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This paper mainly concentrates on obtaining solutions and other exact traveling wave solutions using the generalized G-expansion method. Some new exact solutions of the coupled nonlinear Schrödinger system using the mentioned method are extracted. This method is based on the general properties of the nonlinear model of expansion method with the support of the complete discrimination system for polynomial method and computer algebraic system (AS) such as Maple or Mathematica. The nonparaxial solitons with the propagation of ultra-broad nonparaxial pulses in a birefringent optical waveguide is studied. To attain this, an illustrative case of the coupled nonlinear Helmholtz (CNLH) system is given to illustrate the possibility and unwavering quality of the strategy utilized in this research. These solutions can be significant in the use of understanding the behavior of wave guides when studying Kerr medium, optical computing and optical beams in Kerr like nonlinear media. Physical meanings of solutions are simulated by various Figures in 2D and 3D along with density graphs. The constraint conditions of the existence of solutions are also reported in detail. Finally, the modulation instability analysis of the CNLH equation is presented in detail. [ABSTRACT FROM AUTHOR]

The generalized q-deformed sinh Gordon equation (GDSGE) serves as a significant nonlinear partial differential equation with profound applications in physics. This study investigates the GDSGE's mathematical and physical properties, examining its solutions and clarifying the essence of the q-deformation parameter. The Sardar sub-equation method (SSEM) and sine-Gordon expansion method (SGEM) are employed to solve this GDSGE. The synergistic application of these techniques improves our knowledge of the GDSGE and provides a thorough foundation for investigating different evolution models arising in various branches of mathematics and physics. A positive aspect of the proposed methods is that they offer a wide variety of solitons, including bright, singular, dark, combination dark-singular, combined dark-bright, and periodic singular solitons. Obtained solutions demonstrate the method's high degree of reliability, simplicity, and functionalization for various nonlinear equations. To better describe the physical characterization of solutions, a few 2D and 3D visualizations are generated by taking precise values for parameters using mathematical software, Mathematica. [ABSTRACT FROM AUTHOR]

The present work aims to investigate solitary wave solutions for two recently developed extended equations in the context of (2 + 1) -dimensional and (3 + 1) -dimensional structures. The equations under consideration are of the Korteweg–de Vries (KdV) type, which are well-recognized as significant aspects of fluid dynamics. These equations have broad applications in physics, mathematics, and other scientific disciplines, particularly in the study of waves, soliton theory, plasma physics, biology and chemistry, and nonlinear phenomena. Its soliton solutions and integrability properties make it a fundamental model in various areas of research. This serves as the main motivation for our research work. To analyze these equations, we employ an advanced direct algebraic equation method capable of generating several sorts of solutions, including solitary and shock wave solutions, as well as their combination. In addition to these wave phenomena, singular solitons and solutions expressed in Jacobi and Weierstrass doubly periodic types have also been observed. The utilization of this outstanding technique and the subsequent acquisition of novel solutions demonstrate the originality of our study. This also allows further exploration of nonlinear models that accurately depict significant physical processes in our everyday existence. • Solitary wave solutions for two recently developed extended equations. • The (2+1)-dimensional and (3+1)-dimensional structures. • The equations under consideration are of the Korteweg–de Vries (KdV) type. • The KdV equations are well-recognized as significant aspects of fluid dynamics. [ABSTRACT FROM AUTHOR]