Abstract:
The $\mathrm{K}(f^m, g^n)$ equation is studied, which generalizes the modified Korteweg–de Vries equation $\mathrm{K}(u^3, u^1)$ and the Rosenau–Hyman equation $\mathrm{K}(u^m, u^n)$ to other dependences of nonlinearity and dispersion on the solution. The considered functions $f(u)$ and $g(u)$ can be linear or can have the form of a smoothed step. It is found numerically that, depending on the form of nonlinearity and dispersion, the given equation has compacton and kovaton solutions, Riemann-wave solutions, and oscillating wave packets of two types. It is shown that the interaction between solutions of all found types occurs with the preservation of their parameters.
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