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Exploring novel solitary wave phenomena in Klein-Gordon equation using model expansion method.

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The -model expansion method effectively finds solitary wave solutions for the Klein-Gordon equation, yielding diverse exact solutions and enhancing understanding of nonlinear wave dynamics.

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Area of Science:

  • Theoretical Physics
  • Nonlinear Dynamics
  • Mathematical Physics

Background:

  • The Klein-Gordon (KG) equation is a fundamental model in theoretical physics, crucial for understanding relativistic wave-particle dynamics.
  • Nonlinear wave phenomena and their exact solutions are vital in fields like quantum field theory, cosmology, and nonlinear optics.
  • Existing methods for solving nonlinear partial differential equations (PDEs) have limitations in scope and applicability.

Purpose of the Study:

  • To demonstrate the utility of the -model expansion method for solving the Klein-Gordon equation.
  • To generate a wide range of exact solitary wave solutions, including Jacobi elliptic, hyperbolic, and trigonometric forms.
  • To analyze and visualize various types of solitons (bright, dark, singular, periodic) using 2D, 3D, and contour plots.

Main Methods:

  • Application of the -model expansion technique to the Klein-Gordon equation.
  • Derivation of diverse exact solutions in different functional forms.
  • Computational visualization of solitary wave behaviors through plotting.

Main Results:

  • Successful identification of numerous solitary wave solutions for the KG equation using the -model expansion method.
  • Visualization of bright, dark, singular, and periodic solitons, illustrating complex nonlinear dynamics.
  • The -model expansion method proves to be a powerful and adaptable tool for nonlinear wave analysis.

Conclusions:

  • The -model expansion method significantly expands the repertoire of exact solutions for nonlinear wave equations like the KG equation.
  • This approach enhances the understanding of complex wave behaviors and their physical implications.
  • The method's adaptability suggests broad potential for application to other nonlinear PDEs in physics and applied mathematics.