Extended (G′G)-expansion method for solving the coupled KdV equations with two arbitrary constants and its application to MEMS system

The coupled Korteweg-de Vries (cKdV) equations with two arbitrary constants hold significant importance in the field of micro-electro-mechanical systems (MEMS). These equations describe the behavior of nonlinear waves in MEMS devices. In MEMS applications, the cKdV equations can be used to analyze the dynamics of microstructures such as cantilevers, membranes, and resonators. By solving these equations, researchers can predict the behavior of MEMS devices under different operating conditions. In this paper, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> -expansion method is extended to seek more general travelling solutions of the cKdV equations with two arbitrary constants. The two arbitrary constants offer flexibility in modeling different physical phenomena and boundary conditions. As a result, many new and more general exact travelling wave solutions are obtained including soliton solutions, hyperbolic function solutions, trigonometric function solutions and rational solutions. They help in understanding the complex interactions between mechanical and electrical properties. Additionally, the study of these equations provides insights into the nonlinear behavior of MEMS systems, which is crucial for improving their performance and reliability. Overall, the cKdV equations with two arbitrary constants play a vital role in advancing the design and understanding of MEMS applications.