⇦Chinese Physics Letters, 2017, Vol. 34, No. 1, Article code 010202 Rogue Waves in the (2+1)-Dimensional Nonlinear Schrödinger Equation with a Parity-Time-Symmetric Potential * Yun-Kai Liu(刘芸恺), Biao Li(李彪)** Affiliations Ningbo Collaborative Innovation Center of Nonlinear Harzard System of Ocean and Atmosphere, and Department of Mathematics, Ningbo University, Ningbo 315211 Received 29 August 2016 *Supported by the National Natural Science Foundation of China under Grant Nos 11271211, 11275072 and 11435005, the Ningbo Natural Science Foundation under Grant No 2015A610159, the Opening Project of Zhejiang Provincial Top Key Discipline of Physics Sciences in Ningbo University under Grant No xkzw11502, and the K. C. Wong Magna Fund in Ningbo University.
**Corresponding author. Email: libiao@nbu.edu.cn Citation Text: Liu Y K and Li B 2017 Chin. Phys. Lett. 34 010202 Abstract The (2+1)-dimension nonlocal nonlinear Schrödinger (NLS) equation with the self-induced parity-time symmetric potential is introduced, which provides spatially two-dimensional analogues of the nonlocal NLS equation introduced by Ablowitz et al. [Phys. Rev. Lett. 110 (2013) 064105]. General periodic solutions are derived by the bilinear method. These periodic solutions behave as growing and decaying periodic line waves arising from the constant background and decaying back to the constant background again. By taking long wave limits of the obtained periodic solutions, rogue waves are obtained. It is also shown that these line rogue waves arise from the constant background with a line profile and disappear into the constant background again in the $(x,y)$ plane. DOI:10.1088/0256-307X/34/1/010202 PACS:02.30.Ik, 02.30.Jr, 05.45.Yv © 2017 Chinese Physics Society Article Text Rogue waves, mostly known as large and spontaneous ocean surface waves, have been responsible for numerous maritime disasters. In past decades, both the experimental observation and theoretical analysis on rogue waves have been devoted to physical areas as diverse as the Bose–Einstein condensates,[1,2] optical system,[3-5] ocean,[6] superfluids,[7] plasma[8,9] and so on.[10] Mathematically, Peregrine is the first one who obtained the fundamental rogue wave solution (i.e., first-order rogue wave solution) in the nonlinear Schrödinger (NLS) equation,[11] which is located in both space and time. Thus the first-order rogue wave solution of the NLS equation is also called the Peregrine solution. Recently, the higher-order rogue wave solutions in the NLS equation were studied in a series of works by different methods.[12-18] Generally, higher-order rogue waves can be treated as superpositions of several fundamental rogue waves, and these superpositions can create higher amplitudes which still keep both space and time local. Moreover, the hierarchy of other soliton equations have also been verified possessing rogue waves.[19-25] However, rogue waves studied before are mostly one-dimensional, while ocean surface waves are always two-dimensional rogue waves. In addition to the one-dimensional rogue wave studied so far, two-dimensional rogue waves have also derived lots of attention. Furthermore, the two-dimensional analogue of rogue waves, expressed by more complicated rational forms, has been recently reported in the Davey–Stewartson (DS) systems,[26,27] the Kadomtsev–Petviashvili–I equation,[28,29] the Yajima–Oikawa system,[30], the Fokas system[24] and so on.[31-35] Recently, Ablowitz et al.[36] proposed a nonlocal NLS equation $$\begin{align} &q_{t}(x,t)-iq_{xx}(x,t)\pm 2iV(x,t)q(x,t)=0,\\ &V(x,t)=q(x,t)q^{*}(-x,t),~~ \tag {1} \end{align} $$ which implies the parity-time (PT) symmetry, namely, $x \rightarrow-x$, $t \rightarrow -t$ and complex conjugate. Here $q(x, t)$ and $V(x,t)$ (called PT potential) imply the electric-field envelope of the optical beam and complex refractive-index distribution or an optical potential respectively, and the asterisk denotes the complex conjugate. In optics, the real part of $V(x,t)$, defined by $V_{\rm R}(x, t)$, denotes the refractive-index profile, and the imaginary part of $V(x, t)$ denotes the gain/loss distribution, defined by $V_{\rm I}(x, t)$. Based on $V_{\rm R}(x, t)=V_{\rm R}(-x,t)$ and $V_{\rm I}(x,t)=-V_{\rm I}(-x, t)$, a PT-symmetric system can be designed. The concept of the PT symmetry, based on the non-hermitian Hamiltonians,[37-41] has recently attracted much attention, in particular in the fields of optics and photonics,[42-45] which provide a fertile ground for PT-related notions and experiments. Lately, numerous important works have been carried out on the NLS equation with a PT potential by Yan et al.[46-48] Specifically, Horikis et al. studied rogue waves in nonlocal media.[49] Motivated by the work of Horikis et al., we propose a (2+1)-dimensional (2D) nonlocal NLS equation satisfying the two-dimensional PT symmetry condition $V(x,y)=V^{*}(-x,-y)$ $$\begin{align} &i q_{t}+q_{xy}+qr=0,\\ &r_{y}=[q(x,y,t)q(-x,-y,t)^{*}]_{x},~~ \tag {2} \end{align} $$ where $V(x,y,t)=q(x,y,t)q^{*}(x,y,t)$. We consider the rogue waves of this 2D nonlocal NLS equation, and extend the rogue waves of nonlocal media[49] into multi-dimensional versions. Firstly, we consider breathers and rogue waves in the 2D nonlocal NLS equation. The 2D nonlocal NLS equation is translated into the bilinear form $$\begin{align} &(iD_{t}+D_{x}D_{y})g \cdot f =0,\\ &(D_{y}^{2}+1)f \cdot f =gg(-x,-y)^{*},~~ \tag {3} \end{align} $$ through the variable transformation $$\begin{align} q=\frac{g}{f},~~r=2(\log f)_{xy},~~ \tag {4} \end{align} $$ where $f$ and $g$ are functions with respect to three variables $x$, $y$ and $t$, and satisfy the condition $$\begin{align} f(-x,-y,t)=f(x,y,t),~~ \tag {5} \end{align} $$ and the operator $D$ is Hirota's bilinear differential operator[50] defined by $P(D_{x},D_{y},D_{t})F(x,y,t,\ldots)\cdot G(x, y, t, \ldots)=P(\partial_{x}-\partial_{x'},\partial_{y}-\partial_{y'}, \partial_{t}-\partial_{t'}, \ldots)F(x, y, t, \ldots)G(x', y', t', \ldots)|_{x'=x, y'=y, t'=t, \ldots}$, where $P$ is a polynomial of $D_{x}$, $D_{y}$, $D_{t}$, $\ldots$. Next we generate breathers and rogue waves in the 2D nonlocal NLS Eq. (2). Following earlier works,[51-53] a family of periodic solutions termed breathers can typically be derived by an expansion scheme $$\begin{align} g=\,&1+a_{1}\exp(ikx+imy+{\it \Omega}t+\xi_{01})\\ &+a_{2}\exp(-ikx-imy+{\it \Omega} t+\xi_{02})\\ &+Ma_{1}a_{2}\exp(2{\it \Omega}t+\xi_{01}+\xi_{02}),\\ f=\,&1+\exp(ikx+imy+{\it \Omega}t+\xi_{01})\\ &+\exp(-ikx-imy+{\it \Omega}t+\xi_{02})\\ &+M\exp(2{\it \Omega}t+\xi_{01}+\xi_{02}).~~ \tag {6} \end{align} $$ As functions $f$ and $g$ have to hold the special nonlocal nature of Eq. (2), one must restrict the parameters $k$, $m$ and ${\it \Omega}$ to be real and $\xi_{01}=\xi_{02}$. Based on some simple calculation by means of symbolic computation software such as Maple, the parameters $a_{1}$, $a_{2}$ (complex), ${\it \Omega}$, $M$ (real) are given by $$\begin{align} {\it \Omega}=-k\sqrt{-m^{2}+2},~~M=\frac{2}{-m^{2}+2},\\ a_{1}=a_{2}=(-m^{2}+1)+im\sqrt{-m^{2}+2}.~~ \tag {7} \end{align} $$ It is not difficult to find $|a_{1}|=|a_{2}|=1$, and the constraint $-m^{2}+2>0$ must hold for ${\it \Omega}$ to be real. Hence, the periodic solution can also be expressed in terms of hyperbolic and trigonometric functions as $$\begin{align} q=g_{1}/f_{1},~~r=2(\log f_{1})_{xy},~~ \tag {8} \end{align} $$ where $$\begin{align} f_{1}=\,&\sqrt{M}\cosh {\it \Theta}+\cos(kx+my),\\ g_{1}=\,&\sqrt{M}[\cos^{2}\beta\cosh{\it \Theta}+\sin^{2}{\beta}\sinh{\it \Theta}\\ &+i\cos\beta\sin\beta(\cosh{\it \Theta}-\sinh{\it \Theta})]\\ &+\cos(kx+my)(\cos\beta+i\sin\beta),~~ \tag {9} \end{align} $$ $a_{1}=a_{2}=\exp(i\beta)$, ${\it \Theta}={\it \Omega}(t-t_{0})$, $\exp({\it \Omega} t_{0})=\sqrt{M}\exp(\xi)$, and $\xi=\xi_{01}=\xi_{02}$. These periodic solutions for parameter choices $$\begin{align} k=\frac{1}{2},~~p=\frac{1}{3},~~\xi=0,~~ \tag {10} \end{align} $$ are shown in Figs. 1 and 2. As can be seen in Fig. 1, these periodic solutions describe growing and decaying periodic line waves in the $(x,y)$ plane. When $t\rightarrow-\infty$, these solutions go to a uniform constant background. In the intermediate times, periodic line waves which keep in parallel state arise from the constant background, and then they approach much higher amplitudes (see the $t=0$ panel). At larger time, these periodic line waves go back to the constant background (see the $t=5$ panel). Thus these line waves possess the characters appearing from nowhere and disappear without a trace. It is noted that this kind of periodic solution has been studied in the DS equations.[52] Differently, these solutions (8) behave as breathers in the $(x,t)$ and $(y,t)$ planes, which are located in time, see Fig. 2.
cpl-34-1-010202-fig1.png
Fig. 1. Time evolution of periodic solution (Eq. (8)) of the 2D nonlocal NLS Eq. (2) in the $(x,y)$ plane parameters (Eq. (10)).
cpl-34-1-010202-fig2.png
Fig. 2. Dynamics of breathers (Eq. (8)) in the 2D nonlocal NLS Eq. (2) in the $(x,t)$ and $(y,t)$ planes.
Next, we consider rogue waves in the 2D nonlocal NLS Eq. (2). To generate the rogue waves, one can take a long wave limit of $f_{1}$ and $g_{1}$, i.e., in Eq. (9) taking $$\begin{alignat}{1} \exp(\xi_{01})=\exp(\xi_{02})=-1,~m=\lambda k,~ k\rightarrow 0,~~ \tag {11} \end{alignat} $$ where $\lambda$ is an arbitrary real parameter. Then the rogue waves of the 2D nonlocal NLS Eq. (2) can be expressed in rational functions as $$\begin{align} q=\,&1-\frac{4i\lambda t+2\lambda^{2}}{(x+\lambda y)^2+2t^2+\frac{1}{2}\lambda^2},\\ r=\,&\frac{-2\lambda(x+\lambda y)^2+4\lambda t^2+2\lambda^3}{[(x+\lambda y)^2+2t^2+\frac{1}{2}\lambda^2]^2}.~~ \tag {12} \end{align} $$ These solutions describe a decaying line wave with the line oriented in the $(\lambda,-1)$ direction of the $(x,y)$ plane, see Fig. 3. This line wave is different from the moving line solitons of the (2+1)-dimensional soliton equations, since the line solitons maintain a perfect profile without any decay during their propagation in the $(x,y)$ plane. What is more, when $t\rightarrow \pm\infty$, the solution $q$ uniform approaches to the constant background 1, while in the intermediate time, $|q|$ attains the maximum amplitude 3 (i.e., three times the constant background amplitude) at the center $(x+ \lambda y=0)$ of the line wave at $t=0$. Thus this line wave has the characteristics: appears from nowhere and disappears without a trace, hence it is defined as a line rogue wave.
cpl-34-1-010202-fig3.png
Fig. 3. Time evolution of rogue waves (Eq. (12)) in the 2D nonlocal NLS Eq. (2) in the $(x,y)$ plane. Here the parameter is $\lambda=3$.
cpl-34-1-010202-fig4.png
Fig. 4. Dynamics of rogue waves (Eq. (12)) in the 2D nonlocal NLS Eq. (2) in the $(x,t)$ and $(y,t)$ planes. Here the parameter is $\lambda=3$.
It is noted that the orientation of this line rogue wave is arbitrary as the parameter $\lambda$ can be an arbitrary real parameter. Moreover, these solutions possess different dynamics in the $(x,t)$ plane or the $(y,t)$ plane. As demonstrated in Figs. 3 and 4, it is structurally very similar to the Peregrine breather of the intensively studied NLS equation, and it is located in both space and time. In summary, we have proposed a 2D nonlocal NLS equation under 2D PT symmetry conditions, and this equation extends the nonlocal NLS equation introduced by Ablowitz et al. into multidimensional versions. By the bilinear method, a type of periodic solutions are derived, and these periodic solutions possess different dynamics in different planes. In the $(x,y)$ plane, they describe periodic line waves, which arise from the constant background and decay back to the constant background again (see Fig. 1). However, they are breathers in the $(y,t)$ plane or the $(x,t)$ plane, which are located in time (see Fig. 2). By taking appropriate parameter choices and by taking a long wave limit of the obtained periodic solutions, the nonsingular rogue waves solutions are generated. These rational solutions have a line profile with a varying height in $(y,t)$, which is called line rogue wave. It is shown that the line rogue waves arise from the constant background and disappear into the constant background at larger time (see Fig. 3). Differently, these line waves feature the Peregrine rogue waves in the $(x,t)$ and $(y,t)$ planes. Since there are few works about the rogue waves of the PT-symmetry systems, our study may be helpful to promote understanding of the rogue wave phenomenon. References Matter rogue wavesVector rogue waves in binary mixtures of Bose-Einstein condensatesNon-Gaussian Statistics and Extreme Waves in a Nonlinear Optical CavityOptical rogue wavesFreak Waves in the Linear Regime: A Microwave StudyObservation of an Inverse Energy Cascade in Developed Acoustic Turbulence in Superfluid HeliumLangmuir rogue waves in electron-positron plasmasObservation of Peregrine Solitons in a Multicomponent Plasma with Negative IonsFinancial Rogue WavesWater waves, nonlinear Schrödinger equations and their solutionsMulti-rogue waves solutions to the focusing NLS equation and the KP-I equationRogue waves and rational solutions of the nonlinear Schrödinger equationOn multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equationRogue wave tripletsNonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutionsCircular rogue wave clustersGeneral high-order rogue waves and their dynamics in the nonlinear Schrodinger equationGenerating mechanism for higher-order rogue wavesBreather and rogue wave solutions of a generalized nonlinear Schrödinger equationRogue wave solutions of AB systemHigher-order rogue wave solutions of the three-wave resonant interaction equation via the generalized Darboux transformation N -order bright and dark rogue waves in a resonant erbium-doped fiber systemRational Solutions for the Fokas SystemRogue waves and their generating mechanisms in different physical contextsRogue waves in the Davey-Stewartson I equationDynamics of rogue waves in the Davey–Stewartson II equationInstabilities, breathers and rogue waves in opticsMulti-rogue waves solutions: from the NLS to the KP-I equationRational solutions to two- and one-dimensional multicomponent Yajima–Oikawa systemsTwo spatial dimensional N-rogue waves and their dynamics in Mel’nikov equationOptical Rogue Wave Excitation and Modulation on a Bright Soliton BackgroundInertial Lévy Flight with Nonlinear FrictionDarboux Transformations via Lie Point Symmetries: KdV EquationIntegrable Nonlocal Nonlinear Schrödinger EquationReal Spectra in Non-Hermitian Hamiltonians Having P T SymmetryComplex Extension of Quantum MechanicsTheory of coupled optical PT-symmetric structuresSystematic search for 𝒫𝒯-symmetric potentials with real energy spectraSurface plasmon–polariton amplifiers and lasersBeam Dynamics in P T Symmetric Optical LatticesParity–time synthetic photonic latticesObservation of parity–time symmetry in opticsIntegrable PT-symmetric local and nonlocal vector nonlinear Schrödinger equations: A unified two-parameter modelNonlocal general vector nonlinear Schrödinger equations: Integrability, PT symmetribility, and solutionsDynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potentialBreather solutions to the focusing nonlinear Schrödinger equationGrowing-and-decaying mode solution to the Davey-Stewartson equationBreathers and rogue waves for a third order nonlocal partial differential equation by a bilinear transformation
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