In this research, we constructed the exact travelling and solitary wave solutions of the Kudryashov-Sinelshchikov (KS) equation by implementing the modified mathematical method. The KS equation describe the phenomena of pressure waves in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. Our new obtained solutions in the shape of hyperbolic, trigonometric, elliptic functions including dark, bright, singular, combined, kink wave solitons, travelling wave, solitary wave and periodic wave. We showed the physical interpretation of obtained solutions by three-dimensional graphically. These new constructed solutions play vital role in mathematical physics, optical fiber, plasma physics and other various branches of applied sciences.

In this research, we consider the propagation of one-dimensional nonlinear behavior in a unmagnetized plasma. By using the reductive perturbation technique to formulate the nonlinear mathematic model which is modified Kortewege-de Vries (mKdV), we apply the extended form of two methods, which are extended auxiliary equation mapping and extended direct algebraic methods, to investigate the new families of electron-acoustic solitary wave solutions of mKdV. These new exact traveling and solitary wave solutions which represent the electrostatic potential for mKdV and also the graphical representation of electrostatic potential are shown with the aid of Mathematica.

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