Behavior of solutions to the 1D focusing stochastic nonlinear Schrödinger equation with spatially correlated noise

Abstract

We study the focusing stochastic nonlinear Schrödinger equation in one spatial dimension with multiplicative noise, driven by a Wiener process white in time and colored in space, in the $L^2$-critical and supercritical cases. The mass ($L^2$-norm) is conserved due to the multiplicative noise defined via the Stratonovich integral, the energy (Hamiltonian) is not preserved. We first investigate both theoretically and numerically how the energy is affected by various spatially correlated random perturbations and its dependence on the discretization parameters and the schemes. We then perform numerical investigation of the noise influence on the global dynamics measuring the probability of blow-up versus scattering behavior depending on parameters of correlation kernels. Finally, we study numerically the effect of the spatially correlated noise on the blow-up behavior, and conclude that such random perturbations do not influence the blow-up dynamics, except for shifting of the blow-up center location. This is similar to what we observed for a space-time white driving noise in Millet et al. (Numerical study of solutions behavior to the 1d stochastic $L^2$-critical and supercritical nonlinear Schrödinger equation, 2020. arXiv:2006.10695).

Appendices

Appendix A

In this appendix we show the distribution of the locations of blow-up center $x_c$ as its being influenced by the noise. We provide Figs. 22, 23, 24 and 25 for each of our four examples when we run $N_t=1000$ trials in most of them except for the right part in Fig. 25, where we did $N_t=3000$ trials (note how much more accurate the convergence to the normal distribution is, however, this takes significantly larger computational efforts.) In Tables 1, 2, 3 and 4 we record the mean $\mu _{x_c}$ and variance $\sigma ^2_{x_c}$ of the normal distribution that we obtain for the location random variable $x_c$. Observe that the variance noticeably increases in Examples 3 and  4 in the $L^2$-critical case as $\beta$ increases (similar increase is happening in these Examples in the $L^2$-supercritcal case, see Tables 3 and 4).

Table 1 Mean $\mu _{x_c}$ and variance $\sigma ^2_{x_c}$ of the location of blow-up center random variable $x_c$. Example 1 (Gaussian decay) shown in Fig. 22.

Table 2 Table 2 Mean $\mu _{x_c}$ and variance $\sigma ^2_{x_c}$ of the location of blow-up center random variable $x_c$. Example 2 (polynomial decay, $n=2$) shown in Fig. 23

Table 3 Mean $\mu _{x_c}$ and variance $\sigma ^2_{x_c}$ of the location of blow-up center random variable $x_c$. Example 3 (Riesz kernel) shown in Fig. 24.

Table 4 Mean $\mu _{x_c}$ and variance $\sigma ^2_{x_c}$ of the location of blow-up center random variable $x_c$ Example 4 (exponential kernel) shown in Fig. 25

Appendix B

Here we show Figs. 26, 27, 28 and 29 of blow-up dynamics (convergence of profiles and rates) in Examples 1 and 3.

Fig. 26

Formation of blow-up in Example 1 (exponential decay) with $\beta =0.5$ and $\epsilon =0.1$: snapshots of time evolution for $u_0=3\, e^{-x^2}$ (given in pairs of actual and rescaled solution) at different times. Each pair of graphs shows in solid blue the actual solution |u| and its rescaled version $L^{1/\sigma } |u|$, comparing it to the normalized ground state Q in dashed red. Top row: $L^2$-critical ($\sigma =2$) case (blow-up smooths out and converges slowly to the ground state Q). Bottom row: $L^2$-supercritical ($\sigma =3$) case (blow-up profile becomes smooth and converges faster to the profile $Q_{1,0}$)

Fig. 27

Formation of blow-up in Example 3 (Riesz kernel) with $\beta =0.5$ and $\epsilon =0.1$: snapshots of time evolution for $u_0=3\, e^{-x^2}$. For other details, see Fig. 26

Fig. 28

Blow-up rate tracking in Example 1 (exponential decay) with $\beta =0.5$ and $\epsilon = 0.1$. Top row: $L^2$-critical ($\sigma =2$) case. Bottom row: $L^2$-supercritical case. Left: logarithmic dependence of $\log L(t)$ vs. $\log (T-t)$ (note in both cases the slope is 0.50). Middle: $a(\tau _m)$ vs. $\log L(\tau _m)$ (an extremely slow decay to zero in the top plot and rather fast leveling at a constant level in the bottom plot). Right top: dependence $a(\tau )$ vs. $1/ \ln (\tau )$ to confirm the logarithmic correction. Right bottom: fast convergence to a constant of the quantity $\Vert u\Vert _{L^\infty } \left( 2a(T-t)\right) ^{\frac{1}{2\sigma } }$

Fig. 29

Blow-up rate tracking in Example 3 (Riesz kernel) with $\beta =0.5$ and $\epsilon = 0.1$. For details, see Fig. 28