Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model

Syed T. R. Rizvi; Sana Ghafoor; Aly R. Seadawy; Ahmed H. Arnous; Hakim AL Garalleh; Nehad Ali Shah; Syed T. R. Rizvi; Sana Ghafoor; Aly R. Seadawy; Ahmed H. Arnous; Hakim AL Garalleh; Nehad Ali Shah

doi:10.3934/math.20241027

AIMS Mathematics

2024, Volume 9, Issue 8: 21144-21176. doi: 10.3934/math.20241027

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Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model

1.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
2.
Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, 41411, Kingdom of Saudi Arabia
3.
Department of Engineering Mathematics and Physics, Higher Institute of Engineering, El-Shorouk Academy-11837, Egypt
4.
Department of Mathematical Science, College of Engineering, University of Business and Technology, Jeddah 21361, Saudi Arabia
5.
Department of Mathematics, Saveetha School of Engineering, SIMATS, Chennai, 602105, Tamilnadu, India

Received: 29 February 2024 Revised: 01 June 2024 Accepted: 05 June 2024 Published: 01 July 2024
MSC : 35A09, 35A24, 35C08

In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation.
- integrability,
- variational integrators,
- nonlinear Klein-Gordon model (NLKGM)
Citation: Syed T. R. Rizvi, Sana Ghafoor, Aly R. Seadawy, Ahmed H. Arnous, Hakim AL Garalleh, Nehad Ali Shah. Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model[J]. AIMS Mathematics, 2024, 9(8): 21144-21176. doi: 10.3934/math.20241027

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Abstract

In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation.

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