A new model with solitary waves: solution, stability and quasinormal modes

Abstract

We construct solitary wave solutions in a $1+1$-dimensional massless scalar ($\phi$) field theory with a specially chosen potential $V(\phi )$. The equation governing perturbations about this solitary wave has an effective potential which is a simple harmonic well over a region, and a constant beyond. This feature allows us to ensure the stability of the solitary wave through the existence of bound states in the well, which can be found by semi-analytical methods. A further check on stability is performed through our search for quasi-normal modes (QNM) which are defined for purely outgoing boundary conditions. The time-domain profiles of the perturbations and the parametric variation of the QNM values are presented and discussed in some detail. Expectedly, a damped oscillatory temporal behaviour (ringdown) of the fluctuations is clearly seen through our analysis of the quasi-normal modes.

Appendix A: Variation of QNMs with parameters

In this appendix, we discuss the dependence of the QNM frequencies on the parameters of the potential. Such an analysis will help us in appreciating the confined harmonic potential $U(x')$ as an independent, discontinuous potential with its parameters not necessarily obeying the solitary wave criteria. This independent study could be of use in scenarios where a similar potential may arise. We note that the potential $U(x')$ involves parameters: $\alpha$ and $L'$, which are independent if we do not demand its solitary wave connect. Hence, we may vary them freely and see how the QNM frequencies get affected. In Fig. 8, we show how the spectrum of the QNMs obtained differs when we vary each parameter separately.

Fig. 8

$Im(\omega ')$ vs $Re(\omega ')$ plotted for different parameter values

$\bullet$ In Fig. 8a, we observe the effect of variation of $\alpha$ on the QNMs. The value of $\alpha$ determines the depth of the potential well. One must keep in mind while choosing the parameters that we are allowed to take only those values obeying $\alpha (\alpha L'^2-1) < 1$. ( This guarantees that the nature of the confined harmonic oscillator potential is preserved.) As $\alpha$ increases, the well becomes narrow and the imaginary part of the QNM frequency increases.

• Changing the magnitude of discontinuity or jump at $x'= \pm L'$ also affects the QNM spectrum as observed in Fig. 8a. If we go to lower values of $\alpha$, the well becomes shallower, while its width ($2L'$) remains fixed, which results in a larger discontinuity. Hence, we see that small values of $\alpha$ correspond to lower values of $\omega '_i$.

• When $L'$ is varied, i.e. the width of the potential increases (see Fig. 8b), the imaginary part of QNM decreases. For very large $L'$, the imaginary part will keep on getting smaller. Finally, for $L'\rightarrow \infty$ the well vanishes leaving a constant potential with no interesting features.

In general, we find the lower modes to remain almost unaffected by a change of the parameters. Also, the real part of QNMs remain nearly same under parameter variation, whereas it is the imaginary part which shows the effect. Working in dimensionful variables would lead to the introduction of $V_0$ (in inverse length squared units) in the potential. Changing the magnitude of $V_0$ with the other parameters kept fixed will affect the discontinuity. Hence, increasing $V_0$ or decreasing $\alpha$ will have the same effect on the QNM spectrum.