Lie point symmetries and similarity solutions of the time-dependent coefficients Calogero–Degasperis equation

Abstract

In this paper, a Painlevé analysis of the (2 + 1)-dimensional variable-coefficient Calogero–Degasperis (VCCD) equation ψxt + α(t)ψxψxy + β(t)ψyψxx + γ(t)ψxxxy = 0 has been performed to study the Painlevé properties. We applied the Lie-group formalism to deduce the symmetries and to reduce the (2 + 1)-dimensional VCCD equation to lower dimensional equations, which are again investigated by different methods such as the extended (G′/G)-expansion method, hyperbolic rational function expansion method and Jacobi elliptic function method to obtain some new exact solutions.

Shock wave solution of the variable coefficient nonlinear partial differential equations

“…The variable coefficient version of the Calogero–Degasperis (vcCD) equation where f ( t ), g ( t ) and m ( t ) are arbitrary functions of t . This equation is an important mathematical model in nonlinear physics and has been studied for integrability and for the exact solutions using a combination of the Lie classical method and several other methods including the extended ‐expansion method.…”

“…This equation is an important mathematical model in nonlinear physics and has been studied for integrability and for the exact solutions using a combination of the Lie classical method and several other methods including the extended ‐expansion method. Bansal and Gupta checked the Painlevé property of the (2+1)‐dimensional vcCD equation and proved that the vcCD equation is integrable under certain restrictions on the coefficients of equation under study.…”

“…The variable coefficient version of the Calogero-Degasperis (vcCD) equation [27] u xt C f .t/u x u xy C g.t/u y u xx C m.t/u xxxy D 0, (2.1)…”

Auto-Bäcklund Transformation and Extended Tanh-Function Methods to Solve the Time-Dependent Coefficients Calogero-Degasperis Equation

“…One of the most important NPDEs is the time-dependent coefficients Calogero-Degasperis (VCCD) equation [1] ( ) ( and ( ) t γ are arbitrary functions. The VCCD equation describes the (2 + 1)-dimensional interaction of the Riemann wave propagating along the y-axis with a long wave along the x-axis.…”

“…The VCCD equation describes the (2 + 1)-dimensional interaction of the Riemann wave propagating along the y-axis with a long wave along the x-axis. Many exact solutions have been found for Equation (1) by using symmetry method [1]. Equation (1) with , α β and γ as constants was first constructed by Bogoyavlenskii and Schiff in different ways [2]- [4] and called the Calogero-Bogoyavlenskii-Schiff (CBS) equation.…”