Traveling Waves in Two Distinct Equations: The (1+1)-Dimensional cKdV–mKdV Equation and The sinh-Gordon Equation

In this research work, two mathematical models, the (1+1)-dimensional cKdV–mKdV equation and the sinh-Gordon (shG) equation, are studied using an analytical method to obtain solitary wave solutions. The paper presents explicit parameterized traveling wave solutions for these equations, with hyperbolic function solutions resulting in solitary wave solutions when specific parameter values are used. The Hamiltonian function and phase plane are briefly discussed, and the relationship between the phase picture and its corresponding component solutions are depicticted. The phase orbits of the planar dynamical system are also studied to determine all the traveling wave solutions of the analyzed models. In this article we also examines the impact of parameters on wave velocity and profile. According to the research findings, the method employed is efficient and applicable to different mathematical physics nonlinear evolution equations.