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Accelerating self-modulated nonlinear waves in weakly and strongly magnetized relativistic plasmas

Published online by Cambridge University Press:  27 February 2024

Felipe A. Asenjo Open the ORCID record for Felipe A. Asenjo [Opens in a new window]
Felipe A. Asenjo*
Affiliation:
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago 7491169, Chile
*
†Email address for correspondence: felipe.asenjo@uai.cl
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Abstract

It is known that a nonlinear Schrödinger equation describes the self-modulation of a large amplitude circularly polarized wave in relativistic electron–positron plasmas in the weakly and strongly magnetized limits. Here, we show that such an equation can be written as a modified second Painlevé equation, producing accelerated propagating wave solutions for those nonlinear plasmas. This solution even allows the plasma wave to reverse its direction of propagation. The acceleration parameter depends on the plasma magnetization. This accelerating solution is different to the usual soliton solution propagating at constant speed.

Keywords

plasma wavesplasma nonlinear phenomenaplasma properties
Type
Research Article
Information
Journal of Plasma Physics , Volume 90 , Issue 1 , February 2024 , 905900119
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press

1 Introduction

One of the nonlinear effects present in a relativistic hot magnetized electron–positron plasma is the self-modulation of circularly polarized electromagnetic waves or Alfvén waves. The large amplitude of the electromagnetic wave, and a background magnetic field, can modify the relativistic motion of particles in a significant way. Thus, the self-modulation depends on the magnetization of the plasma. This nonlinear process has been thoroughly studied for the weakly magnetized plasma and strongly magnetized plasma cases in Asenjo et al. (Reference Asenjo, Borotto, Chian, Muñoz and Valdivia2012) and López et al. (Reference López, Asenjo, Muñoz, Chian and Valdivia2013), respectively.

Interestingly, the self-modulation gives origin to soliton plasma wave solutions, propagating at constant speed (Asenjo et al. Reference Asenjo, Borotto, Chian, Muñoz and Valdivia2012; López et al. Reference López, Asenjo, Muñoz, Chian and Valdivia2013). However, there exists another kind of accelerating plasma wave solution that deserves to be explored for relativistic plasmas. This work is devoted to showing that there are nonlinear plasma waves that accelerate due to the self-modulation of the magnetized plasma system.

Relativistic plasma wave modes, presenting accelerating behaviour, have been recently introduced in Li, Li & Wang (Reference Li, Li and Wang2016), Winkler, Vásquez-Wilson & Asenjo (Reference Winkler, Vásquez-Wilson and Asenjo2023) and Minovich et al. (Reference Minovich, Klein, Neshev, Pertsch, Kivshar and Christodoulides2014). Differently to those works, here, we study the case of self-modulation of a circularly polarized electromagnetic or Alfvén wave in a weakly or strongly magnetized relativistic electron–positron plasma with finite temperature. For these cases, the nonlinear Schrödinger equation that models such systems has been found to be (Asenjo et al. Reference Asenjo, Borotto, Chian, Muñoz and Valdivia2012; López et al. Reference López, Asenjo, Muñoz, Chian and Valdivia2013)

(1.1)\begin{equation} {\rm i}\frac{\partial a}{\partial t}+P \frac{\partial^2 a}{\partial z^2}+ Q|a|^2 a=0 , \end{equation}

where $a=a(t,z)$ is the time- and space-dependent complex modulational amplitude of the circularly polarized electromagnetic wave, under the approximation of a slowly time-varying modulation (Asenjo et al. Reference Asenjo, Borotto, Chian, Muñoz and Valdivia2012; López et al. Reference López, Asenjo, Muñoz, Chian and Valdivia2013). Here, $P=c^2/(2\omega )$ is independent of the magnetization, where $c$ is the speed of light, and $\omega$ is the frequency of the wave. Differently, $Q$ depends on the limit case of magnetization of the plasma. In the weakly magnetized case, when $\omega \gg \varOmega _c$ (where is $\varOmega _c$ is the plasma cyclotron frequency), we have that (Asenjo et al. Reference Asenjo, Borotto, Chian, Muñoz and Valdivia2012)

(1.2)\begin{equation} Q=\frac{3\lambda \omega_p^2\varOmega_c^2}{\omega^3 f^5}, \end{equation}

where $\lambda = e^2/m^2c^4$ (with the electron charge $e$ and mass $m$), $\omega _p$ is the plasma frequency of the electron–positron plasma and $f$ is the thermodynamic function relating the plasma density enthalpy per unit of mass and unit of number density (being a function of temperature). On the other hand, for the strongly magnetized plasma case, when $\omega \ll \varOmega _c$, we find that (López et al. Reference López, Asenjo, Muñoz, Chian and Valdivia2013)

(1.3)\begin{equation} Q=\frac{f\lambda \omega_p^2\omega^3}{4\varOmega_c^4}. \end{equation}

It is clear that it is the background magnetic field, through the cyclotron frequency, the physical quantity that induces the nonlinear behaviour of the waves.

Furthermore, any other intermediate case for plasma magnetization (when $\omega$ is of the same order as $\varOmega _c$) does not produce a simple nonlinear Schrödinger equation such as (1.1). In these cases, the nonlinear effects cannot be straightforwardly analysed from relativistic factors of the plasma fluid velocities. Thus, the plasma system is more difficult to study as there are no analytical solutions for the plasma velocities (Asenjo et al. Reference Asenjo, Borotto, Chian, Muñoz and Valdivia2012; López et al. Reference López, Asenjo, Muñoz, Chian and Valdivia2013).

2 Non-accelerating soliton

Equation (1.1) is usually solved in terms of a soliton amplitude propagating with constant speed $v$. This solution can be found by requiring that the amplitude of the electromagnetic wave has the form $|a(t,z)|=|a(z-v t)|$. Straightforwardly, it is found that such solution has the form of a soliton (Asenjo et al. Reference Asenjo, Borotto, Chian, Muñoz and Valdivia2012; López et al. Reference López, Asenjo, Muñoz, Chian and Valdivia2013)

(2.1)\begin{equation} a(t,z)= \mathrm{sech}\,\left( \sqrt{\frac{Q}{2 P}}(z- v t)\right) \exp\left({\rm i} \frac{ v }{2P}z-{\rm i}\left(\frac{v^2}{4P}-\frac{Q}{2}\right) t \right). \end{equation}

3 Accelerating solution

However, (1.1) can be also solved for an accelerating wave, i.e. we look for solution with amplitude in the form $|a(t,z)|=|a(z-v t-\beta t^2/2)|$, where $v$ and $\beta$ play the roles of an initial velocity and the acceleration of the propagation.

Similar to the case of the previous section, this kind of solution can be studied by assuming the form of the plasma wavepacket as

(3.1)\begin{equation} a(t,z)= f\left( \xi \right)\exp\left({\rm i} \eta(t,z)\right), \end{equation}

where $f$ is a function to be determined through (1.1), and depending on the accelerated argument

(3.2)\begin{equation} \xi=\sqrt{\frac{Q}{2P}}\left(z-vt- \frac{\beta}{2} t^2\right) . \end{equation}

Notice that we have assumed, in analogy with the constant velocity soliton solution, the same factor $\sqrt {Q/2P}$ for the argument. Also, the phase function $\eta (t,z)$ must be determined.

Using (3.1) in (1.1), we can find that when the acceleration is

(3.3)\begin{equation} \beta=Q\sqrt{\frac{Q P}{2}}, \end{equation}

and the phase is

(3.4)\begin{equation} \eta(t,z)=\frac{Q}{2}\sqrt{\frac{Q}{2P}}\,\left(t z+\frac{v}{Q}\sqrt{\frac{2}{QP}}z-\frac{v^2}{Q^2}\sqrt{\frac{Q}{2P}}t-v t^2-\frac{Q}{3} \sqrt{\frac{QP}{2}} t^3\right) , \end{equation}

then, the function $f$ fulfils a modified second Painlevé equation (Clarkson Reference Clarkson2003)

(3.5)\begin{equation} \frac{{\rm d}^2 f}{{\rm d} \xi^2}-\xi f+2 f^3=0. \end{equation}

Solutions of this equation can be generally studied numerically, but they are no longer solitons. For the current case, they describe an accelerated propagation of a electromagnetic plasma wave train, with non-constant amplitude and with acceleration $\beta =\sqrt {Q^3 P/2}$ along its direction of propagation (the velocity $v$ is arbitrary). The acceleration of this nonlinear plasma wave depends on the magnetization of the plasma through $Q$, given in (1.2) and (1.3). We emphasize that this kind of accelerated behaviour for the plasma is only possible in the weakly and strongly magnetized limits.

3.1 Case for $v=0$

In order to show the accelerating behaviour of this nonlinear plasma solution, we display in figure 1 the density plot for the numerical solution of (3.5) for $f(\xi =0)=1$, $d_\xi f(\xi =0)=0$ and $v=0$. This plot shows the magnitude of function $f$ in the $t-z$ space, in terms of normalized time $t'=Q t/\sqrt {2}$, and normalized distance $z'=\sqrt {Q/2P} z$. The solution shows curved (parabolic) trajectories for any part of the wave (maxima or minima). This can be explicitly seen through the red dashed lines, that are used as examples. Those lines correspond to $\xi =z'- t'^2/2=\xi _0$, for $\xi _0=-10,-4, 0, 5, 10$. All the red dashed parabolic curves coincide with the dynamics of the plasma wave. Therefore, we conclude that the whole plasma wave propagates with acceleration $\beta$ given by (3.3).

Figure 1. Density plot for $f(\xi )$, with $f(\xi =0)=1$ and $d_\xi f(\xi =0)=0$, in terms of time $t'$ and distance $z'$, for $v=0$. Red dashed lines correspond to parabolic trajectories $z'- t'^2/2=\xi _0$, with $\xi _0=-10, -4, 0, 5, 10$.

3.2 Case for velocity $v$ parallel to the acceleration

In this case, the velocity $v$ produces only a modification of the initial behaviour of the solution, accelerating along the same direction. As acceleration (3.3) is positive, we graphically display this solution with $v>0$. In figure 2, we show a numerical solution of (3.5) for $f(\xi =0)=1$, $d_\xi f(\xi =0)=0$, in terms of normalized time $t'=Q t/\sqrt {2}$, normalized distance $z'=\sqrt {Q/2P}\, z$ and normalized velocity $v'=v/\sqrt {P Q}=2$. Any part of the solution follows accelerated curved parabolic trajectories in the $t-z$ space. We display those curved trajectories as dashed lines for $\xi =z'-v't'- t'^2/2=\xi _0$, with $\xi _0=-10,-4, 0, 5, 10$.

Figure 2. Density plot for $f(\xi )$, with $f(\xi =0)=1$ and $d_\xi f(\xi =0)=0$, in terms of time $t'$, distance $z'$ and normalized positive velocity $v'=2$. Red dashed lines correspond to parabolic trajectories $z'- v't'- t'^2/2=\xi _0$, with $\xi _0= -4, 0, 5, 10, 15$.

3.3 Case for velocity $v$ antiparallel to the acceleration

This case is more interesting as it implies that the acceleration of the wave dynamics can change its direction of propagation. To exemplify this case we consider a negative initial velocity $v<0$. In figure 3, we show what occurs for the specific case of normalized velocity equal to $v'=v/\sqrt {P Q}=-2$. In this figure, we display the numerical behaviour of $f$ in a density plot in terms of the same normalized variables $t'$ and $z'$ as above. Because of its accelerating nature, the whole wavepacket changes its initial direction of propagation. This occurs for any part of the wave, as is shown by the parabolic curves in $t - z$ space (in red dashed lines), for $\xi =z'- v't'- t'^2/2=\xi _0$, with $\xi _0= -4, 0, 5, 10, 15$. The acceleration, therefore, allows this nonlinear plasma solution to reverse its propagation direction when the initial velocity has the opposite direction to the direction of acceleration. It is straightforward to calculate that this change in direction takes a time equal to

(3.6)\begin{equation} \Delta t={-}\frac{v}{\beta}>0. \end{equation}

Figure 3. Density plot for $f(\xi )$, with $f(\xi =0)=1$ and $d_\xi f(\xi =0)=0$, in terms of time $t'$, distance $z'$ and normalized negative velocity $v'=-2$. Red dashed lines correspond to parabolic trajectories $z'- v't'- t'^2/2=\xi _0$, with $\xi _0= -4, 0, 5, 10, 15$.

3.4 Asymptotic behaviour

On the other hand, although a numerical study of the whole solution is possible, the analytical properties of the solution for $f$ can be found in the case when $\xi \rightarrow -\infty$. In this limit, the solution of the modified second Painlevé equation (3.5) behaves as (Clarkson Reference Clarkson2003)

(3.7)\begin{equation} f(\xi)\approx \kappa\, {\mbox{Ai}}(\xi) , \end{equation}

where ${\mbox {Ai}}$ is the Airy function, and $\kappa$ is an arbitrary constant. As the accelerating propagating properties of this electromagnetic plasma wave are present for any value of $\xi$, this solution pertains to the same family of other accelerating Airy solutions already found in optics and plasmas (Baumgartl, Mazilu & Dholakia Reference Baumgartl, Mazilu and Dholakia2008; Abdollahpour et al. Reference Abdollahpour, Suntsov, Papazoglou and Tzortzakis2010; Chong et al. Reference Chong, Renninger, Christodoulides and Wise2010; Chávez-Cerda et al. Reference Chávez-Cerda, Ruiz, Arrizón and Moya-Cessa2011; Jiang, Huang & Lu Reference Jiang, Huang and Lu2012; Kaminer et al. Reference Kaminer, Bekenstein, Nemirovsky and Segev2012; Mahalov & Suslov Reference Mahalov and Suslov2012; Panagiotopoulos et al. Reference Panagiotopoulos, Papazoglou, Couairon and Tzortzakis2013; Minovich et al. Reference Minovich, Klein, Neshev, Pertsch, Kivshar and Christodoulides2014; Li et al. Reference Li, Li and Wang2016; Wiersma et al. Reference Wiersma, Marsal, Sciamanna and Wolfersberger2016; Esat Kondakci & Abouraddy Reference Esat Kondakci and Abouraddy2018; Efremidis et al. Reference Efremidis, Chen, Segev and Christodoulides2019; Bouchet et al. Reference Bouchet, Marsal, Sciamanna and Wolfersberger2022; Winkler et al. Reference Winkler, Vásquez-Wilson and Asenjo2023).

4 Final remark

We have presented a new nonlinear plasma solution with accelerating properties. As the initial velocity of the argument (3.2) is arbitrary, a whole set of a new kind of different propagations can be obtained. This is achieved in a relativistic plasma regime, depending on how magnetized the plasma is, producing wavepackets with acceleration (3.3).

As a nonlinear dynamics of electron–positron plasmas can be found in pulsar magnetospheres (Chian & Kennel Reference Chian and Kennel1983; Beskin, Gurevich & Istamin Reference Beskin, Gurevich and Istamin1993), relativistic jets (Iwamoto & Takahara Reference Iwamoto and Takahara2002), the early universe (Lesch & Bisk Reference Lesch and Bisk1998) or supernovae (Hardy & Thoma Reference Hardy and Thoma2000), these electromagnetic propagation solutions can produce new mechanisms for plasma acceleration in those regimes. In addition, as the nonlinear Schrödinger equation also appears in electrostatic plasma propagation (see for instance Kourakis & Shukla Reference Kourakis and Shukla2004; Misra & Shukla Reference Misra and Shukla2011; Rajabi & Mohammadnejad Reference Rajabi and Mohammadnejad2023), the above solution also predicts nonlinear acceleration for such phenomena. In general, it is expected that the propagation of the wavepacket experiences an acceleration along the direction of propagation, even having the possibility to reverse such a direction. Those implications are left for future studies.

Finally, the Painlevé equation has been explored in different realms of plasma physics (Khater et al. Reference Khater, Callebaut, Shamardan and Ibrahim1997; Khater, Callebaut & Ibrahim Reference Khater, Callebaut and Ibrahim1998; Ibrahim Reference Ibrahim2003; Rogers & Clarkson Reference Rogers and Clarkson2018; Kumar, Mohan & Kumar Reference Kumar, Mohan and Kumar2022). Therefore, this work contributes to showing that the Painlevé equation is also a straightforward consequence of accelerating solutions in relativistic nonlinear plasmas.

Acknowledgements

Editor A.C. Bret thanks the referees for their advice in evaluating this article.

Funding

This work has been carried out thanks to FONDECYT grant No. 1230094.

Declaration of interests

The author reports no conflict of interest.

References

Abdollahpour, D., Suntsov, S., Papazoglou, D.G. & Tzortzakis, S. 2010 Spatiotemporal Airy light bullets in the linear and nonlinear regimes. Phys. Rev. Lett. 105, 253901.CrossRefGoogle ScholarPubMed
Asenjo, F.A., Borotto, F.A., Chian, A.C.-L., Muñoz, V. & Valdivia, J.A. 2012 Self-modulation of nonlinear waves in a weakly magnetized relativistic electron-positron plasma with temperature. Phys. Rev. E 85, 046406.CrossRefGoogle Scholar
Baumgartl, J., Mazilu, M. & Dholakia, K. 2008 Optically mediated particle clearing using airy wavepackets. Nat. Photonics 2, 675.CrossRefGoogle Scholar
Beskin, V.S., Gurevich, A.V. & Istamin, Y.N. 1993 Physics of the Pulsar Magnetosphere. Cambridge University Press.Google Scholar
Bouchet, T., Marsal, N., Sciamanna, M. & Wolfersberger, D. 2022 Two dimensional Airy beam soliton. Sci. Rep. 12, 9064.CrossRefGoogle ScholarPubMed
Chávez-Cerda, S., Ruiz, U., Arrizón, V. & Moya-Cessa, H.M. 2011 Generation of Airy solitary-like wave beams by acceleration control in inhomogeneous media. Opt. Exp. 19, 16448.CrossRefGoogle ScholarPubMed
Chian, A.C.-L. & Kennel, C.F. 1983 Self-modulational formation of pulsar microstructures. Astrophys. Space Sci. 97, 9.CrossRefGoogle Scholar
Chong, A., Renninger, W.H., Christodoulides, D.N. & Wise, F.W. 2010 Airy–Bessel wave packets as versatile linear light bullets. Nat. Photonics 4, 103.CrossRefGoogle Scholar
Clarkson, P.A. 2003 Painlevé equations: nonlinear special functions. J. Comput. Appl. Maths 153, 127.CrossRefGoogle Scholar
Efremidis, N.K., Chen, Z., Segev, M. & Christodoulides, D.N. 2019 Airy beams and accelerating waves: an overview of recent advances. Optica 6, 686.CrossRefGoogle Scholar
Esat Kondakci, H. & Abouraddy, A.F. 2018 Airy wave packets accelerating in space–time. Phys. Rev. Lett. 120, 163901.CrossRefGoogle Scholar
Hardy, S.J. & Thoma, M.H. 2000 Neutrino-electron processes in a strongly magnetized thermal plasma. Phys. Rev. D 63, 025014.CrossRefGoogle Scholar
Ibrahim, R.S. 2003 Truncated Painlevé expansion and reduction of sine-Poisson equation to a quadrature in collisionless cold plasma. IMA J. Appl. Maths 68, 523.CrossRefGoogle Scholar
Iwamoto, S. & Takahara, F. 2002 Relativistic outflow of electron-positron pair plasma from a Wien equilibrium state. Astrophys. J. 565, 163.CrossRefGoogle Scholar
Jiang, Y., Huang, K. & Lu, X. 2012 Propagation dynamics of abruptly autofocusing Airy beams with optical vortices. Opt. Express 20, 18579.CrossRefGoogle ScholarPubMed
Kaminer, I., Bekenstein, R., Nemirovsky, J. & Segev, M. 2012 Nondiffracting accelerating wave packets of Maxwell's equations. Phys. Rev. Lett. 108, 163901.CrossRefGoogle ScholarPubMed
Khater, A.H., Callebaut, D.K. & Ibrahim, R.S. 1998 Bäcklund transformations and Painlevé analysis: exact soliton solutions for the unstable nonlinear Schrödinger equation modeling electron beam plasma. Phys. Plasmas 5, 395.CrossRefGoogle Scholar
Khater, A.H., Callebaut, D.K., Shamardan, A.B. & Ibrahim, R.S. 1997 Bäcklund transformations and Painlevé analysis: exact soliton solutions for strongly rarefied relativistic cold plasma. Phys. Plasmas 4, 3910.CrossRefGoogle Scholar
Kourakis, I. & Shukla, P.K. 2004 Electron-acoustic plasma waves: oblique modulation and envelope solitons. Phys. Rev. E 69, 036411.CrossRefGoogle ScholarPubMed
Kumar, S., Mohan, B. & Kumar, A. 2022 Generalized fifth-order nonlinear evolution equation for the Sawada-Kotera, lax, and caudrey-dodd-gibbon equations in plasma physics: Painlevé analysis and multi-soliton solutions. Phys. Scr. 97, 035201.CrossRefGoogle Scholar
Lesch, H. & Bisk, G.T. 1998 Can large-scale magnetic fields survive during the pre-recombination era of the universe? Phys. Plasmas 5, 2773.CrossRefGoogle Scholar
Li, H., Li, X. & Wang, J. 2016 Airy-like electron plasma wave. J. Plasma Phys. 82, 905820103.CrossRefGoogle Scholar
López, R.A., Asenjo, F.A., Muñoz, V., Chian, A.C.-L. & Valdivia, J.A. 2013 Self-modulation of nonlinear Alfvén waves in a strongly magnetized relativistic electron-positron plasma. Phys. Rev. E 88, 023105.CrossRefGoogle Scholar
Mahalov, A. & Suslov, S.K. 2012 An “Airy gun”: Self-accelerating solutions of the time-dependent Schrödinger equation in vacuum. Phys. Lett. A 377, 33.CrossRefGoogle Scholar
Minovich, A.E., Klein, A.E., Neshev, D.N., Pertsch, T., Kivshar, Y.S. & Christodoulides, D.N. 2014 Airy plasmons: non-diffracting optical surface waves. Laser Photon. Rev. 8, 221.CrossRefGoogle Scholar
Misra, A.P. & Shukla, P.K. 2011 Modulational instability and nonlinear evolution of two-dimensional electrostatic wave packets in ultra-relativistic degenerate dense plasmas. Phys. Plasmas 18, 042308.CrossRefGoogle Scholar
Panagiotopoulos, P., Papazoglou, D.G., Couairon, A. & Tzortzakis, S. 2013 Sharply autofocused ring-Airy beams transforming into non-linear intense light bullet. Nat. Commun. 4, 2622.CrossRefGoogle Scholar
Rajabi, B. & Mohammadnejad, M. 2023 Modulational instability of ion-acoustic waves in a dense quantum plasma. Phys. Plasmas 30, 082112.CrossRefGoogle Scholar
Rogers, C. & Clarkson, P.A. 2018 Ermakov-Painlevé II reduction in cold plasma physics. Application of a Bäcklund transformation. J. Nonlinear Math. Phys. 25, 247.CrossRefGoogle Scholar
Wiersma, N., Marsal, N., Sciamanna, M. & Wolfersberger, D. 2016 Airy beam self-focusing in a photorefractive medium. Sci. Rep. 6, 35078.CrossRefGoogle Scholar
Winkler, M.A., Vásquez-Wilson, C. & Asenjo, F.A. 2023 Exact and paraxial airy propagation of relativistic electron plasma wavepackets. Eur. Phys. J. D 77, 97.CrossRefGoogle Scholar
Figure 0

Figure 1. Density plot for $f(\xi )$, with $f(\xi =0)=1$ and $d_\xi f(\xi =0)=0$, in terms of time $t'$ and distance $z'$, for $v=0$. Red dashed lines correspond to parabolic trajectories $z'- t'^2/2=\xi _0$, with $\xi _0=-10, -4, 0, 5, 10$.

Figure 1

Figure 2. Density plot for $f(\xi )$, with $f(\xi =0)=1$ and $d_\xi f(\xi =0)=0$, in terms of time $t'$, distance $z'$ and normalized positive velocity $v'=2$. Red dashed lines correspond to parabolic trajectories $z'- v't'- t'^2/2=\xi _0$, with $\xi _0= -4, 0, 5, 10, 15$.

Figure 2

Figure 3. Density plot for $f(\xi )$, with $f(\xi =0)=1$ and $d_\xi f(\xi =0)=0$, in terms of time $t'$, distance $z'$ and normalized negative velocity $v'=-2$. Red dashed lines correspond to parabolic trajectories $z'- v't'- t'^2/2=\xi _0$, with $\xi _0= -4, 0, 5, 10, 15$.