Introduction
The most general class of nonlinear wave structures is solitary waves or soliton having approximately self-consistent wave amplitudes and phases. Solitons are important and extensively studied because a whole class of nonlinear differential equations encountered in plasma physics, solid-state physics, particle physics, hydrodynamics, nonlinear optics, and biology are seen to support the solutions of solitary waves. In the last few years, solitary waves in plasma have been studied theoretically and experimentally by various authors. Washimi and Taniuti (Reference Washimi and Taniuti1966) first theoretically studied ion acoustic solitary waves (IASWs) in cold collisionless unmagnetized plasma deriving the K–dV equation using the reductive perturbation technique. The solitary wave in plasma was observed experimentally by Ikezi (Reference Ikezi1973), Lonngren (Reference Lonngren1983) and others. A lot of theoretical works on the nonlinear propagation of ion acoustic (IA) waves have been done by various authors incorporating different parameters in the plasma example, ion temperature (Tagare, Reference Tagare1973), two-temperature electrons (Ghosh et al., Reference Ghosh, Paul, Das and Paul2008), resonant electrons (Schamel, Reference Schamel1973), gravitation (Paul et al., Reference Paul, Chatterjee and Paul2017), negative ion (Chattopadhyaya and Paul, Reference Chattopadhyaya and Paul2012), positrons (Paul et al., Reference Paul, Das, Paul, Bandyopadhyay, Chattopadhyaya and De2012), nonthermal electrons (Gill et al., Reference Gill, Bala, Kaur, Saini, Bansal and Kaur2004), kappa-distributed electron (Baluku and Hellberg, Reference Baluku and Hellberg2012), Tsallis-distributed electrons (Bala et al., Reference Bala, Gill, Bains and Kaur2017), etc. and have shown that these plasma parameters have considerable impact on the excitation of solitary waves. However, in recent years, there is considerable interest in the nonlinear propagation of waves in a plasma consisting of electron/ion beams. The studies of the propagation of waves in beam plasma are important in magnetospheric and solar physics (Goldman, Reference Goldman1983; Hoffmann and Evans, Reference Hoffmann and Evans1968). The nonlinear structure of a plasma may change considerably in the presence of electron/ion beams. An important property of an electron beam plasma is that it can change the propagation characteristic of the Trivelpiece–Gould (TG) solitons (Krivoruchko et al., Reference Krivoruchko, Fainberg, Shapiro and Shevchenko1975). The effects of an ion beam on the motion of solitons in an ion beam-plasma system has been studied by Gell and Roth (Reference Gell and Roth1977). The effects of ion beams are more important on the excitation of ion acoustic solitary waves in plasma. Abrol and Tagare (Reference Abrol and Tagare1979) obtained a modified K–dV equation for an IASW in an ion–beam–plasma system with cold ions, beam ions, and nonisothermal electrons. They investigated the effect of the resonant electrons, both trapped and free electrons, of the plasma–ion to beam–ion mass ratio and of the beam–ion concentration on the amplitude and width of the solitary wave. Later, many authors (Karmakar et al., Reference Karmakar, Das and Singh1988; Das and Singh, Reference Das and Singh1991; El-Labany, Reference El-Labany1995; Huibin and Kelin, Reference Huibin and Kelin2009; Zank and McKenzie, Reference Zank and McKenzie1998; Das et al. Reference Das, Singh and Karmakar2011; Misra and Adhikary, Reference Misra and Adhikary2011; Das, Reference Das2012; Das and Deka, Reference Das and Deka2015) have studied solitary waves in ion beam plasma and have obtained some fascinating results which are important in different situations in laboratory and space plasma. Recently, Kaur et al. (Reference Kaur, Singh and Saini2017) have investigated the nonlinear propagation of ion acoustic solitary waves (IASWs) in an unmagnetized plasma composed of a positive warm ion fluid, two-temperature electrons obeying kappa-type distribution, and penetrated by a positive ion beam. They have used the reductive perturbation method to derive the nonlinear equations, namely K–dV, modified K–dV (mK–dV), and Gardner equations. The characteristic features of both compressive and rarefactive nonlinear excitations from the solution of these equations are studied and compared in the context with the observation of the He+ beam in the polar cap region near solar maximum by the Dynamics Explorer 1 satellite. It is observed that the superthermality and density of cold electrons, number density, and temperature of the positive ion beam crucially modify the basic properties of compressive and rarefactive IASWs in the K–dV and mK–dV regimes. It is further analyzed that the amplitude and width of Gardner solitons are appreciably affected by different plasma parameters. The characteristics of double layers (DLs) are also studied in detail below the critical density of cold electrons. The theoretical results may be useful for the observation of nonlinear excitations in laboratory and ion beam-driven plasmas in the polar cap region near solar maximum and polar ionosphere as well in Saturn's magnetosphere, solar wind, pulsar magnetosphere, etc., where the population of two-temperature superthermal electrons is present. More recently, Kaur et al. (Reference Kaur, Singh, Ghai and Saini2018) have investigated the propagation characteristics of dust acoustic solitary and rogue waves in an unmagnetized ion beam plasma with electrons and ions following the kappa-type distribution in nonplanar geometry. The reductive perturbation method is employed to derive the cylindrical/spherical K–dV equation, which is further transformed into a standard K–dV equation by neglecting the geometrical effects. Using new stretching coordinates, the nonlinear Schrödinger equation has also been derived from the standard K–dV equation to study the different order rational solutions of dust acoustic rogue waves. The impact of various physical parameters on the characteristics of dust acoustic solitary waves is elaborated specifically in nonplanar geometry. Further, the effects of ion beam and superthermality of electrons/ions on the characteristics of rogue waves are studied. It is predicted that the results obtained may be useful in comprehending a variety of phenomena in Earth's magnetosphere polar cap region where the presence of positive ion beam has been detected and also in other regions of space/astrophysical environments where dust along with superthermal electrons and ions exists.
However, in recent years, researchers are very much interested to study the propagation of waves in a quantum plasma because quantum effects become important in a variety of environments, such as in intense laser–solid density plasma interaction experiments, dense astrophysical and cosmological environments, ultra-small electronic devices, and metal nanostructures. The recent developments in nanoscience (Manfredi et al., Reference Manfredi, Hervieux, Yin, Crouseilles, Massobrio, Bu-lou and Goyenex2009), biophotonics (Barnes et al., Reference Barnes, Dereux and Ebbesen2003), fiber optics (Agrawal, Reference Agrawal1989), quantum confinement (Ang et al., Reference Ang, Kwan and Lau2003), ultracold plasmas (Killian, Reference Killian2007), etc., the study has gained momentum in a totally new direction. The quantum effects are also important in laboratory-produced ultra-dense plasmas (Kremp et al., Reference Kremp, Bornath, Bonitz and Schlanges1999) and laser-based inertial fusion experiments (Azechi, Reference Azechi2006) as well as the interior of Jovian planets, neutron stars (Shapiro and Teukolsky, Reference Shapiro and Teukolsky1983), and white dwarfs (Madelung, Reference Madelung1927), etc., where the density is very high. Quantum plasmas show numerous nonlinearities which are absent in classical plasmas. This has led to the investigation of waves in quantum plasma more important than ever. There have been many modes of waves in plasmas like ion acoustic waves (IAWs), electron acoustic waves, dust acoustic waves, dust ion acoustic waves, electron plasma waves, and shock waves. Using the Quantum Hydrodynamic (QHD) model, Haas et al. (Reference Haas, Garcia and Goedert2003) has studied the importance of the role of quantum diffraction in a linear and nonlinear regime. They adopted an equation of state pertaining to a zero-temperature Fermi gas for the electrons by disregarding pressure effects for the ions. By an appropriate rescaling of the variables, they identified a nondimensional parameter H, proportional to quantum diffraction effects. The system is shown to support linear waves, which, in the limit of small H, resemble the classical IAWs. In the weakly nonlinear limit, the quantum plasma is shown to support waves described by a deformed K–dV equation which depends in a nontrivial way on the quantum diffraction parameter H. In the fully nonlinear regime, the system also admits traveling waves which can exhibit periodic patterns. The works of Haas et al. (Reference Haas, Garcia and Goedert2003) have encouraged a large number of authors to study the nonlinear propagation of acoustic waves (e.g., solitary waves and modulation instability) in quantum plasma. We have here mentioned some of the important works on solitary waves in quantum plasma. Ali and Shukla (Reference Ali and Shukla2006) have considered a one-dimensional QHD model for a three-species quantum plasma to derive the K–dV equation incorporating quantum corrections and studied the nonlinear properties of dust acoustic solitary waves. They examined the quantum mechanical effects numerically both on the profiles of the amplitude and the width of dust acoustic solitary waves. It is found that the amplitude remains constant but the width shrinks for different values of a dimensionless electron quantum diffraction parameter H. Subsequently, Ali et al. (Reference Ali, Moslem, Shukla and Schlickeiser2007) have investigated the linear and nonlinear properties of the IAWs by using the QHD equations together with the Poisson equation in a three-component quantum electron–positron–ion plasma. They have derived the linear dispersion relation, the K–dV equation, and an energy equation containing quantum corrections. They have also performed the computational investigations to examine the quantum mechanical effects on the linear and nonlinear waves. It is found that both the linear and nonlinear properties of the IAWs are significantly affected by the inclusion of the quantum corrections. Later, Misra and Bhowmik (Reference Misra and Bhowmik2007) have considered a nonplanar spherical geometry to study the nonlinear properties of ion acoustic (IA) waves in an electron–ion quantum plasma with the effects of quantum corrections. Using the QHD model and standard reductive perturbation method, they derived the K–P equation having a variable coefficient and examined numerically the importance of quantum mechanical effects on the compressive and rarefactive solitons. It is found that H plays a significant role in the formation of compressive and rarefactive solitons. A critical value of H is also found which depends on the phase velocity of the wave and the ion to electron Fermi temperature ratio, for which the soliton formation ceases to exist. Subsequently, Mushtaq and Khan (Reference Mushtaq and Khan2007) have studied ion acoustic solitary wave with weakly transverse perturbations in quantum electron–positron–ion plasma using the QHD model. They have derived the linear dispersion relation in the linear regime and the K–P equation in the nonlinear regime. It is found that compressive solitary wave can propagate in this system. The quantum effects are also studied graphically for both the linear and nonlinear profiles of ion acoustic wave. Using the energy consideration method, conditions for the existence of stable solitary waves are obtained. It is found that stable solitary waves depend on quantum corrections, positron concentration, and direction cosine of the wave vector. Later, Sah and Manta (Reference Sah and Manta2009) have investigated the nonlinear wave structure of electro-acoustic waves (EAWs) in a three-component unmagnetized dense quantum plasma consisting of two distinct groups of electrons (one inertial cold electron and other inertialess hot electrons) and immobile ions. Using one-dimensional QHD and the standard reductive perturbation technique, they derived the K–dV equation governing the dynamics of EAWs. Both compressive and rarefactive solitons along with periodical potential structures are found to exist for various ranges of dimensionless quantum parameter H. The quantum mechanical effects are also examined numerically on the profiles of the amplitude and the width of electro-acoustic solitary waves. It is observed that both the amplitude and the width of electro-acoustic solitary waves are significantly affected by the parameter H. Akbari-Moghanjoughi (Reference Akbari-Moghanjoughi2010) has investigated large-amplitude IASW in a degenerate dense electron–positron–ion plasma considering the ion temperature as well as electron/positron degeneracy effects. It is shown that the ion temperature effects play an important role in the existence criteria and allowed a Mach-number range in such plasmas. He has also pointed out the fundamental difference in the existence of supersonic IASW propagations between degenerate plasmas with nonrelativistic and ultra-relativistic electrons and positrons. Chandra et al. (Reference Chandra, Paul and Ghosh2012) have theoretically studied the linear and nonlinear propagation of electron plasma wave in a two-component unmagnetized dense quantum plasma with streaming of ions both analytically and numerically using one-dimensional QHD. It has been shown that the quantum effect modifies the linear dispersion character of the electron plasma waves in the presence of streaming motion and makes the possibility of two distinct modes. They have also derived the K–dV equation and have shown that both compressive and rarefactive solitary waves would be excited in the model plasma on some critical values of the quantum diffraction parameter. Dip et al. (Reference Dip, Hossen, Salahuddin and Mamun2017) have investigated higher-order nonlinearity of the EAWs, specifically IA waves in an unmagnetized, collisionless, quantum electron–positron–ion plasma. They derived the mK–dV equation. The plasma system is supposed to be formed of positively charged inertial heavy ions, inertialess electrons and positrons to analyze the solitary waves (SWs), and the standard Gardner (SG) equation to analyze the higher-order SWs as well as DLs. The basic features (namely, amplitude, width, phase speed, etc.) of the IA SWs and DLs are examined. The comparison between the mK–dV SWs and SG SWs is also made. It is found that the amplitude, width, and phase speed of the IA SWs and DLs are significantly modified by the effects of both Fermi temperatures as well as pressures and Bohm potentials of electrons and positrons.
However, the propagation of ion acoustic and EAWs in quantum (degenerate) plasma having electron beam or ion beam would give fascinating results which are not studied yet critically, though many authors studied the wave propagation in nondegenerate/classical plasma (Yajima and Wadati, Reference Yajima and Wadati1990; Mondal et al., Reference Mondal, Paul and Roychowdhury1998; El-Labany, Reference El-Labany1995; Zank and McKenzie, Reference Zank and McKenzie1998; Bailung et al., Reference Bailung, Sharma and Nakamura2010; Saberian et al., Reference Saberian, Esfandyari-Kalejahi, Rastkar-Ebrahimzadeh and Afsari-Ghazi2013; Danehkar, Reference Danehkar2018). So, in the present paper, we are interested to study the nonlinear propagation of ion acoustic waves in a quantum plasma consisting of warm ion beams. Using the reductive perturbation method, we have derived the K–dV equation and have obtained the solution of ion acoustic solitary in the quantum plasma having nonrelativistic ion beam. The effects of ion beam parameters, that is ion beam density, ion beam temperature, and ion beam velocity, on the solitary waves are critically discussed with graphical representation. The quantum diffraction parameter H has a significant role in the width of solitary waves in the quantum plasma. For H < 2 and H > 2, the width may be increased or decreased depending upon the values of ion beam parameters.
Basic equations
We consider that the quantum plasma is unmagnetized and collissionless and it consists of positive ions, beam ions, and inertialess electrons. All species of the plasma are assumed to follow the Fermi–Dirac statistics. The positive ions and positive beam ions are considered to be singly ionized. Ion acoustic wave is a low-frequency wave, in which the restoring force comes from the pressure of inertialess electrons and the ion masses provide the driving force to maintain the wave. The electrons are considered inertialess by assuming that the phase velocity of ion acoustic wave is much less than the Fermi velocity of electron and much greater than the Fermi velocities of positive and beam ions. We assume that the plasma particles behave as a one-dimensional Fermi gas at zero temperature and therefore the pressure law is:
$$p_{\rm j} = \displaystyle{{m_{\rm j}V_{{\rm Fj}}^2} \over {3n_{{\rm j}0}^2}} n_{\rm j}^3 $$
where j = e for electrons, j = i for ions, and j = b for beam ions; m j is the mass; 
$V_{{\rm Fj}} = \sqrt {2k_{\rm B}T_{{\rm Fj}}/m_{\rm j}}$ is the Fermi speed, T Fj is the Fermi temperature, and k B is the Boltzmann constant; n j is the number density with the equilibrium value n j0. 
${\rm \omega} _{\rm pe} = \sqrt {4{\rm \pi} n_0 {\rm e}^2/m_{\rm e}}$ is the electron plasma oscillation frequency and V Fe is the Fermi thermal speed of electrons.
The normalized equations governing the plasma dynamics are:
$$\eqalign{0 = &\displaystyle{{\partial {\rm \phi}} \over {\partial x}}-n_{\rm e}\displaystyle{{\partial n_{\rm e}} \over {\partial x}} + \displaystyle{{H^2} \over 2} \displaystyle{\partial \over {\partial x}}\left[ {\displaystyle{1 \over {\sqrt {n_{\rm e}}}} \displaystyle{{\partial^2\sqrt {n_{\rm e}}} \over {\partial x^2}}} \right]} $$
$$\displaystyle{{\partial n_{\rm i}} \over {\partial t}} + \displaystyle{{\partial (n_{\rm i}u_{\rm i})} \over {\partial x}} = 0$$
$$\eqalign{&\left( {\displaystyle{\partial \over {\partial t}} + u_{\rm i}\displaystyle{\partial \over {\partial x}}} \right)u_{\rm i} = -\displaystyle{{\partial {\rm \phi}} \over {\partial x}}-{\rm \sigma} _{\rm i}n_{\rm i}\displaystyle{{\partial n_{\rm i}} \over {\partial x}}} $$
$$\displaystyle{{\partial n_{\rm b}} \over {\partial t}} + \displaystyle{{\partial (n_{\rm b}u_{\rm b})} \over {\partial x}} = 0$$
$$\eqalign{&\left( {\displaystyle{\partial \over {\partial t}} + u_{\rm b}\displaystyle{\partial \over {\partial x}}} \right)u_{\rm b} = -{\rm \mu} \displaystyle{{\partial \rm \phi} \over {\partial x}}-{\rm \mu \sigma} _{\rm b}n_{\rm b}\displaystyle{{\partial n_{\rm b}} \over {\partial x}}} $$
$$\eqalign{\displaystyle{{\partial ^2{\rm \phi}} \over {\partial x^2}} = &{\rm {\rm \chi}} n_{\rm e}-n_{\rm i} -(1-{\rm {\rm \chi}} )n_{\rm b}} $$It is to be mentioned that the following normalization has been used in the above equations:
$$\eqalign{& x\to \displaystyle{{x{\rm \omega} _{\rm pi}} \over {V_{{\rm Fi}}}},\,t\to t{\rm \omega} _{\rm pi}, \,{\rm \varphi} \to \displaystyle{{e{\rm \varphi}} \over {2k_{\rm B}T_{{\rm Fj}}}},\,n_{\rm j}\to \displaystyle{{n_{\rm j}} \over {n_0}} \,{\rm and}\,u_{\rm j}\to \displaystyle{{u_{\rm j}} \over {V_{{\rm Fe}}}}}$$
and 
$H = \hbar {\rm \omega} _{\rm pe}/2{\rm } k_{\rm B}T_{{\rm Fe}}$ is a nondimensional quantum parameter proportional to the quantum diffraction and is equal to the ratio between the plasma energy 
$\hbar {\rm \omega} _{\rm pe}$ (energy of an elementary excitation associated with an electron plasma wave) and the Fermi energy k BT Fe; χ = n e0/n i0 is the ratio of unperturbed electron density and ion density; μ = m i/m b is the ratio of positive ion and beam ion masses and σi,b = T Fi,b/T Fe is the ratio of ion (beam) Fermi temperature to electron Fermi temperature.
K–dV equation
In order to investigate the behavior of IA, we make the following perturbation expansions for the field quantities n e, n i, n b, u e, u i, u b, and φ about their equilibrium values:
$$\eqalign{\left[ {\matrix{ {n_{\rm e}} \cr {u_{\rm e}} \cr {n_{\rm i}} \cr \matrix{u_{\rm i} \hfill \cr n_{\rm b} \hfill \cr u_{\rm b} \hfill} \cr \phi \cr}} \right] = &\left[ {\matrix{ {n_{\rm e0}} \cr 0 \cr {n_{\rm i0}} \cr \matrix{0 \hfill \cr n_{\rm b0} \hfill \cr u_{\rm b0} \hfill} \cr 0 \cr}} \right] + {\rm \varepsilon} \left[ {\matrix{ {n_{\rm e1}} \cr {u_{\rm e1}} \cr {n_{\rm i1}} \cr \matrix{u_{\rm i1} \hfill \cr n_{\rm b1} \hfill \cr u_{\rm b1} \hfill} \cr {\rm \phi_1} \cr}} \right] + {\rm \varepsilon} ^2\left[ {\matrix{ {n_{\rm e2}} \cr {u_{\rm e2}} \cr {n_{\rm i2}} \cr \matrix{u_{\rm i2} \hfill \cr n_{\rm b2} \hfill \cr u_{\rm b2} \hfill} \cr {{\rm \phi}_2} \cr}} \right] + \ldots} $$where n e0 = χ, n b0 = 1 − χ, and n i0 = 1 which are obtained from the equilibrium condition of densities after normalization.
To get the desired K–dV equation describing the nonlinear behavior of ion acoustic waves, we use the standard reductive perturbation technique (Washimi and Taniuti, Reference Washimi and Taniuti1966). We introduce the usual stretching of the space and time variables:
$${\rm \xi} = {\rm \varepsilon} ^{1/2}(x-V_{\rm p}t)\quad{\rm and}\quad\tau = {\rm \varepsilon} ^{3/2}t$$where V p is the phase speed normalized by electron Fermi speed V Fi and ε is a smallness parameter measuring the dispersion and nonlinear effects.
Using Eq. (9), Eqs (2)–(7) are written in terms of the stretched coordinates ξ and τ and then the perturbation expansions (8) are substituted. Solving the lowest-order equations in ε (i.e., ε3/2), we obtain the linear long-wave phase speed V p in terms of plasma parameters after using the boundary conditions:
$$\eqalign{&{\rm \chi} -\displaystyle{1 \over {V_{\rm p}^2 -{\rm \sigma} _{\rm i}}} -\displaystyle{{(1-{\rm \chi} ){\rm \mu}} \over {[{(V_{\rm p}-u_{\rm b0})}^2-{\rm \sigma} _{\rm b}{(1-{\rm \chi} )}^2]}} = 0} $$Simplifying Eq. (10), we obtain the following equation:
$$\eqalign{&c_4V_{\rm p}^4 -2c_3V_{\rm p}^3 + c_2V_{\rm p}^2 + 2c_1V_{\rm p} + c_0 = 0} $$where
$$\eqalign{& c_4 = 1 \cr & c_3 = b_2/2 \cr & c_2 = -[b_1-b_2-b_3 + {\rm \mu} (1-{\rm \chi} )]/{\rm \chi} \cr & c_1 = [b_2(b_1-1)]/2{\rm \chi} \cr & c_0 = -[b_3(b_1 + 1)]/{\rm \chi} \cr & b_3 = [u_{\rm b0}^2 -{\rm \sigma}_{\rm b}(1-{\rm \chi} )^2] \cr & b_2 = 2u_{\rm b0} \cr & b_1 = {\rm \sigma}_{\rm i}}$$Equation (11) is a bi-quadratic equation in V p and it gives four values of phase speed of the ion acoustic wave in quantum plasma (real or imaginary). The four values of V p from the above bi-quadratic equation are obtained as follows:
$$\eqalign{V_{\rm p} = &\displaystyle{1 \over 2}[(c_3 + d_1) + \sqrt {(d_1 + c_3)-4(d_2 + d_3)} \cr V_{\rm p} = &\displaystyle{1 \over 2}[(c_3 + d_1) -\sqrt {(d_1 + c_3)-4(d_2 + d_3)} \cr V_{\rm p} = &\displaystyle{1 \over 2}[(c_3-d_1) + \sqrt {(d_1-c_3)-4(d_2-d_3)} \cr V_{\rm p} = &\displaystyle{1 \over 2}[(c_3-d_1) -\sqrt {(d_1-c_3)-4(d_2-d_3)}} $$where
$$\eqalign{& d_1^2 = c_3^2 -c_2 + 2d_2 \cr & d_1d_3 = c_1 + c_3d_2 \cr & d_3^2 = d_2^2 -c_0}$$
$$\eqalign{d_2 = &\left[ {\displaystyle{1 \over 2}(d_4 + \sqrt {(d_4^2} -d_5)} \right]^{1/3} + \left[ {\displaystyle{1 \over 2}(d_4-\sqrt {(d_4^2} -d_5)} \right]^{1/3} + \displaystyle{1 \over 6}c_2}$$
$$\eqalign{&d_4 = \displaystyle{1 \over 6}c_2(c_0 + c_1c_3) + \displaystyle{1 \over {108}}c_2^3 -\displaystyle{1 \over 2}(c_0c_2-c_0c_3^2 -c_1^2 ) \cr &d_5 = \displaystyle{4 \over {729}}\left[ {3(c_0 + c_1c_3) + \displaystyle{1 \over 4}c_2^2} \right]^2}$$From Eq. (12), the phase speed of IAW in quantum plasma in the presence of ion beam can be studied analytically and numerically, and the effects of plasma parameters on the phase speed would be understood.
Now, to derive the K–dV equation, we use the stretching coordinates given by Eq. (9). Taking the terms of ε5/2, we obtain from Eqs (2)–(7),
$$\eqalign{0 = &\displaystyle{{\partial {\rm \varphi} _2} \over {\partial {\rm \xi}}} -n_{\rm e0}\displaystyle{{\partial n_{\rm e2}} \over {\partial {\rm \xi}}} -n_{\rm e1}\displaystyle{{\partial n_{\rm e1}} \over {\partial {\rm \xi}}} + \displaystyle{{H^2} \over 4}\displaystyle{{\partial ^3n_{\rm e}^{(1)}} \over {\partial {\rm \xi} ^3}}} $$
$$\eqalign{&\displaystyle{{\partial n_{\rm i1}} \over {\partial {\rm \tau}}} -V_{\rm p}\displaystyle{{\partial n_{\rm i2}} \over {\partial {\rm \xi}}} + n_{\rm i0}\displaystyle{{\partial u_{\rm i2}} \over {\partial {\rm \xi}}} + \displaystyle{\partial \over {\partial {\rm \xi}}} (n_{\rm i1}u_{\rm i1}) = 0} $$
$$\eqalign{&\displaystyle{{\partial u_{\rm i1}} \over {\partial {\rm \tau}}} -V_{\rm p}\displaystyle{{\partial u_{\rm i2}} \over {\partial {\rm \xi}}} + u_{\rm i1}\displaystyle{{\partial u_{\rm i1}} \over {\partial {\rm \xi}}} = -\displaystyle{{\partial {\rm \varphi} _2} \over {\partial {\rm \xi}}} -\sigma _{\rm i}n_{\rm i0}\displaystyle{{\partial n_{\rm i2}} \over {\partial {\rm \xi}}} -{\rm \sigma} _{\rm i}n_{\rm i1}\displaystyle{{\partial n_{\rm i1}} \over {\partial {\rm \xi}}}} $$
$$\eqalign{&\displaystyle{{\partial n_{\rm b1}} \over {\partial {\rm \tau}}} -(V_{\rm p}-u_{\rm b0})\displaystyle{{\partial n_{\rm i2}} \over {\partial {\rm \xi}}} + n_{\rm b0}\displaystyle{{\partial u_{\rm i2}} \over {\partial {\rm \xi}}} + \displaystyle{\partial \over {\partial {\rm \xi}}} (n_{\rm b1}u_{\rm b1}) = 0} $$
$$\eqalign{&\displaystyle{{\partial u_{\rm b1}} \over {\partial {\rm \tau}}} -(V_{\rm p}-u_{\rm b0})\displaystyle{{\partial u_{\rm b2}} \over {\partial {\rm \xi}}} + u_{\rm b1}\displaystyle{{\partial u_{\rm b1}} \over {\partial {\rm \xi}}} \cr &\quad \quad = -{\rm \mu} \displaystyle{{\partial {\rm \varphi}_2} \over {\partial {\rm \xi}}} -{\rm \sigma} _{\rm b}{\rm \mu} n_{\rm b0}\displaystyle{{\partial n_{\rm b2}} \over {\partial {\rm \xi}}} -{\rm \sigma} _{\rm b}{\rm \mu} n_{\rm b1}\displaystyle{{\partial n_{\rm b1}} \over {\partial {\rm \xi}}}} $$
$$\displaystyle{{\partial ^2{\rm \varphi} _1} \over {\partial {\rm \xi} ^2}} = {\rm \chi} n_{\rm e2}-n_{\rm i2}-(1-{\rm \chi} )n_{\rm b2}$$Differentiating Eq. (18) with respect to ξ once more, we get
$$\eqalign{\displaystyle{{\partial ^3{\rm \varphi} _1} \over {\partial {\rm \xi} ^3}} = &{\rm \chi} \displaystyle{{\partial n_{\rm e2}} \over {\partial {\rm \xi}}} -\displaystyle{{\partial n_{\rm i2}} \over {\partial {\rm \xi}}} -(1-{\rm \chi} )\displaystyle{{\partial n_{\rm b2}} \over {\partial {\rm \xi}}}} $$Putting the first-order values of n e1, n i1, n b1, u i1, u b1 in terms of ϕ1 in Eqs (13)–(18) and then eliminating n e2, n i2, n b2, u e2, u i2, u b2, ϕ2, the following K–dV equation in quantum plasma having ion beams has been obtained from (19):
$$\displaystyle{{\partial {\rm \phi} _1} \over {\partial {\rm \tau}}} + A{\rm \phi} _1\displaystyle{{\partial {\rm \phi} _1} \over {\partial {\rm \xi}}} + B\displaystyle{{\partial ^3{\rm \phi} _1} \over {\partial {\rm \xi} ^3}} = 0$$where
$$ \eqalign{ A = & \displaystyle{{{\rm \chi} {\rm \mu} (\sigma _{\rm i}-V_{\rm p}^2 ){(V_{\rm p}^2 -{\rm \sigma} _{\rm i})}^2{(V_{\rm b}^2 -{\rm \mu} {\rm \sigma} _{\rm b})}^2} \over {(V_{\rm p}^2 -{\rm \sigma} _{\rm i})(V_{\rm b}^2 -{\rm \sigma} _{\rm b})[2{\rm \chi \mu} V_{\rm p}({\rm \sigma} _{\rm i}-V_{\rm p}^2 -(1/{\rm \chi} ))(V_{\rm p}^2 -{\rm \sigma} _{\rm i})-2{\rm \mu} V_{\rm p}(V_{\rm b}^2 -{\rm \mu \sigma} _{\rm b})]}} \cr& -{{{\rm \mu} (\sigma _{\rm i} + 3V_{\rm p}^2 )(V{}_{\rm b}^2 -{\rm \mu} ^2\sigma _{\rm b}) + {\rm \chi} {\rm \mu} ^2({\rm \mu} {\rm \sigma} _{\rm b} + 3V_{\rm b}^2 )({\rm \sigma} _{\rm i}-V_{\rm p}^2 -\displaystyle{(1 / {\rm \chi)} ){(V_{\rm p}^2 -{\rm \sigma} _{\rm i})}^2} \over {(V_{\rm p}^2 -{\rm \sigma} _{\rm i})(V_{\rm b}^2 -{\rm \mu} {\rm \sigma} _{\rm b})[2{\rm \chi} {\rm \mu} V_{\rm p}({\rm \sigma} _{\rm i}-V_{\rm p}^2 -(1/{\rm \chi} ))(V_{\rm p}^2 -{\rm \sigma} _{\rm i})-2{\rm \mu} V_{\rm p}(V_{\rm b}^2 -{\rm \mu} {\rm \sigma} _{\rm b})]}}$$
$$B = \displaystyle{{(H^2-4){(V_{\rm p}^2 -{\rm \sigma} _{\rm i})}^2(V_{\rm b}^2 -{\rm \mu} {\rm \sigma} _{\rm b})} \over {4[2{\rm \chi} V_{\rm p}({\rm \sigma} _{\rm i}-V_{\rm p}^2 -(1/{\rm \chi} ))(V_{\rm p}^2 -{\rm \sigma} _{\rm i})-2V_{\rm p}(V_{\rm b}^2 -{\rm \mu \sigma} _{\rm b})}}$$where V b = V p − u b0 and the parameters μ, σi, σb, χ, u b0, H, and V p are defined earlier.
It is known that the solitary wave structure is formed due to the balance between the dispersive effect and the nonlinear effect. The relative strength of these two effects determines the characteristic of such solitary wave structure. The coefficients A of the nonlinear term and coefficient B of the dispersion term thus play a crucial role in determining the solitary wave structure.
To find the steady-state solution of the K–dV equation [Eq. (20)], the independent variables ξ and τ are transformed into one variable η = ξ−U τ where U is the normalized constant speed of the wave frame. Applying the boundary conditions: as η → ±∞; (i) φ1 → 0, (ii) (dφ1/dη) → 0, (iii) (d 2φ1/dη2) → 0, the solution is obtained as
$${\rm \varphi} _1 = {\rm \varphi} _0\sec h^2\left( {\displaystyle{\rm {\eta} \over \Delta}} \right)$$where the amplitude φ0 and width Δ of the solitary structure are given by:
$${\rm \varphi} _0 = \displaystyle{{3U} \over A},\quad {\rm \Delta} = \sqrt {\displaystyle{{4B} \over U}} $$For the positive value of A, the solitary wave will be compressive and the negative value of A will generate the rarefactive solitary wave. Since A is independent of H, the amplitude of the solitary wave will remain the same for any value of H. On the other hand, the width of the solitary wave depends on H and other plasma parameters σi, σb, n b0, u b0, μ.
Results and discussions
We have numerically analyzed the behavior of ion acoustic solitary waves in quantum plasma in the presence of an ion beam from the K–dV equation [Eq. (20)] and its solution is given by Eq. (21). It is seen that the structure of solitary waves depends on the nonlinear coefficient (A) and the width depends on dispersion coefficient (B). The characteristics of solitary waves depend on the various plasma parameters, namely, velocity (u b0), concentration (n b0), and temperature (σb) of the ion beam. For the numerical estimation we have considered a model quantum plasma and have used the value of quantum diffraction diffraction (H) from Hass et al. (Reference Haas, Garcia and Goedert2003) and Chandra et al. (Reference Chandra, Paul and Ghosh2013); and for ion beam velocity from Kaur et al. (Reference Kaur, Singh and Saini2017). The results obtained from numerical estimation for the variations of A and B, the structure and width of solitary waves are given below.
Variation of the coefficients of the nonlinear term and the dispersive term
It is seen from the K–dV equation [Eq. (20)] that the coefficient A of the nonlinear term depends on the plasma parameters, that is μ, σi, σb, χ, u b0 but diffraction parameter H has no effect on A; whereas the coefficient B of dispersive term depends on the plasma parameters including quantum diffraction parameter H. Thus, the nonlinear and dispersion coefficients get modified by the ion beam, quantum diffraction effect, and the quantum statistical effect through electron–ion Fermi temperatures. The nonlinear coefficient A has been numerically estimated for a model quantum plasma for plasma parameters related to ion beam and is graphically shown in Figure 1a–1c.
(a) Variation of Nonlinear coefficient with ion beam temperature (σb) for different value of ion beam density (n b0) in quantum plasma with fixed values of plasma parameters σi = 0.01, u b0 = 0.5, V p =1.6, μ = 1 (H+ ion beam). (b) Variation of nonlinear coefficient with ion beam velocity (u b0) for different value of ion beam density (n b0) in quantum plasma with fixed values of plasma parameters V p = 1.6, σi = 0.1, σb = 0.1, μ = 1 (H+ ion beam). (c) Variation of nonlinear coefficient with ion beam velocity (u b0) for different value of ion beam temperature (σb) in quantum plasma with fixed values of plasma parameters σi = 0.01, n b0 = 0.5, V p = 1.6, μ = 1 (H+ ion beam).
Figure 1a shows the variation in the nonlinear coefficient (A) with ion beam temperature (σb) for different values of ion beam density (n b0) in a quantum plasma having fixed values of other parameters as σi = 0.01, u b0 = 0.5, H = 0.1, V p = 1.6, μ = 1 (H+ ion beam).
It is seen that the nonlinear coefficient (A) becomes negative or positive depending on the values of ion beam temperature. The negative values of A indicate that the solitary waves are rarefactive and, for positive values of A, the solitary waves will be compressive. The nonlinear coefficient A decreases up to the certain value of ion beam temperature (σb ≈ 1.2) and then it is suddenly increased to a maximum positive value and finally it decreases slowly with the increase of ion beam temperature keeping a positive value.
Similarly, the variation of A with ion beam velocity (u b0) for different values of velocity ion beam density (n b0) is shown in Figure 1b in a quantum plasma with fixed values of other parameters as H = 0.1. V p = 1.6, σi = 0.1, σb = 0.1, μ = 1.
It is seen from Figure 1b that the nonlinear coefficient (A) becomes negative or positive depending on the values of ion beam velocity. Therefore, the solitary waves will be rarefactive (compressive) when A < 0 (A > 0). It is interesting to see that the nonlinear coefficient A decreases up to the certain value of ion beam velocity (u b0 ≈ 1.3) and then it is suddenly increased to a maximum positive value and finally it starts to decrease with the increase of the ion beam velocity keeping a positive value. Moreover, the nonlinear term is seen to increase with the increase of the ion beam density (n b0).
The variation in the nonlinear coefficient (A) with ion beam velocity (u b0) for different ion beam temperatures (σb) can be understood from Figure 1c for a quantum plasma having fixed values of σi = 0.01, n b0 = 0.5, H = 0.1, V p = 1.6, μ = 1.
It is observed that the nonlinear coefficient A becomes positive or negative depending upon the values of ion beam velocity and ion beam temperature. The positive values of A indicate that the solitary wave will be compressive and the negative value of A means the rarefactive solitary will be excited.
Similarly, the dispersion coefficient (B) is numerically estimated for a quantum plasma and its variation with the quantum diffraction parameter H and the parameters related to the ion beam are graphically shown in Figure 2a–2c.
(a) Variation of dispersion coefficient with ion beam velocity (u b0) for different value of quantum diffraction parameter (H) in quantum plasma with fixed values of plasma parameters σi = 0.1, σb = 0.1, n b0 = 0.5, V p = 1.6, μ = 1 (H+ ion beam). (b) Variation of the dispersion coefficient with ion beam temperature (σb) and quantum diffraction parameter (H) in quantum plasma with fixed values of plasma parameters σi = 0.01, u b0 = 0.5, n b0 = 0.5, V p = 1.2, μ = 1 (H+ ion beam). (c) Variation of the dispersion coefficient with ion beam density (n b0) and quantum diffraction parameter (H) in quantum plasma with fixed values of plasma parameters having σi = 0.01, σb = 0.1, u b0 = 0.5, V p = 1.2, μ = 1.4 (H+ ion beam).
Figure 2(a) shows the variation of B with ion beam velocity (u b0) and H in a quantum plasma having fixed values of σi = 0.1, σb = 0.1, u b0 = 0.5, V p = 1.6, μ = 1.
It is seen that B is significantly affected by H. The coefficient B is negative and it increases with ion beam velocity for H < 2 and B = 0 for H = 2. But, B is positive and it decreases with ion beam velocity for H > 2.
In Figure 2b, the variation of B with ion beam temperature for different values of H is shown for a quantum plasma having fixed values of σi = 0.01, u b0 = 0.5, n b0 = 0.5, V p = 1.2, μ = 1.
In this case also, B is negative and it increases with ion beam temperature for H < 2 and B = 0 for H = 2. But, B becomes positive and it decreases with ion beam temperature for H > 2.
The variation of B with n b0 and H is shown in Figure 2c in a quantum plasma having fixed values of σi = 0.01, σb = 0.1, n b0 = 0.5, V p = 1.2, μ = 1.
It is seen that B is negative for H < 2 and it increases with the increase of n b0. The coefficient B = 0 when H = 2. For H > 2, the diffraction coefficient B is positive and it decreases with the increase of n b0.
Ion acoustic solitary waves
To understand the nature of ion acoustic solitary waves, the profiles of solitary waves are drawn in the quantum plasma in the presence of an ion beam for different values of velocity, density, and temperature of the beam. In Figure 3a, the structure of solitary waves is shown with a variation of u b0 in the quantum plasma having parameters (dimensionless) σi = σb = 0.1, n b0 = 0.5, V p = 1.6, μ = 1.
(a) The profiles of compressive and rarefactive solitary waves in quantum plasma with variation of ion-beam velocity (u b0) with fixed values of plasma parameters σi = σb = 0.1, n b0 = 0.5, V p = 1.6, μ = 1 (H+ ion beam), U = 0.1. (b) The profiles of compressive and rarefactive solitary waves in quantum plasma with variation of ion-beam density (n b0) with fixed values of plasma parameters σi = 0.01, σb = 0.1, u b0 = 0.5, n b0 = 0.5, V p = 1.2, μ = 1 (H+ ion beam), U = 0.1. (c) The profiles of compressive and rarefactive solitary waves in quantum plasma with variation of ion-beam temperature (σb) with fixed values of plasma parameters σi = 0.01, u b0 = 0.5, n b0 = 0.5, V p = 1.2, μ = 1 (H+ ion beam), U = 0.1.
It is observed that both compressive and rarefactive solitary waves may be excited in the quantum plasma depending upon the values of u b0. For u b0 = 0.1 − 1.2, the solitary wave will be rarefactive and the amplitude will decrease with the increase of u b0, but for u b0 = 1.3 and 1.4, the compressive solitary wave will be excited and the amplitude will increase with the increase of u b0.
The effect of ion beam density on solitary waves is shown in Figure 3b in the quantum plasma with fixed values of σi = 0.01, σb = 0.1, u b0 = 0.5, V p = 1.2, μ = 1.
It is seen that the values of n b0 decide to excite the compressive or rarefactive solitary waves. For low values of n b0 (=0.1, 0.2, and 0.3), the solitary waves are compressive and the amplitudes are increased with the increase of n b0, but for large values of n b0 (0.7, 0.8), the solitary wave will be rarefactive in nature and the amplitudes are decreased with the increase of n b0.
In Figure 3c, the structure of solitary waves for different values of ion beam temperature (σb) is shown for a quantum plasma having fixed values of σi = 0.01, n b0 = 0.5, u b0 = 0.5, V p = 1.6, μ = 1.
It is seen that both compressive and rarefactive solitary waves may be excited in the quantum plasma and these are dependent on the values of σb. For σb = 0.0, 0.2, 0.25, the solitary waves are compressive and the amplitude increases with the increase of σb. But when σb = 0.3 and 0.35, the rarefactive solitary waves are excited and the amplitudes are decreased with the increase of σb.
Since the width is very important for the study of solitary waves in quantum plasma, we have drawn the profiles of width for different values of ion beam parameters. From Eq. (22), we see the width of solitary depends on H and other plasma parameters σi, σb, n b0, u b0.
In Figure 4a, the variation in width with σb for different values of H is plotted for a quantum plasma having fixed values of σi = 0.1, σb = 0.1, n b0 = 0.5, V p = 1.2, μ = 1.
(a) Variation of width of solitary wave with ion beam temperature (σb) and quantum diffraction (H) in quantum plasma with fixed values of plasma parameters σi = 0.01, n b0 = 0.5, u b0 = 0.5, V p = 1.2, μ = 1 (H+ ion beam). (b) Variation of width of solitary waves with ion beam velocity (u b0) for different value of quantum diffraction parameter (H) in quantum plasma with fixed values of plasma parameters σi = 0.1, σb = 0.1, n b0 = 0.5, V p = 1.2, μ = 1 (H+ ion beam). (c) Variation of width of solitary waves with ion beam density (n b0) for different value of quantum diffraction parameter (H) in quantum plasma with fixed values of plasma parameters σi = 0.01, σb = 0.1, u b0 = 0.5, V p = 1.2, μ = 1 (H+ ion beam).
It is seen from Figure 4a that the width of solitary waves decreases with the increase of ion beam temperature and it becomes zero at σb ≈ 0.58. Moreover, the widths are decreased with the increase of H when H < 2. But the widths are increased with the increase of H when H > 2.
In Figure 4b, the variation of widths with ion beam velocity (u b0) for different values of H are depicted for a quantum plasma with fixed values of σi = 0.1, σb = 0.1, u b0 = 0.5, V p = 1.2, μ = 1.
It is observed from Figure 4b that the width decreases with the increase of ion beam velocity and it becomes zero at u b0 ≈ 1.3. When u b0 <1.3, the width decreases with the increase of H when H < 2 and it increases with the increase of H when H > 2. But, when u b0 >1.3, the width increases with the increase of H.
The variation in the width of solitary waves with ion beam density (n b0) for different values of quantum diffraction parameter (H) is shown in Figure 4c with fixed values of plasma parameters σi = 0.01, σb = 0.1, u b0 = 0.5, V p = 1.2, μ = 1.
Figure 4c shows that the width of solitary waves decreases with the increase of ion beam density. Moreover, the width decreases with increase of H when H < 2 and it increases with the increase of H when H > 2.
It is important to point out that the solitary wave will not exist when the coefficient of the nonlinear term of the K–dV equation [Eq. (20)] vanishes for some critical values of the plasma parameters μ, σi, σb, χ, ub0, and H. Assuming A ≈ 0, we obtain
$$\eqalign{&{\rm \chi} [({\rm \sigma} _{\rm i}-V_{\rm p}^2 )({\rm \sigma} _{\rm b}^2 -{\rm \mu} V_{\rm b})^2-{\rm \mu} ({\rm \mu \sigma} _{\rm b}-3V_{\rm b}^2 )({\rm \sigma} _{\rm i}-V_{\rm p}^2 )] \cr &\quad = ({\rm \sigma} _{\rm i}-3V_{\rm p}^2 )({\rm \sigma} _{\rm b}^2 -{\rm \mu} V_{\rm b}^2 )-{\rm \mu} ({\rm \mu \sigma} _{\rm b}-V_{\rm b}^2 )} $$From Eq. (23), the critical values of the plasma parameters can be obtained for the nonexistence of the solitary waves in the quantum plasma. The critical value of ion beam density (N bc) is
$$\eqalign{N_{\rm bc} = 1-\displaystyle{{[({\rm \sigma} _{\rm i}-3V_{\rm p}^2 )({\rm \sigma} _{\rm b}^2 -{\rm \mu} V_{\rm b}^2 )-{\rm \mu} ({\rm \mu \sigma} _{\rm b}-V_{\rm b}^2 )]} \over {[({\rm \sigma} _{\rm i}-V_{\rm p}^2 ){({\rm \sigma} _{\rm b}^2 -{\rm \mu} V_{\rm b})}^2-{\rm \mu} ({\rm \mu \sigma} _{\rm b}-3V_{\rm b}^2 )({\rm \sigma} _{\rm i}-V_{\rm p}^2 )]}}}$$ Again, for some critical values H, V p, and V b, the solitary wave will be non-dispersive if the coefficient B ≈ 0 in K–dV equation [Eq. (20)]. In this case, H = 2, or 
$V_{\rm p} = \sqrt {{\rm \sigma} _{\rm i}}$, or 
$V_{\rm b} = \sqrt {{\rm \mu \sigma} _{\rm b}}$.
The variation of critical ion beam density (N bc) with ion beam velocity and ion beam temperature is shown in Figure 5.
Variation of critical value of ion beam density with ion velocity (n b0) for different ion beam temperature (σb) in quantum plasma with fixed values of plasma parameters V p = 1.6, σi = 0.1, μ = 1 (H+ ion beam).
It is observed that N bc does not depend on H and increases slowly with the increase of u b0 and after a certain value of u b0, it is suddenly increased to attain a maximum value. Then, N bc decreases rapidly with the increase of u b0. Moreover, N bc is small for large values of u b0.
But, the solitary waves near the critical density of ion beam in the quantum plasma can be obtained through the derivation of a modified K–dV equation using different stretching coordinates
$${\rm \xi} = (x-V_{\rm p}t)\;{\rm and}\;{\rm \tau} = {\rm \varepsilon} ^3t$$The solution of the modified K–dV equation near critical ion beam density will be represented by Paul et al. (Reference Paul, Mondal and Roychowdhury1996) and Mondal et al. (Reference Mondal, Paul and Roychowdhury1998)
$${\rm \varphi} _{1{\rm c}} = {\rm \varphi} _{0{\rm c}}\sec h({\rm \eta} /{\rm {\Delta}\!\!\ ^{\prime}})$$where φ0c and Δ′ are the amplitude and width of solitary wave near critical ion beam density. Moreover, double-layer (shock-like) solution of ion acoustic wave in the quantum plasma near critical ion beam density may be obtained from the nonlinear equation combining the K–dV equation and the modified K–dV equation (Paul et al., Reference Paul, Mondal and Roychowdhury1996; Mondal et al., Reference Mondal, Paul and Roychowdhury1998; Kaur et al., Reference Kaur, Singh and Saini2017),
$${\rm \varphi} _{\rm d} = \displaystyle{{{\rm \varphi} _{0{\rm d}}} \over 2}[1-{\rm \tanh} ({\rm \eta} /{\Delta} ^{\prime \prime})]$$where φ0d and Δ′′ are the amplitude and the width of DLs.
Summary and conclusions
In this paper, we have used the one-dimensional QHD model to investigate ion acoustic solitary waves in a dense quantum plasma in the presence of ion beam. For the study of the nonlinear behavior of ion acoustic waves, the K–dV equation has been derived by using the reductive perturbation technique. It is seen that the formation and structure of solitary waves are significantly affected by the presence of ion beam in quantum plasma. Ion acoustic solitary waves in the quantum plasma may be compressive or rarefactive depending upon the values of ion beam parameters, that is velocity, density, and temperature of the ion beam. Our results may be useful for understanding the beam–plasma interactions and the formation of nonlinear wave structures in dense quantum plasma. The model includes a quantum statistical effect through the equation of state and quantum diffraction effect through the parameter H. It is known that quantum effects in plasma become important when thermal de Broglie wavelength becomes much larger than the average interparticle distance. In most practical situations, quantum effects for the ions may be neglected because of their heavier mass than the electrons.
To the best of our knowledge, no work on the ion acoustic solitary waves in degenerate plasma in the presence of ion beam has been reported till now. In our analysis, the explicit form of solitary waves in the quantum plasma is obtained from the K–dV equation and the structure of solitary waves is graphically discussed for different values of density, velocity, and temperature of the ion beam.
Our main findings are:
i) Both compressive and rarefactive solitary waves may be excited in the quantum plasma depending upon the values of ion beam parameters but the quantum diffraction has no role in it.
ii) Compressive solitary wave will be excited for ion beam velocity u b0 = 1.3 and 1.4, and amplitude will increase with the increase of u b0; but the solitary wave will be rarefactive for ion beam velocity u b0 = 0.1–1.2 and the amplitude will decrease with the increase of u b0.
iii) For low values of ion beam density (n b0 = 0.1, 0.2, and 0.3), the solitary waves will be compressive and the amplitudes are increased with the increase of beam density; but for large values of n b0 (0.7, 0.8), the solitary wave will be rarefactive and amplitude decreases with the increase of beam density.
iv) Compressive solitary waves will be excited for ion beam temperature σb = 0.0, 0.2, 0.25 and the amplitude increases with the increase of σb; but for σb = 0.3 and 0.35, the solitary waves will be rarefactive in nature and amplitude decreases with the increase of σb.
v) For the numerical estimation of A, B, and the electrostatic potential of solitary waves, we have considered H+ ion beam and the value of μ = 1(μ = m i/m b). For He+ beam (μ = 1/4) and Ar+ beam (μ = 1/40), numerical estimation can be made and in these cases, the structure of solitary will be different. From numerical estimation and graphical representation of solitary waves, it is seen that the solitary waves will be always rarefactive.
vi) The quantum diffraction parameter H plays important roles in the width of solitary waves in the quantum plasma. For H < 2 and H > 2, the width may be increased or decreased depending upon the values of ion beam parameters.
However, with reference to laboratory and space plasma, interesting results would be obtained on the propagation of electrostatic waves in a bounded system like plasma-filled waveguides. The boundaries in such a system may add additional effects. The wave with no dispersion in an unbounded system may become dispersive in the bounded system; and the stable wave of an unbounded system may become unstable in bounded geometry (Ghosh et al., Reference Ghosh, Paul and Banerjee2014; Paul et al., Reference Paul, Bhattacharya, Paul and Chatterjee2016). Moreover, relativistic effect in plasma is important in many practical situations, for example, in space–plasma phenomena (Vette, Reference Vette1970), the plasma sheet boundary of Earth's magnetosphere (Grabbe, Reference Grabbe1989), Van Allen radiation belts (Shprits et al., Reference Shprits, Subbotin, Drozdov and Usanova2013), and laser–plasma interaction experiments (Marklund and Shukla, Reference Marklund and Shukla2006). Following the works of Das and Paul (Reference Das and Paul1985) in nondegenerate plasma, ion acoustic solitary waves in degenerate (quantum) plasma have been studied by Sahu (Reference Sahu2011). Assuming the relativistically degenerate plasma, the nonlinear propagation of electrostatic solitary has been studied by Mamun and Shukla (Reference Mamun and Shukla2010) and others (Akbari-Moghanjoughi, Reference Akbari-Moghanjoughi2011; Masood and Eliasson, Reference Masood and Eliasson2011). Later, Chandra et al. (Reference Chandra, Paul and Ghosh2013) have considered two-temperature electrons in a relativistically degenerate plasma and have theoretically investigated the electro-acoustic solitary waves. It has been shown that the relativistic degeneracy parameter significantly influences the conditions of the formation and properties of solitary waves. Recently, Hossen and Mamun (Reference Hossen and Mamun2014) have studied theoretically and numerically the nonlinear propagation of modified EAWs in an unmagnetized, collisionless, relativistic degenerate quantum plasma through the derivation of the nonplanar K–dV equation which admits a wave solution for the solitary wave profile. More recently, Paul et al. (Reference Paul, Roychowdury and Paul2019) have considered an ultra-relativistic degenerate dense electron–ion–positron plasma for the study of envelope solitary waves and rogue waves and have obtained significant results which can be further studied in the presence of an electron beam or ion beam following the concept of the present paper.
Some works on the nonlinear propagation of ion and electron acoustic waves in a fully relativistic quantum plasma with ion and electron beams are in progress and these will be reported later.
Acknowledgment
The authors are thankful to the referee for his valuable comments which helped to modify the paper in the present form.
