Time-invariant and time-varying discontinuity structures for models described by the generalized Korteweg–Burgers equation

Taking the generalized Korteweg–Burgers equation as the example, it is established by numerical analysis that three types of discontinuity structures are encountered for weakly dissipative media with dispersion and non-linearity: time-invariant structures, time-periodic structures and stochastic structures. Time-invariant weakly dissipative structures contain internal non-dissipative discontinuity structures of the type of transitions between homogeneous or wave states. The structure of a discontinuity can be non-unique. Hystereses arise on account of this, that is, the type of discontinuity depends on the evolutionary path of the system. The dependence of the type of discontinuity on its amplitude and the dissipation parameter has been investigated. The time-invariant solutions of the generalized Korteweg – de Vries equation: the periodic solutions, soliton solutions and the structures of the discontinuities were studied in order to explain the observed phenomena and to predict the type of discontinuity. A technique is developed for analysing the branches of the biperiodic solutions. A correspondence between the structural types of weakly dissipative discontinuity and the branch arrangement pattern is revealed.