The nonlinear dynamics of the DIA SWs propagating in such a multi-component dusty plasma is governed by

$$\frac{\partial {n}_{ni}}{\partial t}+\nabla \cdot \mathrm{(}{n}_{ni}{u}_{ni}\mathrm{)}=\mathrm{0,}$$(2)

$$\frac{\partial {n}_{pi}}{\partial t}+\nabla \cdot \mathrm{(}{n}_{pi}{u}_{pi}\mathrm{)}=\mathrm{0,}$$(3)

$$\frac{\partial {u}_{ni}}{\partial t}+\mathrm{(}{u}_{ni}\cdot \nabla \mathrm{)}{u}_{ni}=\nabla \psi \mathrm{,}$$(4)

$$\frac{\partial {u}_{pi}}{\partial t}+\mathrm{(}{u}_{pi}\cdot \nabla \mathrm{)}{u}_{pi}=-\beta \nabla \psi \mathrm{,}$$(5)

$${\nabla}^{2}\psi ={\mu}_{0}{n}_{e}+{n}_{ni}-{\mu}_{pi}{n}_{pi}-j{\mu}_{d}\mathrm{,}$$(6)

where *n*_{ni}(*n*_{pi}) is the negative (positive) ion number density normalised by its equilibrium value *n*_{ni0}(*n*_{pi0}), *u*_{ni}(*u*_{pi}) is the negative (positive) ion fluid speed normalised by *C*_{ni}=(*k*_{B}*T*_{e}/*m*_{ni})^{1/2}, with *k*_{B} as the Boltzman constant, *T*_{e} is the temperature of electrons, and *m*_{ni} being the rest mass of negative ions. *ψ* is the DIA wave potential normalised by *k*_{B}*T*_{e}/*e*, with *e* being the magnitude of the charge of an electron. *ω*_{cni} is the ion (negative) cyclotron frequency (*eB*_{0}/*m*_{ni}*c*) normalised by plasma frequency *ω*_{p}=(4*πn*_{n}_{0}*e*^{2}/*m*_{ni})^{1/2} with *c* being the speed of light. *β* is the mass ratio of negative ions to positive ion (*m*_{ni}/*m*_{pi}). The time variable (*t*) is normalised by *ω*_{p}^{−1} and the space variables are normalised by Debye radius *λ*_{D}=(*k*_{B}/*T*_{e}/4*πn*_{ni}*e*^{2})^{1/2}.

At equilibrium,

${n}_{pi}+j{n}_{d0}={n}_{e0}+{n}_{ni0}\mathrm{,}$

where, *jn*_{d}_{0}=*n*_{pd}−*n*_{nd} with *n*_{pd} being the positive dust number density and *n*_{nd} being the number density of negative dust. *j*=1 for *n*_{pd}>*n*_{nd} and *j*=−1 for *n*_{pd}<*n*_{nd}, i.e. the value of *j* is dependent on the net charge of the dust grain. We can also write

$${\mu}_{0}={\mu}_{pi}+j{\mu}_{d}-\mathrm{1,}$$(7)

where, *μ*_{0}=*n*_{e}_{0}/*n*_{ni0}, *μ*_{pi}=*n*_{pi0}/*n*_{ni0}, and *μ*_{d}=*n*_{d}/*n*_{ni0}.

To derive a dynamical equation for the nonlinear propagation of the electrostatic waves in a dusty plasma using the reductive perturbation technique [44] the following stretched coordinates [21] are introduced

$$\xi ={\u03f5}^{\frac{1}{2}}\mathrm{(}x-{V}_{0}t\mathrm{}\mathrm{)}\mathrm{,}$$(8)

$$\tau ={\u03f5}^{\frac{3}{2}}t\mathrm{,}$$(9)

where ϵ is a smallness parameter (0<ϵ<1) measuring the weakness of the dispersion and *V*_{0} is the Mach number (the phase speed normalised by *C*_{ni}). *n*_{ni}, *u*_{ni}, and *ψ* can be expanded about their equilibrium values in a power series of ϵ, viz.,

$${n}_{ni}=1+\u03f5{n}_{ni}^{\mathrm{(}1\mathrm{)}}+{\u03f5}^{2}{n}_{ni}^{\mathrm{(}2\mathrm{)}}+\cdots \mathrm{,}$$(10)

$${n}_{pi}=1+\u03f5{n}_{pi}^{\mathrm{(}1\mathrm{)}}+{\u03f5}^{2}{n}_{pi}^{\mathrm{(}2\mathrm{)}}+\cdots \mathrm{,}$$(11)

$${u}_{ni}=\u03f5{u}_{ni}^{\mathrm{(}1\mathrm{)}}+{\u03f5}^{2}{u}_{ni}^{\mathrm{(}2\mathrm{)}}+\cdots \mathrm{,}$$(12)

$${u}_{pi}=\u03f5{u}_{pi}^{\mathrm{(}1\mathrm{)}}+{\u03f5}^{2}{u}_{pi}^{\mathrm{(}2\mathrm{)}}+\cdots \mathrm{,}$$(13)

$$\psi =\u03f5{\psi}^{\mathrm{(}1\mathrm{)}}+{\u03f5}^{3/2}{\psi}^{\mathrm{(}2\mathrm{)}}+\cdots \mathrm{.}$$(14)

Using the stretched coordinates and power series expansion [as shown in (10)–(14)] in (2)–(6); and equating the coefficients of ϵ^{3/2} from the continuity and momentum equation and coefficients of ϵ from Poisson’s equation one can obtain the linear dispersion relation for the DIA SWs in unmagnetised plasmas as

$${V}_{0}=\sqrt{\frac{1}{{\mu}_{0}}\mathrm{(}\frac{k-\frac{3}{2}}{k-\frac{1}{2}}\mathrm{)}\mathrm{(}\beta {\mu}_{pi}+1\mathrm{)}\mathrm{}}\mathrm{.}$$(15)

It can be seen from (15) that the presence of the super-thermal electrons significantly modifies the linear dispersion relation. Compering with the dispersion relation (16) of Haider et al. [27] it can be said that the presence of super-thermal electrons reduce the phase speed of the DIA waves.

Equating the next higher order co-efficient of ϵ from above equations and using the parameters we can finally obtain a K-dV equation describing the nonlinear propagation of the DIA SWs in the dusty plasma

$$\frac{\partial {\psi}^{\mathrm{(}1\mathrm{)}}}{\partial \tau}+A{\psi}^{\mathrm{(}1\mathrm{)}}\frac{\partial {\psi}^{\mathrm{(}1\mathrm{)}}}{\partial \xi}+B\frac{{\partial}^{3}{\psi}^{\mathrm{(}1\mathrm{)}}}{\partial {\xi}^{3}}=\mathrm{0,}$$(16)

where the nonlinear coefficient *A* and the dissipation coefficient *B* are given by

$$A=\frac{3\mathrm{(}\beta {\mu}_{pi}-1\mathrm{)}\mathrm{}}{2{V}_{0}\mathrm{(}\beta {\mu}_{pi}+1\mathrm{)}}-\frac{{V}_{0}}{2}\mathrm{(}\kappa +\frac{1}{2}\mathrm{)}\mathrm{,}$$(17)

$$B=\frac{1}{2}\frac{{\mu}_{0}}{{V}_{0}}\frac{\mathrm{(}\kappa -\frac{3}{2}\mathrm{)}}{\mathrm{(}\kappa -\frac{1}{2}\mathrm{)}}\mathrm{.}$$(18)

The stationary SW solution of (16) can be found by introducing a new coordinate

$$\zeta =\xi -u\tau \mathrm{,}$$(19)

where, *u* is the wave speed (in the reference frame) normalised by *C*_{ni} and *ζ* is normalised by *λ*_{D}.

By imposing the appropriate boundary conditions, namely *ψ*^{(1)}→0, *dψ*^{(1)}/*dζ*→0, *d*^{2}*ψ*^{(1)}/*dζ*^{2}→0 at *ζ*→±∞. Thus, the steady state solution [21] of the K-dV equation, shown in (16), can be expressed as

$${\psi}^{\mathrm{(}1\mathrm{)}}={\psi}_{m}{\text{sech}}^{2}\mathrm{(}\zeta \mathrm{/}\Delta \mathrm{}\mathrm{)}\mathrm{,}$$(20)

where, amplitude (*ψ*_{m}) and width (Δ) of the SWs respectively are

$${\psi}_{m}=\frac{2{V}_{0}u}{\frac{\beta {\mu}_{pi}-1}{\beta {\mu}_{pi}+1}-\frac{{V}_{0}^{2}}{3}\mathrm{(}\kappa +\frac{1}{2}\mathrm{)}}\mathrm{,}$$(21)

$$\Delta =\sqrt{\frac{2{\mu}_{0}}{{V}_{0}u}\frac{\mathrm{(}\kappa -\frac{3}{2}\mathrm{)}}{\mathrm{(}\kappa -\frac{1}{2}\mathrm{)}}}\mathrm{.}$$(22)

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