On symmetries, conservation laws and invariant solutions of the foam-drainage equation

Lie group analysis, exact solutions and conservation laws of (3 1) dimensional a B-type KP equation

“…We consider the multipliers of the form Q (x, y, z,t, u, u x , u y , u z , u t ) for the equation (1). The determining equation (29) for the multipliers takes the form…”

Conserved quantities, optimal system and explicit solutions of a (1 + 1)-dimensional generalised coupled mKdV-type system

“…Nonlinear partial differential equations (NLPDEs) have rapidly become indispensable in the quest to conceptualise the world around us [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] , [33] , [34] , [35] , [36] , [37] , [38] , [39] , [40] , [41] , [42] , [43] , [44] , [45] , [46] , [47] , [48] , [49] , [50] . We give a few recent studies of NLPDEs presented in the literature.…”

A review of some geometric integrators

“…Yet, many of them have important invariance properties, called symmetries [54], under some transformations. Symmetries play a fundamental role since they may encode exact model solutions (self-similar, vortex, shock solutions, …), conservation laws via Nother's theorem or physical principles (Galilean invariance, scale invariance, …) [55][56][57][58][59][60][61][62][63][64][65][66][67][68]. They have also been used for modelling purposes such as the establishment of wall laws in turbulent flows or the development of turbulence models [69][70][71][72][73][74][75][76].…”