Nonclassical symmetries and similarity solutions of variable coefficient coupled KdV system using compatibility method

Differential Equations Invariant Under Conditional Symmetries

“…There are several works devoted to using the non-classical method to construct solutions of PDEs that are different from the ones obtained by classical method using the Lie point symmetries. Among them, let us cite as an example, [2,3,10,16,20,21,24,33,36]. In the case of integrable equations let us mention the works of Sergyeyev [38,39] where he considered the classification of all (1 + 1)-dimensional evolution systems that admit a generalized (Lie-Bäcklund) vector field as a generalized conditional symmetry.…”

On multiple soliton similariton‐pair solutions, conservation laws via multiplier and stability analysis for the Whitham–Broer–Kaup equations in weakly dispersive media

“…This case represents more realistic models than those which are considered with constant coefficients . The soliton wave propagation in the above‐mode has been called “non‐autonomous soliton.” Hence, N‐soliton nonautonomous soliton interactions in various play a definitive role in the formation of the structure of wave and also the propagation direction with a phase shift in dispersive media …”

“…Hence, N-soliton nonautonomous soliton interactions in various play a definitive role in the formation of the structure of wave and also the propagation direction with a phase shift in dispersive media. [18][19][20][21][22][23][24] The dispersive long (surface) water waves with time-dependent coefficient are given by the equations…”

New Similarity Solutions of a Generalized Variable-Coefficient Gardner Equation with Forcing Term

“…In 2006, a systematic method (named the compatibility method) was developed in [27] to seek non-classical symmetries and similarity solutions of a class of variable coefficients Zakharov-Kuznetsov equations. After that, the method was extended to investigate the high-dimensional breaking soliton equation [28], the variable coefficients Broer-Kaup system [29], the Wick-type stochastic Korteweg-de Vries equation [30] and the variable coefficients coupled KdVsystem [31], consecutively. As shown in [27][28][29][30][31], the method is capable of obtaining abundant symmetry reductions and similarity solutions of the considered nonlinear evolution equations.…”

“…After that, the method was extended to investigate the high-dimensional breaking soliton equation [28], the variable coefficients Broer-Kaup system [29], the Wick-type stochastic Korteweg-de Vries equation [30] and the variable coefficients coupled KdVsystem [31], consecutively. As shown in [27][28][29][30][31], the method is capable of obtaining abundant symmetry reductions and similarity solutions of the considered nonlinear evolution equations. In addition, it is able to greatly reduce the computational complexity in comparison to the non-classical group methods (see, for instance, [17,18]).…”