Exponential-asymptotics treatment of steady radiating waves from sources of finite strength

We re-visit the steady forced Korteweg–deVries (fKdV) equation with a smooth locally confined forcing term and the radiation condition of no waves far upstream. When dispersion is small ($\mu \ll 1$), nonlinearity cannot be ignored in computing the radiating wave downstream even though the wave amplitude is exponentially small relative to $\mu$. Earlier studies computed this wave in the limit $\mu \to 0$ via exponential asymptotics, on the assumption that nonlinearity, which is controlled by the strength of the forcing term, is weak and balances with dispersion. Here, we develop a separate exponential asymptotics theory for radiating waves due to forcing of $O\left(1\right)$ strength. A fundamental difference from previous work is that the exponent of the exponentially small factor of the wave amplitude is controlled by the forcing amplitude $A$, which is bound from above by a certain critical value $A_{\mathrm{crit}}=O\left(1\right)$. Furthermore, this exponent tends to zero as $A$ approaches $A_{\mathrm{crit}}$. Thus, the present theory bridges the gap between the exponentially small waves computed earlier for $A\ll 1$ and the steep (cnoidal) waves that arise when $A$ is near $A_{\mathrm{crit}}$. The asymptotic results are supported by direct numerical solutions of the fKdV equation. Furthermore, the numerical results confirm that $A_{\mathrm{crit}}$ is a limiting forcing amplitude, beyond which steady solutions diverge downstream.