Codimension two Lie invariant solutions of the modified Khokhlov–Zabolotskaya–Kuznetsov equation

Abstract

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.

Conservation laws and new exact solutions to the maccari’s modulation equations

“…The last equation shows that T = (T x , T y , T t ) satisfies the local conservation law of (13), where T x , T y , T t are given in (25). Thus we have obtained conserved vector T = (T x , T y , T t ) of (13), i.e.…”

“…Among these techniques, Hirota's bilinear method [22] is most effective in finding multi-soliton solutions of nonlinear PDEs. The Lie symmetry analysis [23][24][25], developed by Sophus Lie is one of the celebrated classic approaches for analytic solutions [26][27][28][29] and qualitative analysis of differential equations. This theory not only works for obtaining solutions but also plays a central role in constructing conservation laws.…”

Integrability properties and invariant solutions of some biological models

“…The theory of Lie groups 1–3 is one of the most effective methods to analyze differential equations, and there are many examples of the application of Lie groups to ordinary, partial, and integro‐differential equations in the literature 4–12 . In addition, the Lie groups theory can be applied to the nonlinear dynamical systems as well by considering the approach called the artificial Hamiltonian, a method that provides a gateway by which to compute the exact solution of a variety of coupled nonlinear systems of ordinary differential equations (ODEs) 13 .…”

“…The theory of Lie groups [1][2][3] is one of the most effective methods to analyze differential equations, and there are many examples of the application of Lie groups to ordinary, partial, and integro-differential equations in the literature. [4][5][6][7][8][9][10][11][12] In addition, the Lie groups theory can be applied to the nonlinear dynamical systems as well by considering the approach called the artificial Hamiltonian, a method that provides a gateway by which to compute the exact solution of a variety of coupled nonlinear systems of ordinary differential equations (ODEs). 13 In addition, it is a fact that every first-order system of ODEs can be written in the form of an artificial Hamiltonian system, and then the concept of artificial Hamiltonian presents a novel approach for finding the solutions to dynamical systems of first-order ODEs, and this approach is very applicable to determining first integrals of the aforementioned systems.…”

Adapting a new formula to generalize multidimensional transforms

“…Partial differential equations are used to create the mathematical solution of physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, and electrodynamics. Solutions to partial differential equations can be achieved by some methods such as Laplace transforms, Homotopy method, the Chebyshev wavelet operational matrix method, the finite difference method, Adomian decomposition method, the residual power series method, and others [17][18][19][20][21][22][23][24][25].…”