This analysis derives explicit solutions for the KdV equation with interface, suggesting local well-posedness.
In this paper, we consider the Korteweg–de Vries (KdV) equation on the real line with an interface. Using Fokas's unified transform method, the explicit solution formulas for the linear forced KdV equation with an interface are derived. Building on these solution formulas, we establish standard estimates for the linear solution and a bilinear estimate for the nonlinear term in a suitable Sobolev space. Using these estimates and a contraction mapping argument, we prove the local well‐posedness for the KdV equation with an interface.
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Huang et al. (2025) studied this question.
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