Introduction
Nonlinear wave propagation in plasmas has become one of the most important subjects of plasma studies and many works have been devoted to the nonlinear interactions of high frequency electromagnetic (EM) waves in an electron-ion plasma\QCITE{cite}{}{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. However, when an electron-ion plasma contains extremely massive, micrometer-size charged dust grains, there appears the possibility of new normal modes\QCITE{cite}{}{16,17,18,19,20,21,22,23,24,25}. The latter include the dust-acoustic (DA) waves which have extremely low phase velocities (in comparison with the electron and ion thermal velocities) and which appears as a normal mode of a three-component dusty plasma comprising of electrons, ions and dust grains. In a dusty plasma$,$ both the electrons and the ions can at times be considered to be Boltzmann-distributed, whereas the charged dust particles are always inertial. Thus, the pressures of the electrons and ions provide the restoring force, whereas the inertia comes from the dust mass. Experimental confirmations about the existence of DA waves have also been made in several laboratory experiments\QCITE{cite}{}{28,29,30}. In addition, a number of studies have discussed the properties of complex (dusty) plasmas\QCITE{cite}{}{26,27}.
Many studies have been conducted to examine nonlinear coupling of high frequency EM waves and DA or dust-ion acoustic (DIA) waves in both unmagnetized and magnetized dusty plasmas and it has been found that the presence of extremely massive charged dust grains modifies the strength of the coupling coefficient, because the number densities of the ions and electrons are not equal. It is well known that the slow modulation of a monochromatic EM plane wave can be described by the nonlinear Schr\"{o}dinger (NLS) equation. For a medium with a positive coefficient of cubic nonlinearity, the instability that arises in the transverse direction is known as self-focusing, while that in the longitudinal direction is referred to as the modulational instability. Several theoretical attempts have been made to investigate modulational instability and to search for nonlinear structures in dusty plasmas.
In this paper, we consider the nonlinear propagation of high-frequency, long wavelength transverse (EM) waves in a collisionless dusty plasma. A NLS-like equation with an additional term is derived and its modulational instability is investigated, leading to the excitation of DA waves. It is observed that when the extra nonlinear term compensates the diffraction term, the growth rate becomes maximum. In this paper we assume that the size of the dust grains is much smaller than the Debye radii and the wavelengths of the EM waves. Our basic emphasis is on the focusing of EM waves and we show how, in the area of localization of wave energy, the density of grains increases and the ions, following the grains, start clustering around them while electrons are pushed away from that region. Further, this localization of EM waves is dependent on the shape of the EM pulse. Considering the example of pancake and bullet shaped pulses, we find that only the latter leads to compression of grains in the supersonic regime of the focusing region. Here we might speculate on the possible existence of dust atoms which Tsintsadze\QTR{it}{\ et al.}\QCITE{cite}{}{31}\QTR{it}{\ }have recently proposed, by deriving a Thomas-Fermi type equation for dust grains in plasmas, where the negatively charged dust grain acts as the positive nucleus and the positively charged trapped ions circumnavigate around it.
The manuscript is organized in the following fashion. In Sec. II, we present the basic formulation for the motion of dust grains and discuss the condition for compression of the dust grains. In Sec.III, we derive a NLS equation governing the dynamics of modulated DA waves. The discussion of the compression phenomenon and the possibility of crystallization of the dust atoms is presented in Sec.IV. Section V deals with the nonstationary solution of the NLSE. Finally, conclusions are given in Sec.VII.
Mathematical Formalism
We consider a three-component fully ionized plasma composed of electrons (with mass $m$ and charge $e$), ions (mass $m_{i}$ and charge $q_{i}=Z_{i}e$) and heavy charged dust particulates with mass $m_{d}$ and charge $q_{d}=-Z_{d}e$ in thermodynamic equilibrium. Our aim here is to consider some phenomena which can arise during the propagation of electromagnetic waves in such plasmas.
We shall consider some specific properties of such interaction in a collisionless dusty plasma, by assuming the oscillation time $t_{0}$ of EM waves $(t_{0}\thicksim \frac{2\pi }{\omega _{0}}$ here $\omega _{0}$is some characteristic frequency associated with the EM wave) to be much less than that of all other particulates. Here we are interested in the motion of dust grains so we consider that the time with which velocity and density of grains changes to be much larger than that of electrons and ions, i.e., \EQN{6}{1}{}{0}{\RD{\CELL{v_{d}\left( \frac{\partial v_{d}}{\partial t}\right) ^{-1},\text{ }n_{d}\left( \frac{\partial n_{d}}{\partial t}\right) ^{-1}>>t_{e}\backsim \frac{1}{\omega _{pe}},\text{ }t_{i}\backsim \frac{1}{\omega _{pi}} }}{1}{}{}{}} where $\omega _{pe}$, $\omega _{pi}$ are the Langmuir frequencies for electrons and ions, respectively. It is for this reason that electrons and ions are taken to be inertialess. We also note as the electrons being lighter than the ions, are most effected by the electromagnetic field via the ponderomotive force. For the spacial scale we assume inequalities $a<
The Boltzmann distribution for the electrons and ions is expressed as \EQN{0}{1}{}{}{\RD{\CELL{n_{e}=n_{oe}\exp \left( \frac{e\varphi -\frac{e^{2}|\QTR{bf}{A}|^{2}}{2m_{o}c^{2}}}{T_{e}}\right)}}{2}{1}{}{}}\EQN{0}{1}{}{}{\RD{\CELL{n_{i}=n_{oi}\exp \left( \frac{-Z_{i}e\varphi }{T_{i}}\right)}}{2}{2}{}{}}We note that the effect of the ponderomotive force is taken into account only for the electrons for the reason given above. The ponderomotive force here has been expressed through the vector potential $\QTR{bf}{A}$ which has the form \EQN{0}{1}{}{}{\RD{\CELL{\QTR{bf}{A}=\QTR{bf}{A}_{o}(\QTR{bf}{r},t)\exp i(\QTR{bf}{k}_{o}\QTR{bf}{\cdot r}-\omega _{o}t)}}{2}{3}{}{}}here $\QTR{bf}{A}_{0}(\QTR{bf}{r},t)$ is the amplitude of vector potential of the EM wave which is slowly varying in space and time.
The continuity and momentum equations of dust grains are \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial n_{d}}{\partial t}+\QTR{bf}{\nabla \cdot (}n_{d}\QTR{bf}{v}_{\QTR{bf}{d}})=0}}{2}{4}{}{}}\EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial \QTR{bf}{v}_{d}}{\partial t}+(\QTR{bf}{v}_{\QTR{bf}{d}}\QTR{bf}{\cdot \nabla })\QTR{bf}{v}_{\QTR{bf}{d}}=\frac{\QTR{bf}{\nabla }(Z_{D}e\varphi )}{m_{d}}}}{2}{5}{}{}}In Eq.(5), we have neglected the pressure term, by considering that for the temperatures the following inequality $T_{d}<
It is important to emphasize here that the structure of Eq.(8) is quite different from the case of a two component plasma, where the ponderomotive force is added to the equation of motion of the ions as a usual pressure term with a minus sign. In our case $\beta $ is positive and we can say that the ponderomotive force acts on dust grains as a negative pressure \EQN{0}{1}{}{}{\RD{\CELL{P_{pond}=-\beta m_{d}n_{d}U_{pond}}}{2}{9}{}{}}We now assume that the ponderomotive force due to EM waves is not strong enough to effect the nonlinearity of the dust grains i.e., we can neglect the second term of l.h.s in Eq.(8) and linearize the continuity Eq.(4). Combining the two equations, we obtain \EQN{0}{1}{}{}{\RD{\CELL{\left( \frac{\partial ^{2}}{\partial t^{2}}-\QTR{bf}{u}_{d}^{2}\QTR{bf}{\nabla }^{2}\right) \frac{\delta n_{d}}{n_{0d}}=-\beta \QTR{bf}{\nabla }^{2}U_{pond}}}{2}{10}{}{}}Eq.(10) shows that the r.h.s contains a negative ponderomotive pressure unlike Zakharov's equation\QCITE{cite}{}{10}, where a positive ponderomotive force always appears.
Schr\UNICODE{0xf6}dinger Equation and Modulation instability
In spite of the large number of publications devoted to the derivation of the NLS equation and the investigation of modulational and filamenational instabilities, envelop solitons, self focusing etc. in dusty plasmas, we will show that in a three component dusty plasma the structure of the nonlinear Schr\"{o}dinger equation is physically different from that in a two component plasma.
In order to construct the nonlinear Schr\"{o}dinger Equation and then to investigate the excitation of dust acoustic waves we start with Maxwells equations, assuming the amplitude of the electromagnetic waves to be nonrelativistic. Thus we can obtain the following equation for the vector potential \EQN{0}{1}{}{}{\RD{\CELL{\nabla ^{2}\QTR{bf}{A}-\frac{\partial ^{2}\QTR{bf}{A}}{\partial t^{2}}=\frac{n_{e}}{n_{oe}}\QTR{bf}{A}}}{2}{11}{}{}}Here we have introduced the following dimensionless variables $\QTR{bf}{r}\rightarrow \frac{\omega _{pe}}{c}\QTR{bf}{r},$ $t\rightarrow \omega _{pe}t$, where $\omega _{pe}=\sqrt{\frac{4\pi e^{2}n_{0}}{m_{e}}}$ is the Langmuir frequency. Substituting expression (6) into Eq.(11) and expressing the density deviation of the equilibrium density $(n_{e}-n_{0e}\simeq \delta n_{e})$ and the density variation of dust grains and ponderomotive potential, we obtain a new type of nonlinear Schr\"{o}dinger equation \EQN{0}{1}{}{}{\RD{\CELL{2i\omega _{o}\left( \frac{\partial }{\partial t}+\QTR{bf}{v}_{g}\QTR{bf}{\cdot }\frac{\partial }{\partial \QTR{bf}{r}}\right) \QTR{bf}{A}_{o}+\QTR{bf}{\nabla }^{2}\QTR{bf}{A}_{o}+\left( Z_{D}\alpha \frac{\delta n_{d}}{n_{od}}+\frac{Z_{i}n_{0i}}{n_{0d}}\alpha \frac{U_{pond}}{T_{i}}\right) \QTR{bf}{A}_{o}=0}}{2}{12}{}{}}here $\omega _{0}$ is dimensionless frequency i.e., $\frac{\omega _{0}}{\omega _{p}}$ and we have made use of the linear dispersion relation for EM wave \EQN{0}{1}{}{}{\RD{\CELL{\omega _{o}^{2}=\omega _{p}^{2}+k_{0}^{2}c^{2}}}{2}{13}{}{}}We note that in Eq.(12) $\QTR{bf}{v}_{g}$ is the dimensionless group velocity given by $\frac{1}{c}\frac{\partial \omega _{o}}{\partial \QTR{bf}{k}_{o}}=\frac{\QTR{bf}{k}_{0}c}{\omega _{0}}.$
It is important to note that the Schr\"{o}dinger equation $\left[ \text{Eq.(12)}\right] $ looks quite different than that obtained in the usual electron ion plasma, since the last term of Eq.(12) is new and we will show this term introduces new physics, which contributes to the development of a strong modulation instability. Using Madelung's representation of the complex amplitude of the EMW \EQN{0}{1}{}{}{\RD{\CELL{A_{0}=a(\QTR{bf}{r},t)e^{i\psi (\QTR{bf}{r},t)}}}{2}{14}{}{}}where the amplitude $a$ and the phase $\psi $ are real and substituting this into the nonlinear Schr\"{o}dinger equation, we obtain from the imaginary part of Eq.(12) after multiplying through by $a$ the following equation \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial }{\partial t}a^{2}+\QTR{bf}{(v}_{g}\cdot \QTR{bf}{\nabla })a^{2}+\frac{1}{\omega }\QTR{bf}{\nabla }(a^{2}\QTR{bf}{\nabla }\psi )=0}}{2}{15}{}{}}and from the real part we get \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial \psi }{\partial t}+\left( \QTR{bf}{v}_{g}\cdot \QTR{bf}{\nabla }\right) \psi +\frac{1}{2\omega }\left( \QTR{bf}{\nabla }\psi \right) ^{2}-\frac{1}{2a\omega }\QTR{bf}{\nabla }^{2}a-\frac{\alpha }{2\omega }\left( Z_{d}\frac{\delta n_{d}}{n_{0d}}+\frac{Z_{i}n_{0i}}{n_{0d}}\frac{U_{pond}}{T_{i}}\right) =0}}{2}{16}{}{}}We will now show how the ponderomotive potential in Eq.(16) changes the dispersion relation of the linear perturbation. To this end, we linearize Eq.(15) and (16) with respect to the perturbations, which are represented as $a=a_{0}+\delta a$, $\psi =\psi _{0}+\delta \psi ,$ where $a_{0\text{ }},\psi _{0}$ denote the equilibrium values and $\delta a$, $\delta \psi $ are small perturbations. Upon solving Eq.(16) we observe that $\psi _{0}$ is time dependent and is expressed as \EQN{0}{1}{}{}{\RD{\CELL{\psi _{_{0}}(t)=\frac{\alpha }{2\omega }\frac{Z_{i}n_{0i}}{n_{0d}}\frac{e^{2}a_{0}^{2}}{2mc^{2}T_{i}}t}}{2}{17}{}{}}After linearization of Eqs.(10), (15) and (16), we seek plane wave solutions proportional to $\exp \left[ i(\QTR{bf}{q\cdot r}-\Omega t)\right] .$ Finally, we obtain the following dispersion relation \EQN{0}{1}{}{}{\RD{\CELL{\left\{ \left( \Omega -\QTR{bf}{q\cdot v}_{g}\right) ^{2}-\frac{q^{2}}{4\omega ^{2}}\left( q^{2}-\alpha \frac{Z_{i}n_{0i}}{n_{0d}}\frac{e^{2}a_{0}^{2}}{m_{o}c^{2}T_{i}}\right) \right\} \left[ \Omega ^{2}-q^{2}\QTR{bf}{u}_{d}^{2}\right] =\frac{\alpha \beta q^{4}}{4\omega ^{2}}\frac{e^{2}a_{0}^{2}}{m_{0}c^{2}}}}{2}{18}{}{}}The third term on the l.h.s of Eq.(18) is an additional term, which appears only for a three component dusty plasma. This additional term can increase the growth rate, when the diffraction term becomes of the same order as this term, or conversely stabilizes the instability, when the first and second terms on l.h.s are smaller than the term on the r.h.s of Eq.(18) and that $\Omega ^{2}>q^{2}u_{d}^{2}.$ We first consider the case when the third term on the l.h.s of Eq.(18) compensates the diffraction term $\frac{q^{4}}{4\omega ^{2}}.$ By taking $\omega _{pe}>>k_{0}c,($in a dense plasma, the group velocity becomes small and can be of the same order as $\QTR{bf}{u}_{d}),$ we obtain from the dispersion relation (18) having coincidental roots $\Omega =\QTR{bf}{q\cdot v}_{g}+\gamma $ and $\Omega =\QTR{bf}{q\cdot u}_{d}+\gamma ,$ the following expression for the growth rate \EQN{0}{1}{}{}{\RD{\CELL{\func{Im} \gamma =\frac{\sqrt{3}}{4}qc\left[ \frac{n_{0e}}{Z_{d}Z_{i}n_{0i}}\left( \frac{T_{i}}{m_{0}c^{2}}\right) \left( \frac{ea_{0}}{T_{e}}\right) ^{2}\right] ^{1/3}}}{2}{19}{}{}}Now we assume the plasma to be tenuous $(\omega _{pe}<
Compression Phenomenon of dust grains by EMW
In the present section we investigate the phenomenon of clustering of dust grains in the region where the EM wave is localized. If the nonlinear terms (last two terms) in the Schr\"{o}dinger equation$\left[ \text{Eq.(12})\right] $ are positive then the EM wave can be focused at the focal point. We will now show the existence of compression of dust grains in the localized area of electromagnetic wave. First we suppose that the dust grains are inertialess so that Eq.(8) reduces to \EQN{0}{1}{}{}{\RD{\CELL{\frac{\delta n_{d}}{n_{0d}}=\frac{\beta }{u_{d}^{2}}U_{pond}>0}}{2}{21}{}{}}which means that when the energy density of the EM wave increases then at that point the density of the dust gains also increases. Further while the ions are attracted towards the grains, the electrons are expelled from that region. Thus, in the localized region of EM wave, dust grains and ions are found in abundance suggesting the possibility of crystallization of dust atoms\QCITE{cite}{}{31}. We may also note that Eq.(10) describes subsonic motion when the inequality $\frac{\partial ^{2}}{\partial t^{2}}<
In the opposite case, i.e., when the regime is supersonic $\left( \frac{\partial ^{2}}{\partial t^{2}}>>\QTR{bf}{u}_{d}^{2}\QTR{bf}{\nabla }^{2}\right) ,$ then Eq.(10) becomes \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial ^{2}}{\partial t^{2}}\left( \frac{\delta n_{d}}{n_{od}}\right) =-\beta \QTR{bf}{\nabla }^{2}U_{pond}}}{2}{22}{}{}}If we assume here that the EM wave propagates along the z-axis and introduce $\QTR{bf}{r}_{\bot }$ and $\tau =t-z/\QTR{bf}{v}_{g}$ then integrating Eq.(22) twice we obtain \EQN{0}{1}{}{}{\RD{\CELL{\frac{\delta n_{d}}{n_{od}}=-\frac{\beta }{v_{g}^{2}}\left[ U_{pond}+v_{g}^{2}\Delta _{\bot }\int_{\tau _{\shortmid }}^{\tau }d\tau ^{\shortmid }\int_{\tau _{\shortmid }}^{\tau ^{\shortmid }}d\tau ^{_{\shortmid \shortmid }}U_{pond}+C\tau +C_{1}\right]}}{2}{23}{}{}}If we further assume that $U_{pond}(\QTR{bf}{r}_{\bot },\tau _{\shortmid })=0,$ then we obtain the condition that $C=0=C_{1}.$
Let us now consider the example of radiation pulse (EM wave) with a unit step function having a profile of the form \EQN{0}{1}{}{}{\RD{\CELL{U_{pond}=U_{0}e^{-\frac{\QTR{bf}{r}_{\bot }^{2}}{2r_{o}^{2}}}\left\{ \Theta (\tau -\tau _{1})-\Theta (\tau -\tau _{2})\right\}}}{2}{24}{}{}}where $\tau _{2}-\tau _{1}$ is the pulse width, $r_{o}$ characterizes the pulse length and $\Theta (x)$ is the unit step function, which has the property that when $x>0,$ $\Theta (x)=1$ and if $x\leqslant 0$ then $\Theta (x)=0.$ Substituting Eq.(24) into Eq.(23) and integrating, we obtain the following expression for the fluctuating density of the dust grains \EQN{0}{1}{}{}{\RD{\CELL{\frac{\delta n_{d}}{n_{od}}=-\frac{\beta }{2v_{g}^{2}}U_{pond}\left\{ 1-\frac{v_{g}^{2}(\tau -\tau _{\shortmid })^{2}}{r_{o}^{2}}\left( 1-\frac{r_{\bot }^{2}}{2r_{o}^{2}})\right) \right\}}}{2}{25}{}{}}If we consider in the above expression $r_{\bot }=0$ and $\tau =\tau _{2},$ then in this case $U_{pond}=U_{0}=$ cons$\tan $t and \EQN{0}{1}{}{}{\RD{\CELL{\frac{\delta n_{d}}{n_{od}}=-\frac{\beta U_{0}}{2v_{g}^{2}}\left( 1-\frac{v_{g}^{2}\tau _{o}^{2}}{r_{o}^{2}}\right)}}{2}{26}{}{}}where $\tau _{0}=\tau _{2}-\tau _{1}.$ The above equation shows that if the initial shape of the EM wave pulse has the form of a light bullet, i.e., $v_{g}\tau _{0}>r_{0},$ then the density of dust grains increases in the area where focusing of the EM wave takes place. However when the shape of pulse has a pancake form i.e., $r_{0}>v_{g}\tau _{0},$ then the opposite occurs i.e., $\delta n_{d}<0$.>
Envelope of EM Wave
We have shown in Sec.(III) that the modulated amplitude of EM waves leads to the excitation of dust acoustic waves, whose amplitude grows exponentially. After a certain time the wave stops growing due to the appearance of the nonlinear terms which did not exist in the linear analysis.
Now we will take into account nonlinear terms but retain only the quadratic nonlinearities which is a satisfactory approximation for nonrelativistic amplitude of the EM waves. In this case the nonlinearities enter only through the ponderomotive force, which redistributes the particles and changes the density of the plasma, (the hydrodynamic nonlinearities, i.e., the convective derivative term $\left\{ \left( \QTR{bf}{v}_{d}.\QTR{bf}{\nabla }\right) \QTR{bf}{v}_{d}\right\} ,$ remain irrelevant, at least as long as the wave does not steepen too much).
Introducing the notation $F=\left( \frac{1}{2m_{0}c^{2}T_{e}}\right) ^{1/2}eA_{0}(\QTR{bf}{r},t)$, $\delta n=\frac{Z_{d}\alpha }{n_{0d}}\delta n_{d}$ and rewriting the equations ($10)$ and (12) we obtain the following coupled equations: \EQN{0}{1}{}{}{\RD{\CELL{2i\omega _{o}\left( \frac{\partial }{\partial t}+\QTR{bf}{v}_{g}\QTR{bf}{\cdot \nabla }\right) F+\nabla ^{2}F+(\delta n+|F|^{2})F=0}}{2}{27}{}{}}\EQN{0}{1}{}{}{\RD{\CELL{\left( \frac{\partial ^{2}}{\partial t^{2}}-u_{d}^{2}\QTR{bf}{\nabla }^{2}\right) \delta n=-u_{d}^{2}\left( \frac{n_{0e}}{n_{od}}\right) \alpha \nabla ^{2}|F|^{2}}}{2}{28}{}{}}First, we shall consider the one dimensional steady state problem. To this end, we introduce the new variable $\xi =x-v_{g}t,$ and write $F=F_{0}(\xi ,t)\exp \left[ -i(\omega -\Delta \omega )t\right] $. The function $F_{0}(\xi ,t)$ is real and $\Delta \omega $ is a negative correction to the frequency which in dimensionless form is \EQN{0}{1}{}{}{\RD{\CELL{\Delta \omega =\frac{\omega _{p}^{2}+k_{0}^{2}c^{2}-\omega _{0}^{2}}{2\omega _{0}}}}{2}{29}{}{}}In this case, from Eq.(27) and Eq.(28) we get the following coupled equations \EQN{0}{1}{}{}{\RD{\CELL{\frac{d^{2}F_{0}}{d\xi ^{2}}-2\omega _{0}\Delta \omega F_{0}+(\delta n+F_{0}^{2})F_{0}=0}}{2}{30}{}{}}\EQN{0}{1}{}{}{\RD{\CELL{\delta n=\frac{1}{1-M^{2}}\left( \frac{n_{0e}}{n_{0d}}\right) \alpha F_{0}^{2}}}{2}{31}{}{}}where $M=\frac{v_{g}}{u_{d}}$ is the Mach number of the DA wave. Substituting Eq.(31) into Eq.(30), we obtain the NLS equation with a cubic nonlinearity, the solution of which is well known. But our case is physically different compared to the case of an electron ion plasma, and to obtain a soliton solution, it is necessary that the density perturbation of the dust grains must be positive. In our case $M<1$,>
We now consider the quasi-nonstationary regime by introducing new variables $\tau $ and $\xi $ and also assume that $v_{g}\thickapprox u_{d}$, in this case Eq.(28) will reduce to \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial \delta n}{\partial \tau }=\frac{\alpha u_{d}}{2}\left( \frac{n_{0e}}{n_{0d}}\right) \frac{\partial }{\partial \xi }F_{0}{}^{2}}}{2}{32}{}{}}We may rewrite Eq.(30) and Eq.(32) as \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial ^{2}\chi }{\partial z^{2}}-\chi +(\delta N+f_{c}^{2}\chi ^{2})\chi =0}}{2}{33}{}{}}\EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial \delta N}{\partial \tau }=\frac{\partial }{\partial z}\chi ^{2}}}{2}{34}{}{}}where $\chi =\frac{1}{\sqrt{2\omega \Delta \omega }}\frac{F_{0}}{f_{c}},$ $f_{c}^{2}=\frac{2}{\alpha u_{d}\sqrt{2\omega \Delta \omega }}\left( \frac{n_{0e}}{n_{0d}}\right) $, $z=\sqrt{2\omega \Delta \omega }\xi $ and $\delta N=\frac{Z_{d}\alpha }{2\omega \Delta \omega }\frac{\delta n_{d}}{n_{0d}}.$
We now examine the general solution of Eq.(33) and Eq.(34) by following ref \QCITE{cite}{}{32} for arbitrary initial distribution $\chi (z,0).$ Substituting $\delta N$ from Eq.(33) into Eq.(34) we obtain the nonlinear equation for the amplitude of the EM field \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial }{\partial \tau }\left( \frac{1}{\chi }\frac{\partial ^{2}\chi }{\partial z^{2}}\right) +f_{c}^{2}\frac{\partial \chi ^{2}}{\partial \tau }=-\frac{\partial }{\partial z}\chi ^{2}}}{2}{35}{}{}}Since $\chi (z,\tau )$ is a slowly varying function of time $\tau $, we can neglect the second term on the l.h.s in comparison with the r.h.s and further by multiplying both sides of Eq.(35) by $\chi ^{2},$we rewrite the first term in Eq.(35) in the form as $\frac{\partial }{\partial z}\left[ \chi ^{2}\frac{\partial }{\partial z}\left( \frac{1}{\chi }\frac{\partial \chi }{\partial \tau }\right) \right] ,$ integrating once and using boundary conditions $|z|\rightarrow \infty $ and $\chi (\pm \infty )=0,$ we obtain \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial }{\partial z}\left( \frac{1}{\chi }\frac{\partial \chi }{\partial \tau }\right) =-\frac{\chi ^{2}}{2}}}{2}{36}{}{}}We now introduce the function \EQN{0}{1}{}{}{\RD{\CELL{\Phi (z,\tau )=\int_{-\infty }^{z}dz^{^{\prime }}\chi ^{2}(z^{^{\prime }},\tau )}}{2}{37}{}{}}so that Eq.(36) on integration yields \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial \Phi }{\partial \tau }=-\frac{\Phi ^{2}}{2}+C\Phi +C_{1}}}{2}{38}{}{}}where $C_{1}=0$, because $\Phi (-\infty ,\tau )=\frac{\partial \Phi (-\infty ,\tau )}{\partial \tau }=0.$
For $z\rightarrow \infty ,$ we obtain from Eq.(38) the following \EQN{0}{1}{}{}{\RD{\CELL{\Phi (+\infty ,\tau )=\int_{-\infty }^{\infty }dz^{^{\prime }}\chi ^{2}(z^{^{\prime }},\tau )=\Phi _{o}}}{2}{39}{}{}}where $\Phi _{o}$ is a constant. In this case from Eq.(36) we have $C=\frac{\Phi _{o}}{2}$, thus Eq. (36) can be written as \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial \Phi }{\partial t}=-\frac{\Phi }{2}(\Phi -\Phi _{o})}}{2}{40}{}{}}The above equation has two different solutions corresponding to $\Phi >\Phi _{o}$ or $\Phi <\Phi _{o}.$
We first consider the case $\Phi _{o}>\Phi $ then the solution of Eq.(38) is \EQN{0}{1}{}{}{\RD{\CELL{\Phi =\frac{\Phi _{o}}{2}\left[ 1+\tanh \frac{\Phi _{o}}{4}\left[ \tau +\tau _{o}(z)\right] \right]}}{2}{41}{}{}}From Eq.(41) at $\tau =0,$ we obtain for $\tau _{o}(z)$\EQN{0}{1}{}{}{\RD{\CELL{\tau _{o}(z)=-\frac{2}{\Phi _{o}}\ln \left( \frac{\Phi _{o}}{\Phi (z,0)}-1\right)}}{2}{42}{}{}}for which we have $\tau _{0}(+\infty )=\infty $ when $\Phi =\Phi _{o}$ and $\tau _{0}(-\infty )=-\infty $ when $\Phi (-\infty )=0.$
The expression for the energy density of EM wave can be obtained by differentiating the function $\Phi (z,\tau )$ with respect to $z$ to obtain \EQN{0}{1}{}{}{\RD{\CELL{\chi ^{2}(z,\tau )=\frac{\partial \Phi }{\partial z}=\frac{\Phi _{o}^{2}}{8}\frac{\partial \tau _{o}(z)}{\partial z}\left[ \sec \text{h}^{2}\left( \frac{\Phi _{o}}{4}\right) \left( \tau +\tau _{o}(z)\right) \right]}}{2}{43}{}{}}Now we analyze Eq.(43) for the special case when the initial function is \EQN{0}{1}{}{}{\RD{\CELL{\chi (z,\tau =0)=\frac{\chi _{o}}{\cosh z}}}{2}{44}{}{}}In this case $\tau _{o}(z)=\frac{4}{\Phi _{o}}z$ and $\frac{\Phi _{o}}{2}=\chi _{o}^{2}$
Finally we have \EQN{0}{1}{}{}{\RD{\CELL{\chi ^{2}=\chi _{o}^{2}\sec h^{2}\left( z+\frac{\chi _{o}^{2}}{2}\tau \right)}}{2}{45}{}{}}For the density of dust grains in this case we obtain \EQN{0}{1}{}{}{\RD{\CELL{Z_{d}\frac{\delta n_{d}}{n_{0d}}=\frac{2}{\alpha }\left( \frac{\omega _{p}^{2}+k_{0}^{2}c^{2}-\omega _{o}^{2}}{\omega _{p}^{2}}\right) \frac{1}{\cosh ^{2}\left( \frac{\chi _{o}^{2}}{2}\tau +z\right) }}}{2}{46}{}{}}We now investigate the second case, when $\Phi >\Phi _{o}.$ In this case the term $\Phi _{o}$ dominates the r.h.s in Eq.(38) and leads to $\Phi (z,\tau )\rightarrow \infty $ in finite time. The solution of Eq.(40) \EQN{0}{1}{}{}{\RD{\CELL{\Phi (z,\tau )=\frac{\Phi _{o}}{2}\left[ 1+\coth (\frac{\Phi _{o}}{4})\left( \tau +\tau _{o}(z)\right) \right]}}{2}{47}{}{}}In this it follows that at $\tau +\tau _{o}(z)\rightarrow 0,$ when $\Phi (z,\tau )\rightarrow \infty .$
At $\tau =0,$ we obtain from Eq.(47) that \EQN{0}{1}{}{}{\RD{\CELL{\tau _{o}(z)=\frac{2}{\Phi _{o}}\ln \left( 1-\frac{\Phi _{o}}{\Phi (z,0)}\right)}}{2}{48}{}{}}By differentiating the function $\Phi (z,\tau )$ with respect to $z$, we find \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial \Phi }{\partial z}=\chi ^{2}=-\frac{\Phi _{o}^{2}}{8}\frac{\partial \tau _{o}}{\partial z}\left[ \frac{1}{\sinh ^{2}\left[ (\frac{\Phi _{o}}{4})\left( \tau +\tau _{o}(z)\right) \right] }\right]}}{2}{49}{}{}}Expression (47) and (49) describe the nonlinear breaking of the wave front and for time $\tau =-\tau _{o}(z)$ a shock wave must appear.
Conclusions
In the present work we have investigated some novel aspects of the nonlinear wave propagation in a dusty plasma, which to the best of our knowledge have not been considered before. We have shown in Section II that we are able to obtain an equation which resembles Zakharov's equation but with an important difference that our equation has a negative ponderomotive pressure term. This negative ponderomotive pressure term introduces new physics having important consequences which have subsequently been discussed in later Sections. In Sec.III, we obtained a NLS type equation for the amplitude of EM waves in a dusty plasma with an additional term which leads to a strong modulational instability. The growth rates of this strong modulated instability have been obtained for the cases of a dense and a tenuous plasma, respectively.
In Sec.IV, we have shown that in our case the EM wave can be focused and that the dust grains cluster in the focus region. We have also shown that when the EM wave pulse has the form of a light bullet the dust grain density increases in the focusing region of the EM wave while the opposite happens when the EM wave pulse has a pancake shape.
As shown in Sec.III, the nonlinear interaction is governed by a Schr\"{o}dinger-like equation for the EM wave envelopes and a driven DA wave equation. The coupled nonlinear equations admit both stationary and nonstationary solutions. In Sec.V, the stationary solutions are characterized as EM wave crests which propagate with a velocity close to the dust sonic speed and lead to the localization of dust grains which in turn lead to crystallization. On the other hand, a nonstationary density response to DA wave admits shock like structures.
We believe that the results obtained here are both new and important for the physics associated with the dusty plasmas and will help in the better understanding of non linear phenomena in such complex plasmas.
\QTR{bf}{Acknowledgment: }One of us (Z.E) acknowledges the financial support provided by the Salam Chair in Physics, GCU Lahore, Pakistan.
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