The Two Modified G′G-Expansion Methods for Obtaining Solitons of the Korteweg-De Vries Equation

The Korteweg-de Vries (KdV) equation plays an important role in describing the propagation of water waves in shallow channels. Several analytical methods, such as tanh-function methods (TFMs), exponential function method, and Jacobi elliptic function method, provide the exact solitons to the KdV equation. Contrarily, the G′G-expansion method, whose strength depends on the embedded Ricati equation for obtaining solitons of the KdV equation. There are a lot of Riccati equations that can be embedded in the G′G-expansion method; however, these Riccati equations; GG″+λG2=0 and G′−μG2=0, have not been embedded in the G′G-expansion method to obtain the multiple solitons of the KdV equations. To bridge this gap in the literature, two different modified G′G-expansion methods, each embedded with a Riccati equation, are therein this paper. The modified G′G-expansion method endowed with a second-order Riccati equation GG″+λG2=0, produced 15 solitons of the KdV equation comprising 10 singular dark solitons and 5 singular multidark solitons, each with different amplitude, kinetic energy, and potential energy. On the other hand, the new G′G-expansion method embedded with a first-order Riccati equation G′−μG2=0 yielded only four solitons of the KdV equation. These solitons are mainly two singular bright solitons and two singular dark solitons. The fifteen solitons obtained indicate that embedding the second-order Riccati equation in the G′G-expansion method is more effective than using the first-order Riccati equation. Surprisingly, all the solitons of the modified G′G-expansion method embedded with a second-order Riccati equation GG″+λG2=0 do not yield any bright soliton. On a new development, the shape and colour of each soliton of the KdV equation indicate its stability as the solitary wave propagates through a medium. This shape changes depending on the position and time it takes to propagate, as observed in the KdV equation using each of the introduced modified G′G-expansion methods. The results demonstrate the complex dynamical behavior of the KdV equation and emphasize its wide-ranging applications in fluid dynamics, plasma physics, and optical systems.