Constructing solutions of the ‘bad’ Jaulent–Miodek equation based on a relationship with the Burgers equation

The ‘bad’ Jaulent–Miodek (JM) equation describes the wave evolution of inviscid shallow water over a flat bottom in the presence of shear, which is ill-posed and unstable so that its general initial problem on the zero plane is difficult to solve through traditional mesh-based numerical methods. In this paper, using the Darboux transformation, we find a relation between the ‘bad’ JM equation and the well-known Burgers equation. Based on the Burgers equation, we construct the analytical and numerical solutions of the ‘bad’ JM equation via the Hirota bilinear method and the time-splitting Fourier spectral method. Specifically, we numerically present the interaction between two Gaussian packets of the ‘bad’ JM equation. This approach extends the applicability of traditional numerical methods for solving general initial problems of the ‘bad’ JM equation.