Patterns of Emergent Nonlinear Waves in an Initial Discontinuity Problem of the Bad Jaulent-Miodek Equation

By means of the Whitham modulation theory, this paper studies the discontinuous initial value problem of the bad Jaulent-Miodek (JM) equation, which is compared with the good JM equation. The linear stability of the good and bad JM equations is analyzed to show the difference between the two equations. Also the physical significance of the JM equations is discussed by considering the reduction of the Euler’s equation. Then the zero-, one-, two-phase periodic solutions and the corresponding Whitham equations in the framework of the bad JM equation are derived by finite-gap integration approach. The degeneration of the one-phase periodic solution along with the genus-one Whitham equation are analyzed by taking the two sides limits of the modulus m of the Jacobi elliptic functions. The basic rarefaction wave patterns and dispersive shock wave patterns are proposed analytically and graphically, and then the complete classification of solutions and the all possible waveforms evolving from discontinuous initial values in the bad JM equation are established. In the results, various patterns of emergent nonlinear waves that appear to be new and are being detected for the first time