Lie symmetry reduction for \((2+1)\)-dimensional fractional Schrödinger equation (Q6617571)

In this paper, the authors consider fractional Schrödinger equation [\textit{C. Li} et al., Appl. Math. Lett. 94, 238--243 (2019; Zbl 1538.35437)] in the following form:\N\[\NiD_t^{\alpha} q + aq_{xx} - bq_{yy} + \gamma q|q|^2 = 0,\quad t > 0,\ 0 <\alpha \le 1,\tag{1.1}\N\]\N in which \(a, b\), and \(\gamma\) are constants, \(\gamma\) is the coefficient for the self-phase modulation term, \(q = q(x, y, t)\) typifies the complex function, the terms \(aq_{xx}\) and \(bq_{yy}\) denote the group velocity dispersion effect, and the fourth term represents the self-phase modulation effect. The Riemann-Liouville fractional derivative is given by:\N\[\ND_t^{\alpha}v(t,x)=\begin{cases} \N\frac{1}{\Gamma(m-\alpha)}\frac{\partial^m}{\partial t^m}\int_0^t(t-w)^{m-1-\alpha}v(w,x)dw, &0\le m-1<\alpha<m,\ m\in \mathbb{N},\\\N\frac{\partial^mv(t,x)}{\partial t^m}, &\alpha=m \in \mathbb{N},\end{cases} \tag{2.1}\N\]\Nthe Euler gamma function, denoted by \(\Gamma(z)\), is defined as \(\Gamma(z) = \int_0^{\infty}e^{-t}t^{z-1}dt\).\N\NThis work investigates similarity reduction, exact solutions, convergence analysis, and conservation laws of Eq. (1.1). And, this work investigates Lie symmetry analysis of Eq. (1.1) from the standpoint of Eq. (2.1).\N\NInitially, presuming that\N\[\Nq(t, x, y) = iu(t, x, y) + v(t, x, y), \tag{3.1}\N\]\Nafter substituting Eq. (3.1), the following equations are obtained by separating the real and imaginary parts of Eq. (1.1),\N\[\N\begin{cases} \ND_t^{\alpha} u-av_{xx}+bv_{yy}-\gamma (v^3 + vu^2) = 0, \\\ND_t^{\alpha} v+au_{xx}-bu_{yy}+\gamma (uv^2 + u^3 ) = 0. \end{cases}\tag{3.2}\N\]\NAssume that (3.2) is invariant under one-parameter Lie group of point transformations (for example below):\N\begin{align*} \N\hat{t}& = t + \epsilon\tau (t, x, y, v, u) + o(\epsilon^2),\\\N\hat{x} &= x + \epsilon\xi(t, x, y, v, u) + o(\epsilon^2 ),\\\N\hat{y} &= y + \epsilon\rho(t, x, y, v, u) + o(\epsilon^2 ),\\\N\hat{u}&= u + \epsilon\eta(t, x, y, v, u) + o(\epsilon^2),\\\N\hat{v} &= v + \epsilon\phi(t, x, y, v, u) + o(\epsilon^2 ). \N\end{align*}\N\(\tau , \xi, \rho, \eta\), and \(\phi\) are the infinitesimals of the forms for the dependent and independent variables, respectively, and \(\epsilon \ll 1\) is the Lie group parameter. These leads to the following transformed equation,\N\[\N\begin{cases} \ND_t^{\alpha} \hat{u}-a\hat{v}_{xx}+b\hat{v}_{yy}-\gamma (\hat{v}^3 + \hat{v}\hat{u}^2) = 0, \\\ND_t^{\alpha} \hat{v}+a\hat{u}_{xx}-b\hat{u}_{yy}+\gamma (\hat{u}\hat{v}^2 + \hat{u}^3 ) = 0. \N\end{cases}\tag{3.3}\N\]\NThe authors have completed exact solutions of Eq. (3.2) through a power series approach [\textit{M. Inc} et al., Physica A 496, 371--383 (2018; Zbl 1514.35460); \textit{R. Sahadevan} and \textit{T. Bakkyaraj}, J. Math. Anal. Appl. 393, No. 2, 341--347 (2012; Zbl 1245.35142)] and symbolic calculations. Exact solutions of Eq. (1.1) have been found:\N\[\Nq(t,x,y)=i\sum_{m=0}^{\infty}a_m(x-y)^mt^{-\frac{(m+1)\alpha}{2}}+\sum_{m=0}^{\infty}b_m(x-y)^mt^{-\frac{(m+1)\alpha}{2}}, \tag{4.8}\N\]\N\(u=f(z)t^{-\frac{\alpha}{2}},v=g(z)t^{-\frac{\alpha}{2}},f(z)=\sum_{m=0}^{\infty}a_mz^m\), \(g(z)=\sum_{m=0}^{\infty}b_mz^m\), \(z=(x-y)t^{-\frac{\alpha}{2}}\) where the initial conditions are\(f (0) = a_0 , f^{\prime} (0) = a_1 , g(0) = b_0, g^{\prime}(0) = b_1\) .\N\NThe authors construct conservation laws for Eq. (1.1) via the Ibragimov theorem [\textit{N. H. Ibragimov}, J. Math. Anal. Appl. 333, No. 1, 311--328 (2007; Zbl 1160.35008)].\N\NThe authors state the following: ``The invariance properties in the sense of Lie point symmetry of the time fractional Schrödinger equation are presented in this paper. The Lie symmetries of the equations are acquired and the related fractional PDEs are reduced to FODEs.''