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Rogue wave solutions of the (2+1)-dimensional derivative nonlinear Schrödinger equation

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Abstract

In this paper, we focus on the construction of rogue wave solutions for the (2+1)-dimensional derivative nonlinear Schrödinger equation. The N-order generalized Darboux transformation is obtained, and the determinant form of N-order rogue waves is also presented by taking limit on the classical Darboux transformation. On the plane wave solution background, two different kinds of rogue wave solutions (linear rogue wave and parabolic rogue wave) are constructed successively. The characteristics of two types of rogue waves are analyzed by some figures and physical qualities.

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Acknowledgments

This work is supported by the Shanghai Leading Academic Discipline Project under Grant No. XTKX2012, by the Technology Research and Development Program of University of Shanghai for Science and Technology, by Hujiang Foundation of China under Grant No. B14005 and by the National Natural Science Foundation of China under Grant No. 11201302.

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, P. O. Box 253, Shanghai, 200093, China

    Li-Li Wen & Hai-Qiang Zhang

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  1. Li-Li Wen
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Correspondence to Hai-Qiang Zhang.

Appendix: Choose the special solution

Appendix: Choose the special solution

Firstly, we need to solve the linear system (2) with the seed solution by the gauge transformation method:

$$\begin{aligned} \varPsi =M\varPhi , M=diag\left( e^{-\frac{i}{2}\theta },e^{\frac{i}{2}\theta }\right) , \end{aligned}$$
(5)

where

$$\begin{aligned} \varPsi =\left( \begin{array}{c} \varphi _{1} \\ \varphi _{2} \end{array}\right) , \varPhi =\left( \begin{array}{ll} \phi _{1} \\ \phi _{2} \end{array} \right) . \end{aligned}$$
(6)

Then, matrix \(\varPhi \) satifies the following linear system:

$$\begin{aligned}&\varPhi _{x}=N\varPhi ,\end{aligned}$$
(7a)
$$\begin{aligned}&\varPhi _{t}=\lambda ^{2}\varPhi _{y}-\nu N\varPhi , \end{aligned}$$
(7b)

where

$$\begin{aligned} N= & {} \frac{i}{2}\left( \begin{array}{cc} \lambda ^{2}+\mu &{} 2ia\lambda \\ -2ia\lambda &{} -\lambda ^{2}-\mu \end{array}\right) . \end{aligned}$$

And (7a) could be rewrited as

$$\begin{aligned}&2 a \lambda \phi _2 -i \left( \lambda ^2+\mu \right) \phi _1+2 \phi _{1x}=0,\end{aligned}$$
(8a)
$$\begin{aligned}&\quad -2 a \lambda \phi _1+i \left( \lambda ^2+\mu \right) \phi _2+2 \phi _{2x}=0. \end{aligned}$$
(8b)

By some calculations, i.e., (8a)+(8b) and (8a)-(8b), we could have

$$\begin{aligned} 2 \left( \phi _1+\phi _2\right) _{x}+\left[ 2 a \lambda +i \left( \lambda ^2+\mu \right) \right] \left( \phi _2-\phi _1\right) =0,\end{aligned}$$
(9a)
$$\begin{aligned} 2 \left( \phi _1-\phi _2\right) _{x}+\left[ 2 a \lambda -i \left( \lambda ^2+\mu \right) \right] \left( \phi _1+\phi _2\right) =0. \end{aligned}$$
(9b)

For convenience, \(\phi _2-\phi _1\) and \(\phi _2+\phi _1\) are marked as R and K, respectively. Therefore, from (9a) and (9b), we could deriving that

$$\begin{aligned} K&=a_1(t,y) e^{\frac{1}{2} x \sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}}\\&\quad +a_2(t,y) e^{-\frac{1}{2} x \sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}},\\ R&=\frac{\sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}}{2 a \lambda +i \left( \lambda ^2+\mu \right) }\nonumber \\&\quad \times \left[ a_2(t,y)e^{-\frac{1}{2} x \sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}}\right. \\&\quad \left. -\,a_1(t,y) e^{\frac{1}{2}x \sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}}\right] . \end{aligned}$$

Next, inserting K and R into \(\phi _2-\phi _1=R\) and \(\phi _2+\phi _1=K\), we could obtained

$$\begin{aligned} \phi _{1}= & {} \frac{\sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}+2 a \lambda +i \lambda ^2+i \mu }{2 \left[ 2 a \lambda +i \left( \lambda ^2+\mu \right) \right] }\\&\times e^{\frac{1}{2} x \sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}}a_1(t,y)\\&- \frac{\sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}-2 a \lambda -i \lambda ^2-i \mu }{2 \left[ 2 a \lambda +i \left( \lambda ^2+\mu \right) \right] }\\&\times e^{-\frac{1}{2} x \sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}}a_2(t,y),\\ \phi _{2}= & {} \frac{\sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}+2 a \lambda +i \lambda ^2+i \mu }{2 \left[ 2 a \lambda +i \left( \lambda ^2+\mu \right) \right] }\\&\times e^{-\frac{1}{2} x \sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}}a_2(t,y)\\&-\frac{\sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}-2 a \lambda -i \lambda ^2-i \mu }{2 \left[ 2 a \lambda +i \left( \lambda ^2+\mu \right) \right] }\\&\times e^{\frac{1}{2} x \sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}}a_1(t,y). \end{aligned}$$

Now, simultaneous Eq. (7b) and \(\phi _{i}(i=1,2)\), \(a_1(t,y)\) and \(a_2(t,y)\) could be derived:

$$\begin{aligned} a_1(t,y)&=f\left( \lambda ^2 t+y\right) e^{-\frac{1}{2} \nu t \sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}},\\ a_2(t,y)&=g\left( \lambda ^2 t+y\right) e^{\frac{1}{2} \nu t \sqrt{-4 a^2 \lambda ^2-\left( \lambda ^2+\mu \right) ^2}}, \end{aligned}$$

where \(f\left( \lambda ^2 t+y\right) \) and \(g\left( \lambda ^2 t+y\right) \) are arbitrary complex functions about \(\lambda ^2 t+y\). The different solutions could be derived by changing \(f\left( \lambda ^2 t+y\right) \) and \(g\left( \lambda ^2 t+y\right) \). Then, \(\varphi _{1}\) and \(\varphi _{2}\) could be derived.

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Wen, LL., Zhang, HQ. Rogue wave solutions of the (2+1)-dimensional derivative nonlinear Schrödinger equation. Nonlinear Dyn 86, 877–889 (2016). https://doi.org/10.1007/s11071-016-2930-y

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