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In this work, a complete tree of nonlocally related partial differential equations of Chaplygin gas equations is constructed. This tree includes the systems, which are obtained through local conservation laws and the local symmetry based method. We classify the nonlocal symmetries from potential systems as well as inverse potential systems (IPSs). Furthermore, we propose a systematic algorithm for identification of nonlocal symmetries through IPSs by combining the ideas in the studies of Bluman and Yang [J. Math. Phys. 54, 093504 (2013)] and Yang and Cheviakov [J. Math. Phys. 55, 083514 (2014)]. Finally, we obtain a new exact solution through nonlocal symmetry analysis and physical behavior of solution is presented.

Invariant solutions for the modified Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation are obtained by using classical Lie symmetries. The complete set of local point symmetries is established for modified KZK equation governing the propagation of finite amplitude. An optimal set of two‐dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. The optimality among the subalgebras of Lie algebra is proved in a constructive manner by using rank of coefficient matrix of general two‐dimensional element and successive application of adjoint actions. On the basis of these subalgebras, we carry out group invariant reductions and compute exact solutions for different classes of subalgebras in an optimal system. Mathematical and physical behaviors of different invariant solutions are shown graphically demonstrating that classical Lie symmetries are capable of solving the modified KZK equation.

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