Abstract:
The concept of dual-mode equations focuses on nonlinear models that describe the
movement of waves in two directions at the same time, influenced by how their phase
changes. Initially, Korsunsky introduced a dual-mode model that improved the Ko rteweg De Vries equation (KDVe) by turning it into a second-order formulation. In our
research, our aim is to develop a new and simpler dual-mode model, which is based
on the Simplified Modified Camassa-Holm (SMCH) model and the Gardner equation
derived from an ideal fluid model. We will then systematically analyze its solutions and
explore its geometric characteristics. To find analytical solutions for these bidirectional
waves, we’ll use various methods like the extended exponential function expansion, ra tional sine/cosine method, tanh/coth method, direct approach of tan/cot method and
Kudryashov method. Furthermore, we will conduct a thorough examination of how the
speed at which the phase changes affects the behavior of these dual waves, using both
2D and 3D graphs. The solutions we obtain in this study have significant implications
for our understanding of how solitons propagate in the field of nonlinear optics. These
models can be used in many different fields and the solutions we obtain will help make
it clearer how various non-linear phenomena work. This includes areas like plasma
physics, Bose-Einstein condensates, shallow water waves and more.
