Nonlinear Scalar Equations

First we perform a symmetry analysis on the KdV equation, and obtain the traveling-wave solutions in terms of the functions ℘(z),tan2 z,coth2 z, and $\operatorname{cn}^{2}(z\mid m)$ , respectively. The two-soliton solution is also determined via the Hirota bilinear presentation. The KP equation is an extension of the KdV equation. We use symmetry transformations to extend the solutions of the KdV equation to the more sophisticated solutions of the KP equation. Then we solve the KP equation for solutions that are polynomial in a spatial variable, and obtain another type of solution. We also find the Hirota bilinear presentation of the KP equation and obtain the “lump” solution. Using the stable range of the nonlinear term, symmetry transformations, and the generalized power series method, we find a family of singular solutions with seven arbitrary parameter functions of t and a family of analytic solutions with six arbitrary parameter functions of t for the equation of transonic gas flows. Similar solutions are also obtained for the short-wave equation and the Khokhlov–Zabolotskaya equation in nonlinear acoustics of bounded bundles. The symmetry transformations and two new families of exact solutions with multiple parameter functions for the geopotential equation are derived.