This article possesses modulation instability (MI) analysis and new exact wave solutions to unidirectional Dullin–Gottwald–Holm (DGH) system that describes the prorogation of waves in shallow water. The exact wave solutions in single and combined form like shock, singular, and shock‐singular are extracted by means of an innovative integration norm, namely,
()G′false/G2‐expansion scheme. The periodic and plane wave solutions are also emerged. The constraint conditions which ensure the existence of solutions are discussed as well. Moreover, the choice of suitable parameters gives the three‐dimensional and two‐dimensional sketches, and furthermore, their contour plots are also drawn.
In this work, we present a technique for the analytical solution of systems of stiff ordinary differential equations (SODEs) using the power series method (PSM). Three SODEs systems are solved to show that PSM can find analytical solutions of SODEs systems in convergent series form. Additionally, we propose a post-treatment of the power series solutions with the Laplace-Padé (LP) resummation method as a powerful technique to find exact solutions. The proposed method gives a simple procedure based on a few straightforward steps.
This manuscript consist of diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations approach. Furthermore, it includes homoclinic, Ma and Kuznetsov-Ma breather and their relating rogue waves and some interactional solutions, including an interactional approach with the help of the double exponential function. We have elaborated the kink cross-rational (KCR) solutions and periodic cross-rational (KCR) solutions with their graphical slots. We have also constituted some of our solutions in distinct dimensions by means of 3D and contours profiles to anticipate the wave propagation. Parameter domains are delineated in which these exact localized soliton solutions exit in the proposed model.
In this study, under the considerations of symbolic computation with the help of Mathematica software, various types of solitary wave solutions for the (3 + 1)-dimensional Jimo–Miwa (JM) equation are successfully constructed based on the extended modified rational expansion method. The constructed solutions are novel and more general for the JM equation named kink wave solutions, anti-kink wave solutions, bright and dark solutions, mixed solutions in the shape of bright-dark solutions, and periodic waves, which do not exist in the existing literature. The physical phenomena of the demonstrated results is represented graphically by two-dimensional, three-dimensional, and contour images with the help of Mathematica. The obtained results will be widely used to explain the various interesting physical structures in the areas of optics, plasma, gas, acoustics, classical mechanics, fluid dynamics, heat transfer, and many others.
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