Study of the structure of solutions to nonlinear wave equations based on continuous wavelet analysis

An approach based on the continuous wavelet transform is proposed to study the structure of solutions to nonlinear equations and the results are compared with the window Fourier transform. The shock waves governed by the Burgers equation is analyzed followed by the modified Burgers equation with cubic nonlinearity, classical, and dissipative solitons, soliton- like solutions to the modified Korteweg-de Vries (KdV) equation, and the sine-Gordon equation. The four main types of waves analyzed are sawtooth waves, centaur solutions, cnoidal waves, and envelope solitons. The continuous wavelet transform of a signal is based on a mother wavelet function from which a wavelet basis is constructed by using translations and dilations. A format of data representation combining the continuous wavelet transform with the window Fourier transform is developed. The numerical value of the functional are also determined for physical models of nonlinear wave processes with the use of various wavelet bases.