We are interested in studying ion acoustic quasi-soliton in a fully ionized, collisionless unmagnetized plasma, whose components are cold ions, superthermal electrons and superthermal positrons. Adopting a one dimensional fluid formulation, the dynamics of the cold inertial ion component, ignoring the thermal pressure effect, is governed by the following dimensionless set of fluid equations

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{n}_{i}}{\mathrm{\partial}t}+\frac{\mathrm{\partial}({n}_{i}{u}_{i})}{\mathrm{\partial}x}=0,}\end{array}$$(1)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{u}_{i}}{\mathrm{\partial}t}+{u}_{i}\frac{\mathrm{\partial}{u}_{i}}{\mathrm{\partial}x}=-\frac{\mathrm{\partial}\varphi}{\mathrm{\partial}x},}\end{array}$$(2)

$$\begin{array}{}{\displaystyle \frac{{\mathrm{\partial}}^{2}\varphi}{\mathrm{\partial}{x}^{2}}={n}_{e}-p{n}_{p}-(1-p){n}_{i},}\end{array}$$(3)

where *u*_{i} is the dimensionless ion hydrodynamic velocity normalized by the ion acoustic speed *C*_{s} = (*Te/m*_{i})^{(1/2)} with the electron temperature *T*_{e} and the rest ion mass *m*_{i}. *n*_{i}, *n*_{e}, and *n*_{p} are ion, positron and electron number densities normalized by their equilibrium values *n*_{i0}, *n*_{e0}, and *n*_{p0}, respectively. Under the slow ion acoustic wave time scale, annihilation between electrons and positrons is negligible. Thus, we assume that the number density of electrons and positrons satisfies the following distribution functions

$$\begin{array}{}{\displaystyle {n}_{e}={\left(1-\frac{\varphi}{{\kappa}_{e}-\frac{3}{2}}\right)}^{-{\kappa}_{e}+\frac{1}{2}},{n}_{p}=(1+\frac{\sigma \varphi}{{\kappa}_{p}-\frac{3}{2}}{)}^{-{\kappa}_{p}+\frac{1}{2}},}\end{array}$$(4)

with *σ* = *T*_{e}/*T*_{p} being the electron to positron temperature ratio, *ϕ* is the ion acoustic wave potential normalized by *T*_{e}/e, and *p* = *n*_{p0}/*n*_{e0} is the fractional concentration of positrons with respect to electrons in the equilibrium state. It is noted that time variable *t* is normalized ion plasma period
$\begin{array}{}{\displaystyle {\omega}_{pi}^{-1}=\sqrt{{m}_{i}/(4\pi {n}_{e0}{e}^{2})}}\end{array}$
, and space variable *x* is normalized by the Debye length
$\begin{array}{}{\displaystyle {\lambda}_{De}=\sqrt{{T}_{e}/(4\pi {n}_{e0}{e}^{2})}}\end{array}$.

Under the weak perturbation assumption with *eϕ*/ *T*_{e} ≪ 1, we obtain the following approximation of the Poisson’s equation by substituting the distribution functions of electrons and positrons (4) into equation (3)

$$\begin{array}{}{\displaystyle \frac{{\mathrm{\partial}}^{2}\varphi}{\mathrm{\partial}{x}^{2}}\simeq (1-p)(1-{n}_{i})+{a}_{\kappa}\varphi +\frac{1}{2}{b}_{\kappa}{\varphi}^{2},}\end{array}$$(5)

where

$$\begin{array}{}{\displaystyle {a}_{\kappa}=\frac{2{\kappa}_{e}-1}{2{\kappa}_{e}-3}+\frac{(2{\kappa}_{p}-1)p\sigma}{2{\kappa}_{p}-3},}\end{array}$$(6)

$$\begin{array}{}{\displaystyle {b}_{\kappa}=\frac{4{\kappa}_{e}^{2}-1}{(2{\kappa}_{e}-3{)}^{2}}-\frac{(4{\kappa}_{p}^{2}-1)p{\sigma}^{2}}{(2{\kappa}_{p}-3{)}^{2}}.}\end{array}$$(7)

In order to investigate the nonlinear excitations of small-amplitude ion acoustic waves, we employ the reductive perturbation technique. A set of stretched variables are introduced as

$$\begin{array}{}{\displaystyle \xi ={\u03f5}^{1/2}(x-Vt),\phantom{\rule{1em}{0ex}}\tau ={\u03f5}^{3/2}t,}\end{array}$$(8)

with *ϵ* is a small parameter lies in the range 0 < *ϵ* < 1, and *V* is the normalized phase speed to be determined later. The perturbed quantities are expanded about their equilibrium states as

$$\begin{array}{}{\displaystyle {n}_{i}=1+\u03f5{n}_{1}+{\u03f5}^{2}{n}_{2}+\cdots ,}\end{array}$$(9)

$$\begin{array}{}{\displaystyle \varphi =\u03f5{\varphi}_{1}+{\u03f5}^{2}{\varphi}_{2}+\cdots ,}\end{array}$$(10)

$$\begin{array}{}{\displaystyle {u}_{i}=\u03f5{u}_{1}+{\u03f5}^{2}{u}_{2}+\cdots .}\end{array}$$(11)

Substituting equations (8)-(11) into equations (1)-(3) and setting the coefficients of different powers of *ϵ* to zero, one finds a sequence of differential equations. From the order of *ϵ*^{3/2}, we obtain the compatibility conditions about *n*_{1}, *u*_{1} and *ϕ*_{1}, which can be solved as

$$\begin{array}{}{\displaystyle {n}_{1}=\frac{{a}_{\kappa}}{1-p}{\varphi}_{1},\text{\hspace{0.17em}}{u}_{1}=\frac{1}{V}{\varphi}_{1},\text{\hspace{0.17em}}V=\sqrt{\frac{1-p}{{a}_{\kappa}}}.}\end{array}$$(12)

The second order terms in *ϵ* yield a further set of compatibility conditons. Together with the known results (12), we have

$$\begin{array}{}{\displaystyle {n}_{2}=\frac{{b}_{\kappa}}{2(1-p)}{\varphi}_{1}^{2}-\frac{1}{1-p}\frac{{\mathrm{\partial}}^{2}{\varphi}_{1}}{\mathrm{\partial}{\xi}^{2}}+\frac{{a}_{\kappa}}{1-p}{\varphi}_{2},}\end{array}$$(13)

and

$$\begin{array}{}{\displaystyle {u}_{2}=\frac{1}{4}\left[\frac{{b}_{\kappa}}{{a}_{\kappa}V}-\frac{{a}_{\kappa}}{(1-p)V}\right]{\varphi}_{1}^{2}-\frac{1}{2{a}_{\kappa}V}\frac{{\mathrm{\partial}}^{2}{\varphi}_{1}}{\mathrm{\partial}{\xi}^{2}}+\frac{1}{V}{\varphi}_{2}.}\end{array}$$(14)

Eliminating the second order quantities, we obtain the KdV equation

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\varphi}_{1}}{\mathrm{\partial}\tau}+A{\varphi}_{1}\frac{\mathrm{\partial}{\varphi}_{1}}{\mathrm{\partial}\xi}+B\frac{{\mathrm{\partial}}^{3}{\varphi}_{1}}{\mathrm{\partial}{\xi}^{3}}=0,}\end{array}$$(15)

where the nonlinear and dispersive coefficients *A* and *B* are given by

$$\begin{array}{}{\displaystyle A=\frac{3}{2V}-\frac{{b}_{\kappa}{V}^{3}}{2(1-p)},\text{\hspace{0.17em}}\phantom{\rule{1em}{0ex}}B=\frac{{V}^{3}}{2(1-p)}.}\end{array}$$(16)

The soliton-cnoidal wave solution of the KdV equation as well as its quasi-soliton behaviour has been studied in detail [26], which reads

$$\begin{array}{}{\displaystyle {\varphi}_{1}}& =& \frac{3({V}_{1}-{V}_{2})}{2A{G}^{2}}[\frac{({m}^{2}-1+2G{)}^{2}}{({m}^{2}-1)}\mathrm{tanh}(w{)}^{2}\\ & & -2\delta mS({m}^{2}-1+2G)\mathrm{tanh}(w)+{m}^{2}-1]\\ & & -\frac{({m}^{2}+7){V}_{1}-(3{m}^{2}+5){V}_{2}}{2A({m}^{2}-1)},\end{array}$$(17)

with

$$\begin{array}{}{\displaystyle w=\frac{\xi -{V}_{1}\tau}{{W}_{1}}+{c}_{1}\text{arctanh}({c}_{2}S(\eta ,m)),}\\ G=1-{m}^{2}{S}^{2}+\delta mCD,\phantom{\rule{1em}{0ex}}\eta =\frac{\xi -{V}_{2}\tau}{{W}_{2}}\\ S\equiv sn(\eta ,m),\phantom{\rule{1em}{0ex}}C\equiv cn(\eta ,m),\phantom{\rule{1em}{0ex}}D\equiv dn(\eta ,m).\end{array}$$

and the wave parameters are determined as

$$\begin{array}{}{\displaystyle {W}_{1}=\sqrt{\frac{8B(1-{m}^{2})}{{V}_{1}-{V}_{2}}},\phantom{\rule{1em}{0ex}}{W}_{2}=\sqrt{\frac{2B(1-{m}^{2})}{{V}_{1}-{V}_{2}}},}\\ {c}_{1}=\frac{\delta}{2},\phantom{\rule{1em}{0ex}}{c}_{2}=m,\phantom{\rule{1em}{0ex}}{\delta}^{2}=1.\end{array}$$(18)

Obviously, under the ultralimit conditon *m* = 0 (*G* = 1), *V*_{1} = *V* and *V*_{2} = −*V*, the soliton-cnoidal wave solution (17) degenerates to the classical soliton solution of the KdV equation

$$\begin{array}{}{\displaystyle u=\frac{3V}{A}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^{2}\left(\frac{x-Vt}{W}\right),\phantom{\rule{1em}{0ex}}W=\sqrt{\frac{4B}{V}}.}\end{array}$$(19)

Consequently, the solution (17) has an interesting quasi-soliton behaviour under the asymptotic condition *V*_{1} = *V*, *V*_{2} = −*V*, and *m* → 0. It is found that the soliton core profile tends to the classical soliton of the KdV equation while the surrounded conidal wave becomes small amplitude sinusoidal vibrations around zero. Thus, the solution (17) can be viewed as a quasi-soliton solution for its quasi-soliton behaviour.

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