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Rational and semi-rational solutions to the Davey–Stewartson III equation

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Abstract

In this paper, based on the combination of Hirota’s bilinear method and long wave limit technique, we investigate rational and semi-rational solutions to the third-type Davey–Stewartson (DS III) equation and its nonlocal version. Rational solutions to the DS III equation demonstrate to be kinks, lumps and line rogue waves, while semi-rational solutions display hybrids of solitons, lumps and line rogue waves. As to the nonlocal DS III equation, we derive (semi-)rational solutions and breather solutions. Semi-rational solutions show lumps on the periodic line backgrounds, hybrids of breathers and lumps, line rogue waves and line breathers on the periodic line background.

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Funding

The work is supported by National Natural Science Foundation of China (Grant Nos. 11871336, 12175155) and Shanghai Frontier Research Institute for Modern Analysis.

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  1. School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Sheng-Nan Wang & Guo-Fu Yu

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Appendix

Appendix

Rational solution for the DS III with \(N=4\) is expressed in the following

$$\begin{aligned} q&=\Big (108 x^4-576 x^3 y+(-1056 \textrm{i} t\\&\quad +2144 t^2+1056 y^2+648) x^2\\&\quad +768y(\textrm{i} t-\frac{53}{12} t^2-y^2-\frac{39}{16}) x+192 y^4\\&\quad +(384 \textrm{i} t+1664 t^2+864) y^2+1920 \textrm{i} t^3+3600 t^4\\&\quad +1440\textrm{i} t +5928 t^2+2457\Big ) \Big /\Big (108 x^4-576 x^3 y\\&\quad +(2144 t^2+1056 y^2+936)x^2\\&\quad +(-3392 t^2 y-768 y^3-2448 y) x+192 y^4\\&\quad +(1664 t^2+1248) y^2+3600 t^4+4744 t^2+1521\Big ). \end{aligned}$$

Semi-rational solution for the DS III with \(N=3\) admits the explicit form

$$\begin{aligned} q&=\Big ((136 \textrm{i} x^2+(-136 \textrm{i} y+96 \textrm{i}) x+34 \textrm{i}y^2\\&\quad -48 \textrm{i} y-64-19 \textrm{i}+612 \textrm{i}t^2+(-408\\&\quad +192 \textrm{i}) t) e^{x+2 y+3 t-2 \pi }+408 \textrm{i} t+612 t^2+136 x^2\\&\quad -136 x y+34 y^2-51\Big )\Big /\Big ((136 x^2+(-136 y+96)x\\&\quad +612 t^2+34 y^2+192 t-48 y+49) e^{x+2 y+3 t-2 \pi }\\&\quad +612 t^2+136 x^2-136 x y+34 y^2+17\Big ). \end{aligned}$$

Semi-rational solution for DS III with \(N=4\) is written as

$$\begin{aligned} q&=\Big ((2312 \textrm{i} x^2+(-2312 \textrm{i} y-1632 \textrm{i} )x+578\textrm{i} y^2\\&\quad +816 \textrm{i} y-1088-323\textrm{i}+10404 \textrm{i} t^2 \\&\quad +(-6936+3264 \textrm{i}) t) e^{-x-2 y+3 t-\frac{1}{2} x}\\&\quad +(2312 \textrm{i} x^2+(-2312 \textrm{i} y+1632 \textrm{i}) x\\&\quad +578\textrm{i} y^2-816 \textrm{i} y-1088-323 \textrm{i}+10404\textrm{i} t^2 \\&\quad +(-6936+3264 \textrm{i}) t) e^{x+2 y+3 t-\frac{1}{2} \pi }+(-4624 x^2\\&\quad +4624 x y-1156 y^2-314-4352\textrm{i}\\&\quad -20808 t^2+(-13056-13872 \textrm{i}) t)e^{6 t-\pi }\\&\quad +6936 \textrm{i}t+10404 t^2+2312 x^2\\&\quad -2312 x y+578 y^2-867\Big )\Big /\Big ((2312 x^2\\&\quad +(-2312 y-1632)x+10404 t^2\\&\quad +578 y^2+3264 t+816 y\\&\quad +833)e^{-x-2 y+3 t-\frac{1}{2} \pi }\\&\quad +(2312 x^2+(-2312 y+1632)x\\&\quad +10404 t^2+578 y^2+3264t\\&\quad -816 y+833) e^{x+2 y+3 t-\frac{1}{2} \pi }\\&\quad +(20808 t^2+4624 x^2-4624 x y\\&\quad +1156 y^2+13056 t+2626) e^{6t-\pi }+10404 t^2\\&\quad +2312 x^2-2312 x y+578y^2+289\Big ). \end{aligned}$$

Semi-rational solution for the nonlocal DS III with \(N=3\) has the expression

$$\begin{aligned} q&=\Big (((24\textrm{i}x^2+(-24\textrm{i}y-48)x+108\textrm{i}t^2\\&\quad +6\textrm{i}y^2-69\textrm{i}-240t+24y)\sqrt{15}\\&\quad +168x^2+(336\textrm{i}-168y)x+144\textrm{i}t-168\textrm{i}y\\&\quad +756t^2+42y^2+29)e^{\frac{1}{2}\textrm{i}x-\frac{1}{2}\textrm{i}y-\frac{1}{2}\pi }+576\textrm{i}t\\&\quad +864t^2+192x^2-192xy+48y^2\\&\quad -72\Big )\Big /\Big ((-192\sqrt{15}t+192x^2\\&\quad +(384\textrm{i}-192y)x-192\textrm{i}y+864t^2+48y^2-8)\\&\quad e^{\frac{1}{2}\textrm{i}x-\frac{1}{2}\textrm{i}y-\frac{1}{2}\pi }+864t^2+192x^2-192xy+48y^2+24\Big ). \end{aligned}$$

Semi-rational solution for the nonlocal DS III with \(N=4\) owns the form

$$\begin{aligned} q&=\Big (((-1620\textrm{i} t^2-36\textrm{i} x^2+435\textrm{i}+1800 t-480 x)\sqrt{5}\\&\quad -360\textrm{i}t+960\textrm{i}x-3240 t^2-720 x^2\\&\quad -210)e^{-\frac{2}{3} \textrm{i} x-\frac{9}{9} \sqrt{5} t+\eta }+((435 \textrm{i}-360 \textrm{i} x^2\\&\quad -1620 \textrm{i} t^2+1800 t+480 x) \sqrt{5}-360\textrm{i}t\\&\quad -960\textrm{i}x-3240 t^2-720 x^2-210)e^{\frac{2}{3} \textrm{i} x-\frac{5}{9} \sqrt{5} t+\eta }\\&\quad +((-2592 \textrm{i} t^2-576 \textrm{i} x^2-520 \textrm{i}+1440 t) \sqrt{5}+6192 \textrm{i} t\\&\quad +648 t^2+144 x^2-1814)e^{-\frac{10}{9} \sqrt{5} t+2 \eta }-3240\textrm{i} t \\&\quad -4860 t^2-1080 x^2+405\Big ) \Big /\Big ((1440 \textrm{i} x\\&\quad +1080 \sqrt{5} t-4860 t^2-1080 x^2+45)e^{-\frac{2}{3} \textrm{i} x-\frac{5}{9} \sqrt{5} t+\eta }\\&\quad +(-1440 \textrm{i} x-4860 t^2+1080 \sqrt{5} t-1080 x^2\\&\quad +45)e^{\frac{2}{3} \textrm{i} x-\frac{5}{9} \sqrt{5} t+\eta }+(-5832 t^2+2592 \sqrt{5} t-1296 x^2\\&\quad -1602)e^{-\frac{10}{9} \sqrt{5} t+2 \eta }-4860 t^2-1080 x^2-135\Big ). \end{aligned}$$

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Wang, SN., Yu, GF. Rational and semi-rational solutions to the Davey–Stewartson III equation. Nonlinear Dyn 111, 7635–7655 (2023). https://doi.org/10.1007/s11071-022-08219-3

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