Local well-posedness for fourth-order nonlinear dispersive systems on the one-dimensional torus under structural conditions

We study local well-posedness of the initial value problem for a class of fourth-order nonlinear dispersive systems on the one-dimensional torus.

The main difficulty comes from the loss of derivatives in the nonlinear terms.

By introducing suitable structural conditions that compensate for derivative loss through cancellation mechanisms, and constructing modified energies via gauge-type transformations combined with a diagonalization procedure, we establish local well-posedness in high-regularity Sobolev spaces.

The assumptions in the present result relax those in previous works, at the expense of requiring higher regularity.

In particular, our structural conditions extend the range of admissible nonlinearities beyond the scalar case and provide a unified framework for treating multi-component systems.