Higher Time Derivative Theories From Integrable Models

Abstract

Higher Time Derivative Theories are generated by considering space-time rotated KdV and mKdV systems. These systems are then studied to see if/how instabilities, usually associated with higher time derivative theories, manifest on the classical level by presenting both analytic and numerical solutions. For a linearised version of these space-time rotated systems we present a detailed quantisation of the theory that highlights the known dilemma on higher time derivative theories, that we have either negative norm states or the Hamiltonian being unbounded from below.

Figures (5)

• Figure 1: The analytic solutions are compared with the numerical evolution of the exact initial conditions given by these solutions. In panel (a) this is done for the standard KdV equation where the solution is given by $u\left(t,x\right)=2 \kappa ^2 m \text{cn}\left(\left.4 (1-2 m) t \kappa ^3+x \kappa \right|m\right)$ (black dashed lines) and there is one piece of Cauchy data $u\left(0,x\right)$. Panel (b) shows the results for a space-time rotated KdV system. The analytic solution is now $u\left(t,x\right)=\frac{\kappa}{\sqrt{2}} \text{cn}\left(\left.(m-2) t \kappa ^3+x \kappa \right|m\right)^2$ and we have three pieces of Cauchy data $u\left(0,x\right)$, $u_t\left(0,x\right)$, $u_{2t}\left(0,x\right)$, In panel (c) this is done for a soliton solution of the rotated KdV equation where $u\left(t,x\right)=2 \times 2^{1/3}\kappa^{2/3}\sech^2\left[\kappa x -\frac{\kappa^{1/3}}{4 \times 2 ^{1/3}}t\right]$. In all cases the exact solutions are given by dashed lines and the solid lines give the numerical solutions.
• Figure 2: As in figure \ref{['kdvanvnum']} the analytic solution is compared with the numerical evolution of initial conditions, this time for the space-time rotated mKdV equation. In panel (a) the solution $u\left(t,x\right)=\sqrt{\frac{m}{2}}\left(\frac{\kappa}{1-2m}\right)^{\frac{1}{3}}\mathop{\mathrm{cn}}\nolimits\left(\kappa x| m\right)$ where $\kappa=1$ and $m=0.45$ is considered. In panel (b) we plot a comparison between the Cauchy data as given by the exact solution and the numerical evolution of this data. In panel (c) this comparison between analytic and numerical evolution is done for the soliton solution $u\left(t,x\right)=-6^{2/3}\kappa^{1/3}\sech\left[\kappa x -\frac{\kappa^{1/3}}{6 ^{1/3}}t\right]$. As in figure \ref{['kdvanvnum']} the exact solutions are given by dashed lines and the solid lines give the numerical solutions.
• Figure 3: In panel (a) predictions of the amplitude for changing values of sigma are plotted for the indicated combinations of conservation laws. The blue, purple, red, orange regions indicate what we find to be the $n=2$, $n=3$, $n=4$ and $n=5$ regions respectively. The non soliton region is shown in green. The $n=4$ and $n=5$ predictions are found by numerically solving the equations, in all the other cases the equations have been solved analytically. The black crosses indicated the amplitudes of emergent solitons from the numerical evolution of a Gaussian initial pulse. In panels (b) and (c) the numerical evolution of a Gaussian initial pulse with $\sigma=7.22$, $\sigma=8$ respectively are plotted. In panel(b) the orange dotted lines indicate the amplitudes predicted by the relations between $a$ and $\sigma$ plotted in panel (a) in the two soliton region. In panel (c) differently shaded the orange lines indicate the different predictions of the amplitude the same relations make when $\sigma=8$. The pink lines indicate the predictions made by the three soliton region.
• Figure 4: Panel (a) displays the predictions for the square root of the soliton amplitudes for different combinations of conservation laws when a Gaussian initial pulse is evolved under a rotated KdV equation. In panel (b) and (c) this numerical evolution is displayed for $\sigma=3$ and $\sigma=15$ respectively.
• Figure 5: Panel (a) displays the predictions for the square root of the soliton amplitudes for different combinations of conservation laws when a Gaussian initial pulse is evolved under a rotated mKdV equation. In panel (b) and (c) this numerical evolution is displayed for $\sigma=5$ and $\sigma=45$, respectively.