note that in equation (By setting
$\begin{eqnarray}\begin{array}{l}N=2,{\phi }_{2}={\phi }_{1}+\epsilon ,{{\rm{\Omega }}}_{1}^{(0)}={\omega }_{1}^{(0)}\\ +\mathrm{ln}\left(-\displaystyle \frac{\beta }{\epsilon }\right),{{\rm{\Omega }}}_{2}^{(0)}={\omega }_{1}^{(0)}+\mathrm{ln}\left(\displaystyle \frac{\beta }{\epsilon }\right),\end{array}\end{eqnarray}$
in equation ( $\begin{eqnarray}{u}_{2-{db}}={\rho }_{0}\exp \left({\rm{i}}\theta \right)\displaystyle \frac{g}{f},\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{l}g=1+\beta {\omega }_{1,{\phi }_{1}}\exp \left({\omega }_{1}\right)+\beta {\omega }_{3,{\phi }_{3}}\exp \left({\omega }_{3}\right)\\ -\displaystyle \frac{{\beta }^{2}{C}_{11}}{4}\exp \left(2{\omega }_{1}\right)-\displaystyle \frac{{\beta }^{2}{C}_{33}}{4}\exp \left(2{\omega }_{3}\right)\\ +\left({\beta }^{2}{C}_{13}{\omega }_{3,{\phi }_{3}}{\omega }_{1,{\phi }_{1}}+{\beta }^{2}{C}_{13,{\phi }_{1}}{\omega }_{3,{\phi }_{3}}\right.\\ \left.+{\beta }^{2}{C}_{13,{\phi }_{3}}{\omega }_{1,{\phi }_{1}}+{\beta }^{2}{C}_{13,{\phi }_{1}{\phi }_{3}}\right)\exp \left({\omega }_{1}+{\omega }_{3}\right)\\ +\left(-\displaystyle \frac{{\beta }^{3}{{C}_{13}}^{2}{C}_{11}{\omega }_{3,{\phi }_{3}}}{4}-\displaystyle \frac{{\beta }^{3}{C}_{13}{C}_{11}{C}_{13,{\phi }_{3}}}{2}\right)\exp \left(2{\omega }_{1}+{\omega }_{3}\right)\\ +\left(-\displaystyle \frac{{\beta }^{3}{{C}_{13}}^{2}{C}_{33}{\omega }_{1,{\phi }_{1}}}{4}-\displaystyle \frac{{\beta }^{3}{C}_{13}{C}_{33}{C}_{13,{\phi }_{1}}}{2}\right)\\ \times \exp \left({\omega }_{1}+2{\omega }_{3}\right)\\ +\displaystyle \frac{{\beta }^{4}{C}_{11}{{C}_{13}}^{4}{C}_{33}}{16}\exp \left(2{\omega }_{1}+2{\omega }_{3}\right),\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}f=1+\beta {\gamma }_{1,{\phi }_{1}}\exp \left({\gamma }_{1}\right)+\beta {\gamma }_{3,{\phi }_{3}}\exp \left({\gamma }_{3}\right)\\ -\displaystyle \frac{{\beta }^{2}{C}_{11}}{4}\exp \left(2{\gamma }_{1}\right)-\displaystyle \frac{{\beta }^{2}{C}_{33}}{4}\exp \left(2{\gamma }_{3}\right)\\ +\left({\beta }^{2}{C}_{13}{\gamma }_{3,{\phi }_{3}}{\gamma }_{1,{\phi }_{1}}+{\beta }^{2}{C}_{13,{\phi }_{1}}{\gamma }_{3,{\phi }_{3}}\right.\\ \left.+{\beta }^{2}{C}_{13,{\phi }_{3}}{\gamma }_{1,{\phi }_{1}}+{\beta }^{2}{C}_{13,{\phi }_{1}{\phi }_{3}}\right)\exp \left({\gamma }_{1}+{\gamma }_{3}\right)\\ +\left(-\displaystyle \frac{{\beta }^{3}{{C}_{13}}^{2}{C}_{11}{\gamma }_{3,{\phi }_{3}}}{4}-\displaystyle \frac{{\beta }^{3}{C}_{13}{C}_{11}{C}_{13,{\phi }_{3}}}{2}\right)\exp \left(2{\gamma }_{1}+{\gamma }_{3}\right)\\ +\left(-\displaystyle \frac{{\beta }^{3}{{C}_{13}}^{2}{C}_{33}{\gamma }_{1,{\phi }_{1}}}{4}-\displaystyle \frac{{\beta }^{3}{C}_{13}{C}_{33}{C}_{13,{\phi }_{1}}}{2}\right)\\ \times \exp \left({\gamma }_{1}+2{\gamma }_{3}\right)+\displaystyle \frac{{\beta }^{4}{C}_{11}{{C}_{13}}^{4}{C}_{33}}{16}\exp \left(2{\gamma }_{1}+2{\gamma }_{3}\right),\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{l}{\omega }_{j}={\rm{i}}\sqrt{2}{\rho }_{0}x\sin {\phi }_{j}-\left(2{\rm{i}}k\sqrt{2}{\rho }_{0}\sin {\phi }_{j}\right.\\ \left.+2{\rho }_{0}^{2}{\sin }^{2}{\phi }_{j}\cot {\phi }_{j}\right)t+2{\rm{i}}{\phi }_{j}+{\omega }_{j}^{(0)},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{\gamma }_{j}={\omega }_{j}-2{\rm{i}}{\phi }_{j},{\phi }_{3}={\phi }_{1}^{* }+\pi ,{\omega }_{1}^{(0)}={\omega }_{3}^{(0)* },\\ \exp \left({C}_{{jl}}\right)=\left\{\begin{array}{ll}{\left[\displaystyle \frac{\sin \tfrac{1}{2}\left({\phi }_{j}-{\phi }_{l}\right)}{\sin \tfrac{1}{2}\left({\phi }_{j}+{\phi }_{l}\right)}\right]}^{2}, & j\ne l,\\ \displaystyle \frac{1}{{\sin }^{2}{\phi }_{j}}, & j\,=\,l,\end{array}\right..\end{array}\end{eqnarray*}$
The expression of equation (19 ) is verbose, so it is necessary to assign values to the parameters so that readers can easily observe the algebraic structure of u2−db and verify proposition 4.1. By selecting parameters $\left\{k=0,{\rho }_{0}=1,{\phi }_{1}=\tfrac{\pi }{4},\beta =1,{\omega }_{1}^{(0)}=0\right\}$, equation (19 ) can be simplified to20 ) is clearly and intuitively shown in figure 5(a).
$\begin{eqnarray}{u}_{2-{db}}=\exp \left({\rm{i}}t\right)\displaystyle \frac{g}{f},\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{l}g=\left(-4+4{\rm{i}}-2x\right){{\rm{e}}}^{-3t-{\rm{i}}x}+\left(x-2\right){{\rm{e}}}^{-t-{\rm{i}}x}\\ +\left(-4+4{\rm{i}}+2x\right){{\rm{e}}}^{-3t+{\rm{i}}{x}}+\left(-x-2\right){{\rm{e}}}^{-t+{\rm{i}}x}\\ +\left(-2{x}^{2}+2{\cos }^{2}x+2-8{\rm{i}}\right){{\rm{e}}}^{-2t}+4{{\rm{e}}}^{-4t}+1,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}f=\left(-2{\rm{i}}x-4\right){{\rm{e}}}^{-3t-{\rm{i}}x}-{\rm{i}}x{{\rm{e}}}^{-t-{\rm{i}}x}+\left(2{\rm{i}}x-4\right){{\rm{e}}}^{-3t+{\rm{i}}x}\\ +{\rm{i}}x{{\rm{e}}}^{-t+{\rm{i}}x}+\left(2{x}^{2}-2{\cos }^{2}x+6\right){{\rm{e}}}^{-2t}\\ +4{{\rm{e}}}^{-4t}+1.\end{array}\end{eqnarray*}$
The space-time structure of equation (
Figure 5. (a) A degenerate solution of a 2-breather solution $\left|{u}_{2-{db}}\right|$ described by equation ( |
Wang et al obtained degenerate solutions of breather solutions by Darboux transformation and pointed out how degenerate solutions of breather solutions reduce into higher-order rogue waves for the first time [29]. However, on the basis of the bilinear method, only a breather wave can be reduced to a first-order rogue wave by means of the long wave limit [21]. To put it bluntly, equation (17 ) degenerates into a first-order rogue wave by setting {N = 1, φ1 = φ1ε, φ2 = ${\varphi }_{1}^{* }\epsilon +\pi ,{{\rm{\Omega }}}_{1}^{(0)}$ = ${\rm{i}}\pi ,{{\rm{\Omega }}}_{2}^{(0)}$ = −iπ, ε → 0}. Regretfully, there are no relevant studies showing that the long-wave limit method can derive a higher-order rogue wave solution from a N-breather solution.
The mechanism for generating higher-order rogue waves consists of two steps: (i) obtaining a degenerate solution of a N-breather solution; (ii) making the period of the degenerate solution generated in the first step to infinity [29]. These two steps are also the main problems encountered in obtaining higher-order rogue wave solutions using bilinear methods. This study solves the first important step, which makes the idea of deriving higher-order waves from breather waves using bilinear methods a big step closer to becoming a reality. For obtaining a higher-order degenerate solution of a N-breather solution, we have the following proposition:
More generally, by setting
$\begin{eqnarray}\begin{array}{l}{\phi }_{2}={\phi }_{1}+\epsilon ,{\phi }_{3}={\phi }_{1}+2\epsilon ,{\phi }_{4}={\phi }_{1}+3\epsilon ,\cdots ,\\ {\phi }_{N}={\phi }_{1}+\left(N-1\right)\epsilon ,\\ {{\rm{\Omega }}}_{1}^{\left(0\right)}={\omega }_{1}^{\left(0\right)}+\mathrm{ln}\displaystyle \frac{{\left(-1\right)}^{N+1}{C}_{N-1}^{0}}{{\epsilon }^{N-1}},\\ {{\rm{\Omega }}}_{2}^{\left(0\right)}={\omega }_{1}^{\left(0\right)}+\mathrm{ln}\displaystyle \frac{{\left(-1\right)}^{N+2}{C}_{N-1}^{1}}{{\epsilon }^{N-1}},\\ {{\rm{\Omega }}}_{3}^{\left(0\right)}={\omega }_{1}^{\left(0\right)}+\mathrm{ln}\displaystyle \frac{{\left(-1\right)}^{N+3}{C}_{N-1}^{2}}{{\epsilon }^{N-1}},\cdots ,\\ {{\rm{\Omega }}}_{N}^{\left(0\right)}={\omega }_{1}^{\left(0\right)}+\mathrm{ln}\displaystyle \frac{{\left(-1\right)}^{2N}{C}_{N-1}^{N-1}}{{\epsilon }^{N-1}},\end{array}\end{eqnarray}$
in equation (Here ${C}_{n}^{r},\left(0\leqslant r\leqslant n,r,n\in {\mathbb{N}}\right)$ implies binomial coefficients, and ${C}_{n}^{r}=\tfrac{n!}{r!(n-r)!}$.
5. Conclusion
Based on the bilinear method, this paper systematically investigates the multiple-pole solutions and degenerate solutions of focusing nonlinear Schrödinger equation by some skillful limiting means. The greatest innovation of this study is that it provides a simple and fast method to derive the multiple-pole solutions of FNLS equation. Proposition 2.1 and proposition 3.1 give specific approaches to the double-pole solution and the triple-pole solution and give general expressions equation (4 ) and equation (11 ) for the relevant solutions. Proposition 3.2 summarizes proposition 2.1 and proposition 3.1, and points out a general method to derive multiple-pole solutions directly from N-soliton solutions. The space-time structure and dynamical properties of these multi-pole solutions are described by beautiful images (figures 1, 3, 4) and rigorous mathematical expressions (equations (7 ), (14 ), (16 )) respectively. Similarly, this limit method mentioned in this paper can also be used to achieve degeneration of a N-breather solution (see proposition 4.1 and proposition 4.2 for details). There are two limiting steps to convert the higher-order breather solution to the higher-order rogue wave solution. For the first limiting step, proposition 4.1 has been able to derive a degenerate solution from the N-breather solution; however, for the second step, it is worth further thinking how to convert the higher-order degenerate solution obtained at the first stage into a higher-order rogue wave. In addition, this method can be extended to a derivative Schrödinger equation, sine-Gorden equation, mKdV equation and so on.