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Recently, mathematicians, engineers, and scientists have explored the unique characteristics and potential applications of multi-solitons, which is an expanding domain of study. There are various approaches to obtaining multi-soliton solutions of an integrable system, such as the Bäcklund transform, the nonlinear transform method, the Hirota direct method, etc. But each approach has some own characteristics and attributes; however, the Hirota direct method is dominant among them and provides further multi-soliton solutions of nonlinear evolution equations. In this article, we use the Hirota direct method to investigate the single-soliton, double-soliton, and triple-soliton solutions, known as multi-soliton solutions, of the integral Sawada-Kotera equation. We also demonstrate and discuss the effects of amplitude on the fluctuation of wave number in different ranges by comparing 2-D and 3-D plots.

The solitary wave solutions of nonlinear evolution equations, in the recent years is being attractive in the field of physical sciences and engineering. In this article, we have investigated further general solitary wave solutions of three important nonlinear evolution equations, via the simplified MCH equation, the Pochhammer-Chree equation and the Schrödinger-Hirota equation by using modified simple equation method. These equations play an important role in the study of nonlinear sciences. The obtained solutions are expressed in terms of exponential and trigonometric functions including kink, singular kink and periodic soliton solutions. It is shown that the obtained solutions are more general and fresh and can be helpful to analyze the intricate physical incident in mathematical physics.