In this work, we first analyze the semiconservative direct discontinuous Galerkin (DDG) method for the Korteweg–de Vries (KdV) equations. The scheme achieves $$$ r\mathrm{th} $$$ order accuracy in finite element approximation spaces with even degrees $$$ r\ge 2 $$$. Subsequently, we construct and analyze a nonconservative discrete scheme within the framework of the local discontinuous Galerkin (LDG) method. Moreover, this scheme can achieve a suboptimal convergence order of $$$ r+\frac{1}{2} $$$. For temporal discretization, we employ the implicit-explicit additive Runge–Kutta method to achieve high-order accuracy and efficiency. Finally, numerical experiments for the DDG and LDG methods are provided, including the accuracy of solitons, long-term behavior, and conserved quantities. Given the potential for finite-time soliton blowup phenomenon due to the presence of high-order nonlinearity in this model, we also investigate the performance of some discontinuous Galerkin (DG) methods in simulating the instability of solitons while improving the accuracy and efficiency of blowup simulations through the incorporation of the arbitrary Lagrangian–Eulerian (ALE) method for adaptive mesh movement.