Collapse dynamics for two-dimensional space-time nonlocal nonlinear Schrodinger equations

Abstract

The question of collapse (blow-up) in finite time is investigated for the two-dimensional (non-integrable) space-time nonlocal nonlinear Schrodinger equations. Starting from the two-dimensional extension of the well known AKNS q,r system, three different cases are considered: (i) partial and full parity-time (PT) symmetric, (ii) reverse-time (RT) symmetric, and (iii) general q,r system. Through extensive numerical experiments, it is shown that collapse of Gaussian initial conditions depends on the value of its quasi-power. The collapse dynamics (or lack thereof) strongly depends on whether the nonlocality is in space or time. A so-called quasi-variance identity is derived and its relationship to blow-up is discussed. Numerical simulations reveal that this quantity reaching zero in finite time does not (in general) guarantee collapse. An alternative approach to the study of wave collapse is presented via the study of transverse instability of line soliton solutions. In particular, the linear stability problem for perturbed solitons is formulated for the nonlocal RT and PT symmetric nonlinear Schrodinger (NLS) equations. Through a combination of numerical and analytical approaches, the stability spectrum for some stationary one soliton solutions is found. Direct numerical simulations agree with the linear stability analysis which predicts filamentation and subsequent blow-up.

Figures (15)

• Figure 1: The numerical error in the quasi-power (\ref{['eq3']}) and quasi-energy (\ref{['eq5']}) for the RK4 Fourier integrating factor method. The integrals appearing in (\ref{['eq3']}) and (\ref{['eq5']}) were evaluated spectrally. These curves correspond to the partial PT symmetric case ($r(x,y,t) = -q^*(-x,y,t)$) with initial conditions and parameters identical to those in Fig. \ref{['PT_collapse_summary']}.
• Figure 2: Snapshots of the collapsing profile for the LOC NLS system with initial condition given in (\ref{['gaussian']}) and (\ref{['eq16a1']}) with $(x_0,y_0) = (0,0)$ and $A = 2.5,B=C=1$. Time evolves from top-to-bottom. Rows correspond to $t= 0, 0.1, 0.15, 0.198$, respectively. First left most column: top views of $|q(x,y,t)|$ and $|r(x,y,t)|$. Second column: $y = 0$ profile slice, $|q(x,0,t)|$ (solid) and $|r(x,0,t)|$ (dashed). Third column: $x = 0$ profile cut, $|q(0,y,t)|$ (solid) and $|r(0,y,t)|$ (dashed). Fourth column: level curve contours denoting fixed magnitudes $|q(x,y,t)| = c$ with values of $c$: 1, 1.5, 2, 2.25. The level curves for $|r(x,y,t) |$ are identical.
• Figure 3: The singularity time (\ref{['singularity_time_define']}) for RT Gaussian initial data (\ref{['gaussian']}) with $A = 2.5, B = C = 1, y_0 = 0.5, v = \pi / x_0, n = 1$, and different values of (a) $x_0$ and (b) $P$. Blue dots are numerical data.
• Figure 4: Time evolution of the (a) maximum magnitude (\ref{['eqqmax']}) and (b) quasi-variance (\ref{['eq53']}) for the partial PT NLS equation (\ref{['eq11']}) and Gaussian initial condition (\ref{['gaussian']}) with different values of $x_0$. Also shown are the (c) $L^2$ norm (\ref{['l2norm_define']}) and (d) $L^2$ norm of the gradient (\ref{['l2norm_grad_define']}). Blow-up is eventually avoided as $x_0$ increases. The other parameters are $A = 2.5, B = C = 1, y_0 = 0, v = 0$. Note that $H >0$ for each case considered here.
• Figure 5: The singularity time (\ref{['singularity_time_define']}) for partial PT Gaussian initial data (\ref{['gaussian']}) with $A = 2.5, B = C = 1, y_0 = 0$ and for different values of (a) $x_0$ and (b) $P$. Blue dots are numerical data. The red curves are given in (\ref{['collapse_vert_asym']}), respectively, with (a) $C_x = 0.18, K_x = -0.02, x_s = 0.8$ and (b) $C_P = 0.6, K_P = -0.02, P_s = -2.73$.