The similarity forms and invariant solutions of two-layer shallow-water equations

Lie symmetry solutions for two-phase mass flows

“…The Lie group theory has been extensively used to obtain solutions of differential equations arising from wide range of physical problems, including the reduction of differential equations, order reduction of ordinary differential equations, construction of invariant solutions, and mapping between the solutions in mechanics, applied mathematics and mathematical physics, and applied and theoretical physics [11,[19][20][21][22][23][24][25][26][27]. Lie algebra and symmetry group theory is an important and effective method for solving a system of non-linear partial differential equations via similarity forms and similarity solutions [19,23,[27][28][29][30][31][32].…”

“…Lie algebra and symmetry group theory is an important and effective method for solving a system of non-linear partial differential equations via similarity forms and similarity solutions [19,23,[27][28][29][30][31][32]. Particularly related to our interest, Lie theory has been successively applied to construct and investigate self-similar solutions to gravity currents, and relatively simple shallow water or two-layer shallow water flows, impulse modified shallow water equations, dispersive shallow-water flows, Benney system, generalized Burgers equation, etc.…”

“…Particularly related to our interest, Lie theory has been successively applied to construct and investigate self-similar solutions to gravity currents, and relatively simple shallow water or two-layer shallow water flows, impulse modified shallow water equations, dispersive shallow-water flows, Benney system, generalized Burgers equation, etc. [11,[25][26][27][33][34][35][36][37].…”

“…Similarity forms for two-layer shallow water equations are derived and discussed in [27]. In their formulation, the coupling is via the reduced gravity parameter.…”

Lie group analysis of nonlinear inviscid flows with a free surface under gravity

“…Parmar and Timol applied a group theoretic approach to natural heat convection and mass transfer for an inclined surface [21]. Özer and Antar investigated two-layer shallow-water equations [22] and Sekhar and Bira applied group analysis to equations for axisymmetric flow of shallow water [23]. Exact solutions for systems of nonlinear partial differential equations are of great interest; such solutions play a major role in the design, analysis, and testing of numerical methods for solving special initial and boundary value problems [16].…”