In this section, we give some explicit solutions of Eq. (4) via the DT in Section 2. Now we take the nonzero continuous wave (cw) solution \(u=c e^{i(a x+b t)}\) as the initial seed for Eq. (4), where a, b, c are all real parameters. Equation (4) requires the frequency b to satisfy the following nonlinear dispersion relation:
$$ b=\epsilon \bigl(6 a \delta^{2} c^{2}-a^{3}\bigr)+ \alpha \bigl(2 \delta^{2} c^{2}-a^{2} \bigr).$$
(29)
Thus, the cw seed solution of the general Hirota equation (4) is
$$ u=c e^{i(a x+(\epsilon (6 a \delta^{2} c^{2}-a^{3})+\alpha (2 \delta^{2} c^{2}-a^{2})) t)}. $$
(30)
Setting \((f_{1},g_{1})^{T}\) to be the solution of the spectral problem (6), we have
$$\begin{aligned}& f_{1x}=-i \lambda f_{1}+\delta u g_{1}, \qquad g_{1x}=i \lambda g_{1}-\delta u^{*} f_{1}, \end{aligned}$$
(31)
$$\begin{aligned}& f_{1t}=v_{11} f_{1}+v_{12} g_{1}, \qquad g_{1t}=v_{21} f_{1}+v_{22} g_{1}, \end{aligned}$$
(32)
where
$$\begin{aligned}& v_{11}=4 i \epsilon \lambda^{3}-2 i \alpha \lambda^{2}-2 i \epsilon \delta^{2} u^{2} \lambda+i \alpha \delta^{2} u^{2}, \end{aligned}$$
(33)
$$\begin{aligned}& v_{22}=-4 I \epsilon \lambda^{3}+2 i \alpha \lambda^{2}+2 i \epsilon \delta^{2} u^{2} \lambda-i \alpha \delta^{2} u^{2}, \end{aligned}$$
(34)
$$\begin{aligned}& v_{12}=-4 \epsilon \delta u \lambda^{2}+(2 \alpha \delta u-2 i \epsilon \delta u_{x}) \lambda+\epsilon \delta u_{xx}+i \alpha \delta u_{x} +2 \epsilon \delta^{3} u^{3} , \end{aligned}$$
(35)
$$\begin{aligned}& v_{21}=4 \epsilon \delta u \lambda^{2}+(-2 \alpha \delta u-2 i \epsilon \delta u_{x}) \lambda-\epsilon \delta u_{xx} +i \alpha \delta u_{x}-2 \epsilon \delta^{3} u^{3}. \end{aligned}$$
(36)
After tedious computations, we obtain that
$$\begin{aligned}& f_{1}=\bigl(C_{1} e^{A}+C_{2} e^{-A}\bigr) e^{B}, \end{aligned}$$
(37)
$$\begin{aligned}& g_{1}=i \biggl[C_{2} \frac{\sqrt{(a+2\lambda)^{2}+4 \delta^{2} c^{2}}+(a+2 \lambda)}{2 \delta c}-C_{1} \frac{\sqrt{(a+2\lambda)^{2}+4 \delta^{2} c^{2}}-(a+2 \lambda)}{2 \delta c}\biggr] e^{-B}, \end{aligned}$$
(38)
with
$$\begin{aligned}& A=\frac{i \sqrt{(a+2\lambda)^{2}+4 \delta^{2} c^{2}}}{2} \bigl[x-\bigl(a^{2} \epsilon +4 \lambda^{2} \epsilon-2 \delta^{2} c^{2} \epsilon-2 \lambda a \epsilon+\alpha a-2 \lambda \alpha\bigr) t\bigr], \end{aligned}$$
(39)
$$\begin{aligned}& B=\frac{i}{2} \bigl[a x-\bigl(a^{3} \epsilon-6 \epsilon \delta^{2} c^{2} a+a^{2} \alpha-2\alpha \delta^{2} c^{2}\bigr) t\bigr]. \end{aligned}$$
(40)
Multisoliton solutions
In this subsection, we present the multisoliton solutions of Eq. (4) explicitly. To do so, take \(c=0\), that is, zero initial seed solution for Eq. (4). The solutions of the spectral problem (6) in Eqs. (37) and (38) are reduced to the following forms:
$$ f_{1}=e^{-i \lambda x+(4 i \epsilon \lambda^{3}-2 i \alpha \lambda^{2}) t}, \qquad g_{1}=e^{i \lambda x-(4 i \epsilon \lambda ^{3}-2 i \alpha \lambda^{2}) t}. $$
(41)
(1) When \(N=1\) in the Darboux transformation of Theorem 1, after some calculations, we have
$$ u^{(1)}_{ss}=-\frac{2 \eta_{1}}{\delta} e^{-2 i (\varepsilon_{1} x+t (4 \epsilon \varepsilon_{1} (3 \eta_{1}^{2}-\varepsilon _{1}^{2})+2 \alpha (\varepsilon_{1}^{2}-\eta_{1}^{2})))} \operatorname{sech} \bigl(2 \eta_{1} \bigl(x+4 \bigl(\epsilon \eta_{1}^{2}-3 \epsilon \varepsilon _{1}^{2}+ \varepsilon_{1} \alpha\bigr)t \bigr)\bigr), $$
(42)
where \(\lambda=\varepsilon_{1}+i \eta_{1}\), and \(u^{(N)}_{ss}\) is the Nth-order soliton solution.
The solution \(u^{(1)}_{ss}\) represents a bright single soliton whose dynamic features are delineated in Figure 1. We can conclude some physical quantities from this bright single soliton: the maximum amplitude is \(|\frac{2 \eta_{1}}{\delta}|\); the width is \(\frac {1}{2 \eta_{1}}\); the envelop velocity is \(-4 \eta_{1} (\epsilon \eta_{1}^{2}-3 \epsilon \varepsilon_{1}^{2}+\varepsilon_{1} \alpha )\); the frequency is \(-4 \epsilon \varepsilon_{1} (3 \eta _{1}^{2}-\varepsilon_{1}^{2})-2 \alpha (\varepsilon_{1}^{2}-\eta _{1}^{2})\), and the energy E is equal to \(|\frac{4\eta_{1}}{\delta ^{2}}|\).
(2) When \(N=2\) in the Darboux transformation of Theorem 1, we obtain the following two-soliton solution of Eq. (4):
$$ u^{(2)}_{ss}=u^{(1)}_{ss}-\frac{2 i}{\delta} \frac{f_{2}^{(1) *} g_{2}^{(1)} (\lambda_{2}-\lambda _{2}^{*})}{|f_{2}^{(1)}|^{2}+|g_{2}^{(1)}|^{2}}, $$
(43)
where \(u^{(1)}_{ss}\) is given in Eq. (42), and \((f_{2}^{(1)}, g_{2}^{(1)})^{T}\) is the solution of the spectral problem (6) at \(\lambda=\lambda_{2}\).
The dynamics of the two-soliton solution \(u^{(2)}_{ss}\) is delineated in Figure 2 with parameters \(\alpha=1\), \(\epsilon=1\), \(\delta=1\), \(\lambda _{1}=i\), \(\lambda_{2}=2 i\). The solution \(u^{(2)}_{ss}\) represents the elastic interaction between two bright solitons whose dynamic features are delineated in Figure 2. Figure 2 shows the overtaking interaction between two bell-shape solitons for \(u^{(2)}_{ss}\) in Eq. (43), from which we can find that the solitonic shapes and amplitudes have not changed after the interaction.
(3) When \(N=3\), we obtain the following three-soliton solution of Eq. (4):
$$ u^{(3)}_{ss}=u^{(2)}_{ss}- \frac{2 i}{\delta} \frac{f_{3}^{(2) *} g_{3}^{(2)} (\lambda_{3}-\lambda _{3}^{*})}{|f_{3}^{(2)}|^{2}+|g_{3}^{(2)}|^{2}}, $$
(44)
where \(u^{(2)}_{ss}\) is given in Eq. (43), and \((f_{3}^{(2)}, g_{3}^{(2)})^{T}\) is the solution of the spectral problem (6) at \(\lambda=\lambda_{3}\).
The dynamic features of the three-soliton solution \(u^{(3)}_{ss}\) is delineated in Figure 3 with parameters \(\alpha=1\), \(\epsilon=1\), \(\delta =1\), \(\lambda_{1}=i\), \(\lambda_{2}=2 i\), \(\lambda_{3}= 4 i\). The solution \(u^{(3)}_{ss}\) represents the elastic interaction among three bright solitons whose dynamic features are delineated in Figure 3. Figure 3 shows the overtaking interaction among three bell-shape bright solitons for \(u^{(3)}_{ss}\) in Eq. (44), from which we can find that the solitonic shapes and amplitudes have not changed after the interaction.
(4) When \(N=4\), we also obtain the four-soliton solution of Eq. (4) by the same procedure:
$$ u^{(4)}_{ss}=u^{(3)}_{ss}- \frac{2 i}{\delta} \frac{f_{4}^{(3) *} g_{4}^{(3)} (\lambda_{4}-\lambda _{4}^{*})}{|f_{4}^{(3)}|^{2}+|g_{4}^{(3)}|^{2}}, $$
(45)
where \(u^{(3)}_{ss}\) is given in Eq. (44), and \((f_{4}^{(3)}, g_{4}^{(3)})^{T}\) is the solution of the spectral problem (6) at \(\lambda=\lambda_{4}\).
The dynamic features of the four-soliton solution \(u^{(4)}_{ss}\) is delineated in Figure 4 with parameters \(\alpha=1\), \(\epsilon=1\), \(\delta =1\), \(\lambda_{1}=i\), \(\lambda_{2}=2 i\), \(\lambda_{3}= 4 i\), \(\lambda_{4}= 3 i\). The solution \(u^{(4)}_{ss}\) represents the elastic interaction among four bright solitons whose dynamic features are shown in Figure 4. Figure 4 shows the overtaking interaction among four bell-shape bright solitons for solution \(u^{(4)}_{ss}\) in Eq. (45), from which we can find that the solitonic shapes and amplitudes have not changed after the interaction.
We remark that we can also get the high-order soliton solutions of the general Hirota equation (4) by continuing iteration of the DT in Theorem 1.
Multibreather solutions
In this subsection, we consider the case of parameter \(c\neq0\) in the seed solution of the general Hirota equation (4), which we will start with the plane wave seed solution. Without loss of generality, assuming that \(c=1\) and substituting the solution of the spectral problem (6) given by Eqs. (37) and (38) into the 1-fold DT in Eq. (22), the one-breather solution of the general Hirota equation (4) can be obtained immediately. Hereby, iterating the DT by using Theorem 1 we can derive the multibreather solutions of Eq. (4), where \(u^{(N)}_{bs}\) represents the Nth-order breather solution for simplicity. In the following, we only discuss two cases, \(N=1\) and \(N=2\). Moreover, we omit the expressions of the breather solutions because they are too long to write down.
(1) When \(N=1\), the one-breather solution \(u^{(1)}_{bs}\) is derived from the plane wave initial solution and 1-fold DT in Eq. (22), whose dynamic features are delineated in Figures 5 and 6.
Figures 5(a), (b) and 6(a), (b) display the structures of the one-breather solution \(u^{(1)}_{bs}\) for the general Hirota equation (4) with \(\epsilon\neq0\), and Figures 5(c), (d) and 6(c), (d) display the same case except the parameter \(\epsilon=0\). Wes see that the one-breather solution is periodic both in space and time for the parameter \(\epsilon \neq0\). Moreover, comparing Figure 5 with Figure 6, we observe that when \(|\operatorname{Im}(\lambda)|>1\), the evolution and density plots for the one-breather solution \(u^{(1)}_{bs}\) are periodic in time (i.e., Ma soliton), however, they are periodic in space (i.e., Akhmediev soliton) for \(0<|\operatorname{Im}(\lambda)|<1\) under the same parameters. We can see that the parameter ϵ can change the shape and the arrangement of bright solitons.
(2) When \(N=2\), the two-breather solution \(u^{(2)}_{bs}\) is also derived from the plane wave initial solution and 2-fold DT in Theorem 1, whose dynamic features are delineated in Figures 7 and 8.
From Figures 7 and 8 we know that: when \(\epsilon=0\), \(\operatorname{Re}(\lambda _{1})=0\), \(\operatorname{Re}(\lambda_{2})=0\), \(0<|\operatorname{Im}(\lambda_{1})|<1\), \(|\operatorname{Im}(\lambda_{2})|>1\), the solution \(u^{(2)}_{bs}\) represents the elastic interaction between a space periodic breather solution and a time-periodic breather solution (see Figure 7(a), (b)); when \(\epsilon=0\), \(\operatorname{Re}(\lambda_{1})=0\), \(\operatorname{Re}(\lambda_{2})\neq 0\), \(0<|\operatorname{Im}(\lambda_{1})|<1\), the solution \(u^{(2)}_{bs}\) represents the elastic interaction between two breather solutions, in witch one is a space-periodic breather solution, the other is a periodic breather solution in time and space (see Figure 7(c), (d)); when \(\epsilon=0\), \(\operatorname{Re}(\lambda_{1})=0\), \(\operatorname{Re}(\lambda_{2})=0\), \(0<|\operatorname{Im}(\lambda_{1})|<1\), \(0<|\operatorname{Im}(\lambda _{2})|<1\), the solution \(u^{(2)}_{bs}\) represents the elastic interaction between two space periodic breather soliton solutions (see Figure 8(a), (b)); when \(\epsilon=0\), \(\operatorname{Re}(\lambda_{1})=0\), \(\operatorname{Re}(\lambda_{2})=0\), \(|\operatorname{Im}(\lambda_{1})|>1\), \(|\operatorname{Im}(\lambda_{2})|>1\), the solution \(u^{(2)}_{bs}\) represents the interaction between two time-periodic breather soliton solutions (see Figure 8(c), (d)).
Rogue wave solutions
Next, we manage to search a generalized DT. Suppose that \(\varphi _{2}=\varphi_{1} (\lambda_{1}+\sigma)\) is a special solution for Eq. (6). Then after transformation we have \(\varphi _{2}^{(1)}=T_{1}^{(1)} \varphi_{2}\). Expanding \(\varphi_{2}\) at \(\lambda_{1}\), we have from [20]
$$ \varphi_{1} (\lambda_{1}+\sigma)=\varphi_{1}+ \varphi_{1}^{[1]} \sigma +\varphi_{1}^{[2]} \sigma^{2}+\cdots+\varphi_{1}^{[N]} \sigma ^{N}+\cdots, $$
(46)
where \(\varphi_{1}^{[k]}=\frac{1}{k!} \frac{\partial^{k}}{\partial \lambda^{k}} \varphi_{1} (\lambda)|_{\lambda=\lambda_{1}}\), and σ is a small parameter. Through the limit process
$$ \lim_{\sigma\rightarrow0} \frac{[T_{1}^{(1)}|_{\lambda=\lambda _{1}+\sigma}] \varphi_{2}}{\sigma}=\lim_{\sigma\rightarrow0} \frac {[\sigma+T_{1}^{(1)}|_{\lambda=\lambda_{1}}] \varphi_{2}}{\sigma }=\varphi_{1}+T_{1}^{(1)}|_{\lambda=\lambda_{1}} \varphi _{1}^{[1]}\equiv\varphi_{1}^{(1)}, $$
(47)
we can find another solution to the linear system Eq. (6) with \(u^{(1)}\) and spectral parameter \(\lambda=\lambda_{1}\). This allows us to go to the next step of the DT, namely, Eq. (24).
Similarly, the limit
$$\begin{aligned}& \lim_{\sigma\rightarrow0} \frac{[\sigma+T_{1}^{(2)}|_{\lambda=\lambda _{1}}] [\sigma+T_{1}^{(1)}|_{\lambda=\lambda_{1}}] \varphi_{2}}{\sigma^{2}} \\& \quad =\varphi_{1}+ \bigl[T_{1}^{(1)}+T_{1}^{(2)} \bigr]|_{\lambda=\lambda_{1}} \varphi _{1}^{[1]}+T_{1}^{(2)}|_{\lambda=\lambda_{1}} T_{1}^{(1)}|_{\lambda =\lambda_{1}} \varphi_{1}^{[2]} \equiv\varphi_{1}^{(2)} \end{aligned}$$
(48)
provides a nontrivial solution for the linear spectral problem with \(u^{(2)}\) and \(\lambda=\lambda_{1}\). Thus, we may do the third-step iteration of the DT as follows:
$$ \varphi^{(3)}=T_{1}^{(3)} \varphi^{(2)}, \qquad u^{(3)}=u^{(2)}-\frac{2 i s_{21}^{(2)}}{\delta}. $$
(49)
Continuing this process and combining all the DT, a generalized DT is constructed. Thus, we have the following theorem.
Theorem 2
Let
\(\varphi_{1},\varphi_{2},\ldots,\varphi_{N}\)
be
N
distinct solutions of the spectral problem Eq. (6) at
\(\lambda_{1},\ldots,\lambda_{N}\), respectively, and
$$ \varphi_{i} (\lambda_{i}+\sigma)=\varphi_{i}+ \varphi_{i}^{[1]} \sigma +\varphi_{i}^{[2]} \sigma^{2}+\cdots+\varphi_{i}^{[N]} \sigma ^{N}+\cdots\quad (i=1,2,\ldots,n) $$
(50)
be their expansions, where
$$ \varphi_{i}^{[j]}=\frac{1}{j!} \frac{\partial^{j}}{\partial \lambda ^{j}} \varphi_{i} (\lambda)|_{\lambda=\lambda_{i}}\quad (j=1,2,\ldots ). $$
(51)
Define
$$ T=\Gamma_{n} \Gamma_{n-1} \cdots \Gamma_{1} \Gamma_{0}, \qquad \Gamma _{k}=T_{k}^{(m_{k})} \cdots T_{k}^{(1)}\quad (i\geq1), \qquad \Gamma _{0}=I, $$
(52)
where
$$\begin{aligned}& T_{k}^{(j+1)}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} \lambda_{k}-s_{11}^{(j)}&-s_{12}^{(j)}\\ -s_{21}^{(j)}&\lambda_{k}-s_{22}^{(j)} \end{array}\displaystyle \right ), \end{aligned}$$
(53)
$$\begin{aligned}& \varphi_{k}^{(j)}=\lim_{\sigma\rightarrow0} \frac{[\sigma +T_{k}^{(j)}|_{\lambda=\lambda_{k}}] \cdots [\sigma +T_{k}^{(2)}|_{\lambda=\lambda_{k}}] [\sigma+T_{k}^{(1)}|_{\lambda =\lambda_{k}}] \Gamma_{k-1} (\lambda_{k}+\sigma) \cdots \Gamma_{1} (\lambda_{k}+\sigma) \Gamma_{0} \varphi_{k} (\lambda_{k}+\sigma )}{\sigma^{j}} \end{aligned}$$
(54)
(\(1\leq j < m_{i}\)). Then the transformations
$$ \varphi^{(N)}=T \varphi, \qquad u^{(N)}=u+\frac{2 i}{\delta} \sum_{j=0}^{n-1} \sum _{k=1}^{m_{i}} s_{21}^{(j)} [m_{k}] \quad \Biggl(N=n-1+\sum_{k=1}^{n-1} m_{k}\Biggr) $$
(55)
are the generalized DT for the general Hirota equation (4).
We remark that the solution expressed in Eq. (55) is written in terms of summations, which is very easy to understand. In fact, for nonzero \(\varphi^{(k)}\) (\(k=1, 2, \ldots, N \)), all the denominators of \(s_{21}^{(j)}\) are easily seen to be nonzero in these forms; therefore, Eq. (55) provides some nonsingular solutions.
Let us consider an example to illustrate the application of Theorem 2 to the construction of rogue wave solutions. To do so, starting with the seed solution \(u=e^{8 i t}\), the solution of the linear spectral problem (6) at \(\lambda=i h\) is
$$ \varphi_{1} (f)=\left ( \textstyle\begin{array}{@{}c@{}} e^{B} (C_{1} e^{A}+C_{2} e^{-A}) \\ i e^{-B} (C_{1} e^{-A}-C_{2} e^{A})\end{array}\displaystyle \right ), $$
(56)
where
$$ C_{1}=\frac{\sqrt{(2 \sqrt{(4+\lambda^{2})}+2 \lambda)}}{2},\qquad C_{2}=\frac{\sqrt{(2 \sqrt{(4+\lambda^{2})}-2 \lambda)}}{2}, \qquad B=4 i t .$$
(57)
Let \(h=2+f^{2}\). Expanding the vector function \(\varphi_{1} (f)\) at \(f=0\), we have
$$ \varphi_{1} (f)=\varphi_{1}^{[0]}+ \varphi_{1}^{[1]} f^{2}+\varphi _{1}^{[2]} f^{4}+\varphi_{1}^{[3]} f^{6}+\cdots, $$
(58)
where
$$\begin{aligned}& \varphi_{1}^{[0]}=\left ( \textstyle\begin{array}{@{}c@{}} \sqrt{2} e^{4 i t}\\ -\sqrt{2} e^{-4 i t} \end{array}\displaystyle \right ), \end{aligned}$$
(59)
$$\begin{aligned}& \varphi_{1}^{[1]}= \left ( \textstyle\begin{array}{@{}c@{}} \frac{\sqrt{2}}{200} e^{4 i t} ((7\text{,}680+4\text{,}096 i) t^{2}+(800-480 i) t-400 i x^{2}-200 i x+(3\text{,}200-1\text{,}920 i) x t-25 i) \\ -\frac{\sqrt{2}}{200} e^{-4 i t} ((7\text{,}680+4\text{,}096 i) t^{2}+(480 i-800) t-400 i x^{2}+200 i x+(3\text{,}200-1\text{,}920 i) x t-25 i) \end{array}\displaystyle \right ), \end{aligned}$$
(60)
$$\begin{aligned}& \varphi_{1}^{[2]}=\left ( \textstyle\begin{array}{@{}c@{}} P\\ Q \end{array}\displaystyle \right ), \end{aligned}$$
(61)
with
$$\begin{aligned}& P= \frac{\sqrt{2} e^{ i t}}{240\text{,}000} \bigl(1\text{,}875-456\text{,}000 t-30\text{,}000 x-1 \text{,}152\text{,}000 x^{2} t+64\text{,}880\text{,}640 t^{3} x \\& \hphantom{P={}}{}-1\text{,}536\text{,}000 t x^{3}+9\text{,}830\text{,}400 t^{2} x^{2}+4 \text{,}915\text{,}200 x t^{2}-2\text{,}400\text{,}000 t x+5\text{,}836 \text{,}800 t^{2} \\& \hphantom{P={}}{}-180\text{,}000 x^{2}-160\text{,}000 x^{4}-160\text{,}000 x^{3} +16\text{,}220\text{,}160 t^{3}+42\text{,}205 \text{,}184 t^{4} \\& \hphantom{P={}}{}-1\text{,}920\text{,}000 i t x^{2}-4 \text{,}608\text{,}000 i t b_{1}+480\text{,}000 d_{1}-2 \text{,}560\text{,}000 i x^{3} t +1\text{,}920\text{,}000 x d_{1} \\& \hphantom{P={}}{}-3\text{,}360 \text{,}000 i x t-1\text{,}920\text{,}000 i x b_{1}+7\text{,}680 \text{,}000 t b_{1}+4\text{,}608\text{,}000 t d_{1}-819 \text{,}200 i t^{3} \\& \hphantom{P={}}{}-9\text{,}216\text{,}000 i t^{2} x-600\text{,}000 i t-18\text{,}432\text{,}000 i t^{2} x^{2}-3\text{,}276 \text{,}800 i t^{3} x \\& \hphantom{P={}}{}+7\text{,}680\text{,}000 i t d_{1} -14\text{,}208\text{,}000 i t^{2}+62\text{,}914 \text{,}560 i t^{4}-480\text{,}000 i b_{1}\bigr), \end{aligned}$$
(62)
$$\begin{aligned}& Q= \frac{\sqrt{2} e^{ i t}}{240\text{,}000} \bigl(1\text{,}875+456\text{,}000 t+30\text{,}000 x-1 \text{,}920\text{,}000 i x b_{1}-3\text{,}276\text{,}800 i t^{3} x \\& \hphantom{Q={}}{}+480\text{,}000 i b_{1}+1\text{,}152\text{,}000 x^{2} t +64\text{,}880\text{,}640 t^{3} x-1\text{,}536 \text{,}000 t x^{3}+9\text{,}830\text{,}400 t^{2} x^{2} \\& \hphantom{Q={}}{}-4\text{,}915\text{,}200 x t^{2}-2\text{,}400\text{,}000 t x+7\text{,}680\text{,}000 i t d_{1}-3\text{,}360\text{,}000 i x t +5\text{,}836\text{,}800 t^{2} \\& \hphantom{Q={}}{}-180\text{,}000 x^{2}-160\text{,}000 x^{4}+160\text{,}000 x^{3}-16\text{,}220\text{,}160 t^{3}+42\text{,}205\text{,}184 t^{4} \\& \hphantom{Q={}}{}-2\text{,}560\text{,}000 i x^{3} t-4\text{,}608 \text{,}000 i t b_{1}+1\text{,}920\text{,}000 i t x^{2}+819 \text{,}200 i t^{3}+1\text{,}920\text{,}000 x d_{1} \\& \hphantom{Q={}}{}+600 \text{,}000 i t -14\text{,}208\text{,}000 i t^{2} +62\text{,}914 \text{,}560 i t^{4}+9\text{,}216\text{,}000 i t^{2} x \\& \hphantom{Q={}}{} +7 \text{,}680\text{,}000 t b_{1}+4\text{,}608\text{,}000 t d_{1}-18\text{,}432\text{,}000 i t^{2} x^{2}-480\text{,}000 d_{1}\bigr). \end{aligned}$$
(63)
It is clear that \(\varphi_{1}^{[0]}\) is a solution for Eq. (6) at \(\lambda=2 i\). In the following, we only discuss two cases, \(N=1\) and \(N=2\).
(1) When \(N=1\), by means of formula (47) we can obtain
$$ \varphi_{1}^{(1)}=\lim_{f\rightarrow0} \frac{[f^{2}+T_{1}^{(1)}] \varphi _{1} (f)}{f^{2}}=T_{1}^{(1)} \varphi_{1}^{[1]}+i \varphi_{1}^{[0]}. $$
(64)
From Eqs. (59), (60), and (64) we derive a simplified form of the first-order rogue wave solution:
$$ u^{(1)}_{rw}=-\frac{e^{8 i t} (75-8\text{,}704 t^{2}-1\text{,}920 x t-400 x^{2}+1\text{,}600 i t)}{25+8\text{,}704 t^{2}+1\text{,}920 x t+400 x^{2}}, $$
(65)
where \(u^{(N)}_{rw}\) is the Nth-order rogue wave solution.
Figure 9 displays the evolution and density plots of \(u^{(1)}_{rw}\) in Eq. (65), which shows that the first-order rogue wave solution is localized in both space and time.
(2) When \(N=2\), by means of formula (54) we have
$$ \varphi_{1}^{(2)}=\lim_{f\rightarrow0} \frac{[f^{2}+T_{1}^{(2)}] [f^{2}+T_{1}^{(1)}] \varphi_{1} (f)}{f^{4}}= T_{1}^{(2)} T_{1}^{(1)} \varphi_{1}^{[2]}+\bigl(T_{1}^{(1)}+T_{1}^{(2)} \bigr) \varphi_{1}^{[1]}+\varphi_{1}^{[0]}, $$
(66)
which helps to derive the second-order rogue wave solution with parameters \(b_{1}=0\), \(d_{1}=0 \) in Eqs. (62) and (63) as
$$ u^{(2, 1)}_{rw}=\frac{M_{1}}{N_{1}} $$
(67)
with
$$\begin{aligned}& M_{1}= e^{8 i t} \bigl(-659\text{,}411\text{,}697\text{,}664 t^{6}+(363\text{,}646\text{,}156\text{,}800 i-436\text{,}375 \text{,}388\text{,}160 x) t^{5} \\& \hphantom{M_{1}={}}{}+\bigl(160\text{,}432\text{,}128 \text{,}000 i x+ 52\text{,}592\text{,}640\text{,}000-187\text{,}170\text{,}816 \text{,}000 x^{2}\bigr) t^{4} \\& \hphantom{M_{1}={}}{}+\bigl(4\text{,}300\text{,}800 \text{,}000 i-47\text{,}185\text{,}920\text{,}000 x^{3}\bigr) t^{3}+\bigl(8\text{,}110\text{,}080\text{,}000 x^{2} \\& \hphantom{M_{1}={}}{}-8\text{,}601\text{,}600\text{,}000 x^{4}+460 \text{,}800\text{,}000 i x+7\text{,}372\text{,}800\text{,}000 i x^{3}+648\text{,}000\text{,}000\bigr) t^{2} \\& \hphantom{M_{1}={}}{}+\bigl(140 \text{,}400\text{,}000 x+499\text{,}200\text{,}000 x^{3}-45\text{,}000\text{,}000 i-288\text{,}000\text{,}000 i x^{2} \\& \hphantom{M_{1}={}}{}-921\text{,}600\text{,}000 x^{5}\bigr) t-64\text{,}000 \text{,}000 x^{6}-703\text{,}125\bigr), \end{aligned}$$
(68)
$$\begin{aligned}& N_{1}= 659\text{,}411\text{,}697\text{,}664 t^{6}+436 \text{,}375\text{,}388\text{,}160 x t^{5}+\bigl(187\text{,}170 \text{,}816\text{,}000 x^{2} \\& \hphantom{N_{1}={}}{}+36\text{,}981\text{,}964\text{,}800\bigr) t^{4}-\bigl(147\text{,}456\text{,}000 x-47\text{,}185\text{,}920 \text{,}000 x^{3}\bigr) t^{3} \\& \hphantom{N_{1}={}}{}+\bigl(596\text{,}160 \text{,}000-1\text{,}843\text{,}200\text{,}000 x^{2}+8\text{,}601 \text{,}600\text{,}000 x^{4}\bigr) t^{2} \\& \hphantom{N_{1}={}}{}+\bigl(921\text{,}600\text{,}000 x^{5}-38\text{,}400 \text{,}000 x^{3}+61\text{,}200\text{,}000 x\bigr) t \\& \hphantom{N_{1}={}}{}+140\text{,}625+64 \text{,}000\text{,}000 x^{6}+12\text{,}000\text{,}000 x^{4}+6\text{,}750\text{,}000 x^{2}. \end{aligned}$$
(69)
We show the structure of the second-order rogue wave solution Eq. (67) in Figure 10 under the parameters \(\alpha=1\), \(\epsilon=0.1\), \(\delta =2\), \(a=0\), \(\lambda=2 i\).
When choosing parameters \(b_{1}=100\), \(d_{1}=0 \) in Eqs. (62) and (63), we can obtain another second-order rogue wave solution of the general Hirota equation (4) as
$$ u^{(2, 2)}_{rw}=\frac{M_{2}}{N_{2}} $$
(70)
with
$$\begin{aligned}& M_{2}= \bigl(-659\text{,}411\text{,}697\text{,}664 t^{6}+(363\text{,}646\text{,}156\text{,}800 i-436\text{,}375 \text{,}388\text{,}160 x) t^{5} \\& \hphantom{M_{2}={}}{}+\bigl(52\text{,}592\text{,}640 \text{,}000 -187\text{,}170\text{,}816\text{,}000 x^{2}+160 \text{,}432\text{,}128\text{,}000 i x\bigr) t^{4} \\& \hphantom{M_{2}={}}{} +\bigl(98\text{,}304 \text{,}000\text{,}000+24\text{,}920\text{,}064\text{,}000 x -47\text{,}185\text{,}920\text{,}000 x^{3} \\& \hphantom{M_{2}={}}{}+4 \text{,}300\text{,}800\text{,}000 i+51\text{,}118\text{,}080\text{,}000 i x^{2}\bigr) t^{3}+\bigl(648\text{,}000\text{,}000 \\& \hphantom{M_{2}={}}{} +8\text{,}110\text{,}080\text{,}000 x^{2}-8 \text{,}601\text{,}600\text{,}000 x^{4}+1\text{,}105\text{,}920 \text{,}000\text{,}000 x \\& \hphantom{M_{2}={}}{}+294\text{,}912\text{,}000\text{,}000 i+7\text{,}372 \text{,}800\text{,}000 i x^{3} +460\text{,}800\text{,}000 i x\bigr) t^{2} \\& \hphantom{M_{2}={}}{}+\bigl(14 \text{,}400\text{,}000\text{,}000+140\text{,}400\text{,}000 x+499\text{,}200 \text{,}000 x^{3}-921\text{,}600\text{,}000 x^{5} \\& \hphantom{M_{2}={}}{} +230\text{,}400\text{,}000\text{,}000 x^{2}-45 \text{,}000\text{,}000 i+768\text{,}000\text{,}000 i x^{4} \\& \hphantom{M_{2}={}}{}-138 \text{,}240\text{,}000\text{,}000 i x-288\text{,}000\text{,}000 i x^{2} \bigr) t +(1\text{,}125\text{,}000-28\text{,}800\text{,}000\text{,}000 i) x^{2} \\ & \hphantom{M_{2}={}}{}+36\text{,}000\text{,}000 x^{4}-64\text{,}000 \text{,}000 x^{6}-1\text{,}440\text{,}000\text{,}703\text{,}125 \\ & \hphantom{M_{2}={}}{}-1 \text{,}800\text{,}000\text{,}000 i\bigr) e^{8 i t}, \end{aligned}$$
(71)
$$\begin{aligned}& N_{2}= 659\text{,}411\text{,}697\text{,}664 t^{6}+436 \text{,}375\text{,}388\text{,}160 x t^{5}+\bigl(36\text{,}981 \text{,}964\text{,}800 \\ & \hphantom{N_{2}={}}{}+187\text{,}170\text{,}816\text{,}000 x^{2}\bigr) t^{4}-\bigl(98\text{,}304\text{,}000\text{,}000+147\text{,}456 \text{,}000 x \\ & \hphantom{N_{2}={}}{}-47\text{,}185\text{,}920\text{,}000 x^{3}\bigr) t^{3}+\bigl(596\text{,}160\text{,}000-1\text{,}105\text{,}920 \text{,}000\text{,}000 x \\ & \hphantom{N_{2}={}}{}-1\text{,}843\text{,}200\text{,}000 x^{2}+8 \text{,}601\text{,}600\text{,}000 x^{4}\bigr) t^{2} \\ & \hphantom{N_{2}={}}{}+\bigl(921 \text{,}600\text{,}000 x^{5}-230\text{,}400\text{,}000\text{,}000 x^{2}-38\text{,}400\text{,}000 x^{3} \\ & \hphantom{N_{2}={}}{}+61\text{,}200\text{,}000 x+43\text{,}200\text{,}000\text{,}000 \bigr) t+12\text{,}000\text{,}000 x^{4} \\ & \hphantom{N_{2}={}}{}+64\text{,}000\text{,}000 x^{6}+1\text{,}440\text{,}000\text{,}140\text{,}625+6\text{,}750 \text{,}000 x^{2}. \end{aligned}$$
(72)
We show the structure of the second-order rogue wave solution in Eq. (70) by Figure 11 under the parameters \(\alpha=1\), \(\epsilon=0.1\), \(\delta=2\), \(a=0\), \(\lambda=2 i\).