Bifurcation analysis on ion sound and Langmuir solitary waves solutions to the stochastic models with multiplicative noises

Abstract

This article explores on a stochastic couple models of ion sound as well as Langmuir surges propagation involving multiplicative noises. We concentrate on the analytical stochastic solutions including the travelling and solitary waves by using the planner dynamical systematic approach. To apply the method, First effort is to convert the system of equations into the ordinary differential form and present it in form of a dynamic structure. Next analyze the nature of the critical points of the system and obtain the phase portraits on various conditions of the corresponding parameters. The analytic solutions of the system in an account of distinct energy states for each phase orbit are performed. We also show how the results are highly effective and interesting to realize their exciting physical as well as the geometrical phenomena based on the demonstration of the stochastic system involving ion sound as well as Langmuir surges. Descriptions of effectiveness of the multiplicative noise on the obtained solutions of the model, and its corresponding figures are demonstrated numerically.

Keywords: Bifurcation; Ion sound and Langmuir wave; Multiplicative noise; Stochastic model.

Fig-1

(a) Stream flows with directions and (b) Phase orbits for $r_1<0,r_2>0$.

Fig-2

(a) Stream flows with directions and (b) Phase orbits for $r_1>0,r_2<0$.

Fig-3

(a) Stream flows with directions and (b) Phase orbits for $r_1>0,r_2>0$.

Fig-4

(a) Stream flows with directions and (b) Phase orbits for $r_1<0,r_2<0$.

Fig-5

Periodic wave solution of Eq. (14) for the parameters $s=1,w=2,p=2,h=-1,a=0$: (a) absolute, (b) real part, (c) imaginary part for $\kappa =0$; (d) absolute, (e) real part, (f) imaginary part for $\kappa =0.5$; (g) absolute, (h) real part, (i) imaginary part for $\kappa =1\text{.}$.

Fig-6

Periodic wave solution of Eq. (15) for the parameters $s=1,w=2,p=2,h=-1$: (a) absolute, (b) real part, (c) imaginary part for $\kappa =0$; (d) absolute, (e) real part, (f) imaginary part for $\kappa =0.5$; (g) absolute, (h) real part, (i) imaginary part for $\kappa =1\text{.}$.

Fig-7

Periodic wave solution of Eq. (16) for the parameters $s=1,w=2,p=2,h=-1,a=0$: (a) absolute, (b) real part, (c) imaginary part for $\kappa =0$; (d) absolute, (e) real part, (f) imaginary part for $\kappa =0.5$; (g) absolute, (h) real part, (i) imaginary part for $\kappa =1\text{.}$.

Fig-8

Periodic wave solution of Eq. (17) for the parameters $s=2,w=1,p=1,h=-1$: (a) absolute, (b) real part, (c) imaginary part for $\kappa =0$; (d) absolute, (e) real part, (f) imaginary part for $\kappa =0.5$; (g) absolute, (h) real part, (i) imaginary part for $\kappa =1\text{.}$.

Fig-9

Periodic wave solution of Eq. (18) for the parameters $s=2,w=1,p=1,h=-1$: (a) absolute, (b) real part, (c) imaginary part for $\kappa =0$; (d) absolute, (e) real part, (f) imaginary part for $\kappa =0.5$; (g) absolute, (h) real part, (i) imaginary part for $\kappa =1\text{.}$.