Non-compatible fully PT symmetric Davey–Stewartson system: soliton, rogue wave and breather in nonzero wave background

Abstract

In this paper, we study a non-compatible fully PT symmetric Davey–Stewartson system by removing compatible condition of potential field from compatible case. This move makes the first two equations to be inconsistent and brings to the surface a more general Davey–Stewartson system. One major difference between them is that non-compatibility renders more possible solutions. The such system models evolution of optical wave packet in nonlinear optics and provides two spatial dimensional analogues of the integrable nonlocal nonlinear Schrödinger equation. Next, via Hirota’s method and long wave limit, various kinds of solutions are obtained. N-soliton is deduced by Hirota’s bilinear form, while rational and semi-rational solutions are constructed by applying long wave limit to N-soliton solution. The rational solutions can be classified as kink-shaped, W-shaped and two crossed W-shaped rogue waves. The semi-rational solutions display as hybrids of rogue wave in periodic background, two rogue waves in periodic background, lump and Akhmediev breather in constant background, lump and Akhmediev breather in periodic background. Additionally, we give a detailed comparison between compatible and non-compatible Davey–Stewartson systems to clarify their differences. The proposed method might be useful in solving Hamiltonian systems, and it is essential to look into other symmetric reductions.

Appendix

$$\begin{aligned} \displaystyle f=&\left( \vartheta _1 \vartheta _2 \vartheta _3 \vartheta _4 +\tau _{12} \vartheta _3 \vartheta _4+\tau _{13} \vartheta _2 \vartheta _4+\tau _{23} \vartheta _1 \vartheta _4\right. \nonumber \\&\left. +\tau _{14} \vartheta _2 \vartheta _3+\tau _{24} \vartheta _1 \vartheta _3+\tau _{34} \vartheta _1 \vartheta _2\right. \nonumber \\&\left. +\tau _{12} \tau _{34}+\tau _ {14} \tau _{23}+\tau _{13} \tau _{24}\right) \nonumber \\ \displaystyle&+e^{\eta _5}\left( \vartheta _1 \vartheta _2 \vartheta _3 \vartheta _4+\tau _{15} \vartheta _2 \vartheta _3 \vartheta _4\right. \nonumber \\&\left. +\tau _{25} \vartheta _1 \vartheta _3 \vartheta _4+\tau _{35} \vartheta _1 \vartheta _2 \vartheta _4+\tau _{45} \vartheta _1 \vartheta _2 \vartheta _3\right. \nonumber \\&+\left( \tau _{34}+\tau _{35} \tau _{45}\right) \vartheta _1 \vartheta _2\nonumber \\ \displaystyle&+\left( \tau _{23}+\tau _{25} \tau _{35}\right) \vartheta _1 \vartheta _4+\left( \tau _{24}+\tau _{25} \tau _{45}\right) \vartheta _1 \vartheta _3\nonumber \\&+\left( \tau _{13}+\tau _{15} \tau _{35}\right) \vartheta _2 \vartheta _4+\left( \tau _{14}+\tau _{15} \tau _{45}\right) \vartheta _2 \vartheta _3\nonumber \\ \displaystyle&+\left( \tau _{12}+\tau _{15} \tau _{25}\right) \vartheta _3 \vartheta _4+\left( \tau _{25} \tau _{34}+\tau _{24} \tau _{35}+\tau _{23} \tau _{45}\nonumber \right. \\&\left. +\tau _{25} \tau _{35} \tau _{45}\right) \vartheta _1+\left( \tau _{15} \tau _{34}+\tau _{14} \tau _{35} \right. \nonumber \\&\left. +\tau _{13} \tau _{45}+\tau _{15} \tau _{35} \tau _{45}\right) \vartheta _2\nonumber \\ \displaystyle&+\left( \tau _{15} \tau _{24}+\tau _{14} \tau _{25}+\tau _{12} \tau _{45}+\tau _{15} \tau _{25} \tau _{45}\right) \vartheta _3\nonumber \\&+\left( \tau _{15} \tau _{23}+\tau _{13} \tau _{25}+\tau _{12} \tau _{35}+\tau _{15} \tau _{25} \tau _{35}\right) \vartheta _4\nonumber \\ \displaystyle&+\tau _{12} \tau _{34}+\tau _{14} \tau _{23}+\tau _{13} \tau _{24}+\tau _{12} \tau _{35} \tau _{45}\nonumber \\&+\tau _{13} \tau _{25} \tau _{45}+\tau _{14} \tau _{25} \tau _{35}\nonumber \\ \displaystyle&\left. +\tau _{15} \tau _{25} \tau _{34}+\tau _{15} \tau _{24} \tau _{35}+\tau _{15} \tau _{23} \tau _{45}+\tau _{15} \tau _{25} \tau _{35} \tau _{45}\right) ,\nonumber \\ \displaystyle g=&\left( \left( \vartheta _1+b_1\right) \left( \vartheta _2+b_2\right) \left( \vartheta _3+b_3\right) \left( \vartheta _4+b_4\right) \right. \nonumber \\&\left. +\tau _{12}\left( \vartheta _3+b_3\right) \left( \vartheta _4+b_4\right) +\tau _{13} \left( \vartheta _2+b_2\right) \left( \vartheta _4+b_4\right) \right. \nonumber \\ \displaystyle&{+}\tau _{23} \left( \vartheta _1{+}b_1\right) \left( \vartheta _4{+}b_4\right) {+}\tau _{14}\left( \vartheta _2{+}b_2\right) \left( \vartheta _3{+}b_3\right) \nonumber \\&+ \tau _{24}\left( \vartheta _1{+}b_1\right) \left( \vartheta _3{+}b_3\right) {+}\tau _{34}\left( \vartheta _1{+}b_1\right) \left( \vartheta _2{+}b_2\right) \nonumber \\ \displaystyle&\left. +\tau _{12} \tau _{34}+\tau _{14} \tau _{23}+\tau _{13} \tau _{24}\right) \nonumber \\&+e^{\eta _5+i\hbar _5}\left( \left( \vartheta _1+b_1\right) \left( \vartheta _2+b_2\right) \left( \vartheta _3+b_3\right) \left( \vartheta _4+b_4\right) \right. \nonumber \\&\left. +\tau _{15}\left( \vartheta _2+b_2\right) \left( \vartheta _3+b_3\right) \left( \vartheta _4+b_4\right) \right. \nonumber \\ \displaystyle&+\tau _{25}\left( \vartheta _1+b_1\right) \left( \vartheta _3+b_3\right) \left( \vartheta _4+b_4\right) \nonumber \\&+\tau _{35}\left( \vartheta _1+b_1\right) \left( \vartheta _2+b_2\right) \left( \vartheta _4+b_4\right) \nonumber \\&+\tau _{45}\left( \vartheta _1+b_1\right) \left( \vartheta _2+b_2\right) \left( \vartheta _3+b_3\right) \nonumber \\ \displaystyle&+\left( \tau _{34}+\tau _{35} \tau _{45}\right) \left( \vartheta _1+b_1\right) \left( \vartheta _2+b_2\right) \nonumber \\&+\left( \tau _{23}+\tau _{25} \tau _{35}\right) \left( \vartheta _1+b_1\right) \left( \vartheta _4+b_4\right) \nonumber \\&+\left( \tau _{24}+\tau _{25} \tau _{45}\right) \left( \vartheta _1+b_1\right) \left( \vartheta _3+b_3\right) \nonumber \\ \displaystyle&+\left( \tau _{13}+\tau _{15} \tau _{35}\right) \left( \vartheta _2+b_2\right) \left( \vartheta _4+b_4\right) \nonumber \\&+\left( \tau _{14}+\tau _{15} \tau _{45}\right) \left( \vartheta _2+b_2\right) \left( \vartheta _3+b_3\right) \nonumber \\&+\left( \tau _{12}+\tau _{15} \tau _{25}\right) \left( \vartheta _3+b_3\right) \left( \vartheta _4+b_4\right) \nonumber \\ \displaystyle&+\left( \tau _{25} \tau _{34}+\tau _{24} \tau _{35}+\tau _{23} \tau _{45}+\tau _{25} \tau _{35} \tau _{45}\right) \left( \vartheta _1+b_1\right) \nonumber \\&+\left( \tau _{15} \tau _{34}+\tau _{14} \tau _{35}+\tau _{13} \tau _{45}+\tau _{15} \tau _{35} \tau _{45}\right) \left( \vartheta _2+b_2\right) \nonumber \\ \displaystyle&+\left( \tau _{15} \tau _{24}+\tau _{14} \tau _{25}+\tau _{12} \tau _{45}+\tau _{15} \tau _{25} \tau _{45}\right) \left( \vartheta _3+b_3\right) \nonumber \\&+\left( \tau _{15} \tau _{23}+\tau _{13} \tau _{25}+\tau _{12} \tau _{35}\nonumber \right. \\&\left. +\tau _{15} \tau _{25} \tau _{35}\right) \left( \vartheta _4+b_4\right) \nonumber \\ \displaystyle&+\tau _{12} \tau _{34}+\tau _{14} \tau _{23}+\tau _{13} \tau _{24}+\tau _{12} \tau _{35} \tau _{45}\nonumber \\&+\tau _{13} \tau _{25} \tau _{45}+\tau _{14} \tau _{25} \tau _{35}\nonumber \\ \displaystyle&\left. +\tau _{15} \tau _{25} \tau _{34}+\tau _{15} \tau _{24} \tau _{35}\nonumber \right. \\&\left. +\tau _{15} \tau _{23} \tau _{45}+\tau _{15} \tau _{25} \tau _{35} \tau _{45}\right) , \end{aligned}$$

(4.1)

where

$$\begin{aligned}{} & {} \hspace{-20pc}\tau _{s5}=\frac{2\left( -P_5^2+\gamma ^2 Q_5^2\right) \left( 1-\gamma ^2 \lambda _s^2\right) }{-4\epsilon P_5+4 \gamma ^2\epsilon Q_5\lambda _s -\delta _s \sqrt{4 \epsilon +P_5^2-\gamma ^2 Q_5^2}\sqrt{-P_5^2+\gamma ^2 Q_5^2}\sqrt{4 \epsilon } \sqrt{-1+\gamma ^2 \lambda _s^2}}, \end{aligned}$$

$\vartheta _j,b_j,\tau _{ij}(1\le i<j\le 4),\eta _5,\hbar _5$ is defined in (2.3), (2.4), (2.5) and (3.2).

$$\begin{aligned} \displaystyle f=&\tau _{12}+\vartheta _1 \vartheta _2+e^{\eta _3} \left( \tau _{12}+\tau _{13} \tau _{23}+\tau _{13} \vartheta _2\right. \\&\left. +\tau _{23} \vartheta _1 +\vartheta _1 \vartheta _2\right) \\&+e^{\eta _4} \left( \tau _{12}+\tau _{14} \tau _{24}+\tau _{14} \vartheta _2+\tau _{24} \vartheta _1+\vartheta _1 \vartheta _2\right) \\ \displaystyle&+e^{\eta _5} \left( \tau _{12}+\tau _{15} \tau _{25}+\tau _{15} \vartheta _2+\tau _{25} \vartheta _1+\vartheta _1 \vartheta _2\right) \\&+e^{A_{34}}e^{\eta _3+\eta _4}\left( \left( \tau _{23}+\tau _{24}\right) \left( \tau _{13}+\tau _{14}\right) +\left( \tau _{13}+\tau _{14}\right) \vartheta _2\right. \\ \displaystyle&\left. +\left( \tau _{23}+\tau _{24}\right) \vartheta _1+\vartheta _1 \vartheta _2+\tau _{12}\right) \\&+e^{A_{35}}e^{\eta _3+\eta _5} \left( \left( \tau _{13}+\tau _{15}\right) \left( \tau _{23}+\tau _{25}\right) \right. \\&\left. +\left( \tau _{13}+\tau _{15}\right) \vartheta _2+\left( \tau _{23}+\tau _{25}\right) \vartheta _1\right. \\ \displaystyle&\left. +\vartheta _1 \vartheta _2+\tau _{12}\right) +e^{A_{45}}e^{\eta _4+\eta _5} \left( \tau _{12}+\left( \tau _{14}\right. \right. \\&\left. \left. +\tau _{15}\right) \left( \tau _{24}+\tau _{25}\right) \right. \\&\left. +\left( \tau _{14}+\tau _{15}\right) \vartheta _2+\left( \tau _{24}+\tau _{25}\right) \vartheta _1+\vartheta _1 \vartheta _2\right) \\ \displaystyle&+e^{A_{34}}e^{A_{35}}e^{A_{45}}e^{\eta _3+\eta _4+\eta _5}\left( \tau _{12}+\left( \tau _{13}+\tau _{14}\right. \right. \\&\left. \left. +\tau _{15}\right) \left( \tau _{23}+\tau _{24}+\tau _{25}\right) \right. \\&\left. +\left( \tau _{13}+\tau _{14}+\tau _{15}\right) \vartheta _2\right. \\ \displaystyle&\left. +\left( \tau _{23}+\tau _{24}+\tau _{25}\right) \vartheta _1 +\vartheta _1 \vartheta _2\right) ,\\ \displaystyle g=&\tau _{12}+\left( b_1+\vartheta _1\right) \left( b_2+\vartheta _2\right) \\&+e^{\eta _3+i \hbar _3} \left( \tau _{13} \left( b_2+\vartheta _2\right) +\tau _{23} \left( b_1+\vartheta _1\right) \right. \\&\left. +\left( b_1+\vartheta _1\right) \left( b_2+\vartheta _2\right) +\tau _{12}+\tau _{13} \tau _{23}\right) \\ \displaystyle&+e^{\eta _4+i \hbar _4} \left( \tau _{14} \left( b_2+\vartheta _2\right) \right. \\&\left. +\tau _{24} \left( b_1+\vartheta _1\right) +\left( b_1+\vartheta _1\right) \left( b_2+\vartheta _2\right) \right. \\&\left. +\tau _{12}+\tau _{14} \tau _{24}\right) +e^{\eta _5+i \hbar _5}\left( \tau _{15} \left( b_2+\vartheta _2\right) +\tau _{25} \tau _{15}\right. \\ \displaystyle&\left. +\left( b_1+\vartheta _1\right) \tau _{25}+\left( b_1+\vartheta _1\right) \left( b_2+\vartheta _2\right) +\tau _{12}\right) \\&+e^{A_{34}}e^{\eta _3+i\hbar _3+\eta _4+i\hbar _4}\left( \left( \tau _{13}+\tau _{14}\right) \left( \tau _{23}+\tau _{24}\right) \right. \\&\left. +\left( \tau _{13}+\tau _{14}\right) \left( b_2+\vartheta _2\right) \right. \\ \displaystyle&\left. +\left( \tau _{23}+\tau _{24}\right) \left( b_1+\vartheta _1\right) +\left( b_1+\vartheta _1\right) \left( b_2+\vartheta _2\right) \right. \\&\left. +\tau _{12}\right) +e^{A_{35}}e^{\eta _3+i \hbar _3+\eta _5+i\hbar _5}\left( \left( \tau _{13}+\tau _{15}\right) \left( \tau _{23}+\tau _{25}\right) \right. \\ \displaystyle&+\left( \tau _{13}+\tau _{15}\right) \left( b_2+\vartheta _2\right) \left. +\left( \tau _{23}+\tau _{25}\right) \left( b_1+\vartheta _1\right) \right. \\&\left. +\left( b_1+\vartheta _1\right) \left( b_2+\vartheta _2\right) +\tau _{12}\right) \\ \displaystyle&+e^{A_{45}}e^{\eta _4+i\hbar _4+\eta _5+i\hbar _5}\left( \left( \tau _{14}+\tau _{15}\right) \left( \tau _{24}+\tau _{25}\right) \right. \\&+\left( \tau _{14}+\tau _{15}\right) \left( b_2+\vartheta _2\right) +\left( \tau _{24}+\tau _{25}\right) \left( b_1+\vartheta _1\right) \\ \end{aligned}$$

$$\begin{aligned} \displaystyle&\left. +\left( b_1+\vartheta _1\right) \left( b_2+\vartheta _2\right) +\tau _{12}\right) \nonumber \\&+e^{A_{34}}e^{A_{35}}e^{A_{45}}e^{\eta _3+i\hbar _3+\eta _4+i\hbar _4+\eta _5+i\hbar _5}\nonumber \\&\left( \left( \tau _{13}+\tau _{14}+\tau _{15}\right) \left( \tau _{23}+\tau _{24}+\tau _{25}\right) \right. \nonumber \\ \displaystyle&\left. +\left( \tau _{13}+\tau _{14}+\tau _{15}\right) \left( b_2+\vartheta _2\right) \right. \nonumber \\&\left. +\left( \tau _{23}+\tau _{24}+\tau _{25}\right) \left( b_1+\vartheta _1\right) \right. \nonumber \\&\left. +\left( b_1+\vartheta _1\right) \left( b_2+\vartheta _2\right) +\tau _{12}\right) , \end{aligned}$$

(4.2)

where

$$\begin{aligned} \begin{array}{lll} \vspace{0.3cm} \displaystyle &{} \tau _{sl}=\displaystyle \frac{2\left( -P_l^2+\gamma ^2 Q_l^2\right) \left( 1-\gamma ^2 \lambda _s^2\right) }{-4\epsilon P_l+4 \gamma ^2\epsilon Q_l\lambda _s -\delta _s \sqrt{4 \epsilon +P_l^2-\gamma ^2 Q_l^2}\sqrt{-P_l^2+\gamma ^2 Q_l^2}\sqrt{4 \epsilon } \sqrt{-1+\gamma ^2 \lambda _s^2}},\\ \vspace{0.3cm} \displaystyle &{} s=1,2,l=3,4,5. \end{array} \end{aligned}$$

$\vartheta _s,b_s,\tau _{sl},\eta _l,\hbar _l,e^{A_{34}},e^{A_{35}},e^{A_{45}}$ is defined in (2.3), (2.4), (2.5) and (3.2).