Abstract
The theory of nonlinear slow body and surface waves is developed. The conventional long-wavelength approximation is lifted. A new kind of nonlinear evolution equations is obtained. Two complementary equations are obtained. One of them is of the highest accuracy. Another one has less accuracy but its analytical solutions are found. Two analytical solutions of a distinct kind are found. It is revealed that both dark and bright solitons are the solutions of the evolutionary equation for slow waves. One of these two solutions is not singular for arbitrary amplitudes. Another one is not singular only when the critical amplitude is exceeded. The canonic form of the new evolutionary equation is presented. As a by-product of the exploration, the equation of the highest accuracy for body waves in the long-wavelength approximation is obtained. The interplay of shocks and solitons in flux tubes is discussed.
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