We study the numerical solutions of the generalized Korteweg ? Burgers equation, which describes the waves in the presence of ice cover or surface tension, which is also a model equation for a broad class of models of continuum mechanics, in particular for the model of the electron magnetohydrodynamics. There are two cases. In the first case, the tangent to the dispersion branch cuts it away from the origin, in the second case there is no such an intersection. In the second case, the type structure of the discontinuity is determined by solving the system of equations describing the dispersion branch, and the equation describing the line corresponding to the velocity discontinuity. If the roots are purely imaginary, there is a structure with an internal non-dissipative shock of solitary-wave type, if the roots are complex, there is the structure with a discontinuity with the radiation. In the second case, this study does not provide complete information. A technique based on an analysis of the branches of doubly periodic solutions of the generalized Korteweg ? de Vries equation was developed. It was established that the one-wave long-wave branch contains fragments that transit to the two-wave resonant branch with integer ratio of wave periods. For the resonance branches at certain intervals of the intensity of the shock can be found non-dissipative structures that permit us to pass from the resonant branch to the short-wave one-wave branch. Therefore, using the averaged equation for the wave zone, we can construct solutions for weakly dissipative shock structures containing internal non-dissipative resonant discontinuities. Solution with a shock with the radiation can be interpreted as a special case of such solutions with 1/1 resonance. The considered solutions are observed by the direct calculation of the generalized KdV ? Burgers equation using the method of stabilization, and intervals of existence of different internal resonant structures observed in numerical experiment exactly correspond to that is predicted analytically. The study of dependencies of the type of discontinuity from its intensity and the dissipative parameter is fulfilled. Besides stationary structures time-periodic and stochastic structure were found also. For some areas of parameters hysteresis was observed: the presence of two stationary solutions for the same parameters. Type of solution depends on the path of evolution of the system. Observed bifurcation of solutions is typical for the general theory of dynamical systems.
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