Novel waves structures for the nonclassical Sobolev-type equation in unipolar semiconductor with its stability...

Area of Science:

• Mathematical Physics
• Nonlinear Dynamics
• Applied Mathematics

Background:

• Sobolev-type equations model phenomena in ecology, fluid dynamics, soil mechanics, and thermodynamics.
• Understanding solitary wave solutions is crucial for analyzing complex physical systems.
• Accurate soliton solutions aid in constructing specific physical problems with defined boundary and initial conditions.

Purpose of the Study:

• To analytically investigate solitary wave solutions for Sobolev-type equations.
• To explore novel techniques for uncovering diverse solitary wave structures.
• To analyze the linearized stability of the derived solutions.

Main Methods:

• Generalized Riccati equation mapping method.
• Modified Auxiliary Equation (MAE) method.
• Linear stability analysis.
• Visualization using Mathematica for 3D graphs, line graphs, and contours.

Main Results:

• Abundant families of solutions were discovered, including dark solitons, bright solitons, solitary waves, mixed singular solitons, mixed dark-bright solitons, periodic waves, and mixed periodic solutions.
• The linearized stability of the model was investigated.
• Physical behaviors of state variables were illustrated through graphical representations.

Conclusions:

• The study provides a comprehensive analytical framework for Sobolev-type equations, yielding a wide array of solitary wave solutions.
• The findings enhance the understanding of soliton dynamics and their physical implications.
• The presented solutions and visualizations offer valuable insights for researchers in applied mathematics and physics.