In this paper, we study the inverse nodal problem and the eigenvalue gap for the one-dimensional sloshing 4 problem with the p -Laplacian operator. By applying the Prüfer substitution, we first derive the reconstruction formula 5 of the depth function by using the information of the nodal data. Furthermore, we employ the Tikhonov regularization 6 method to consider how to reconstruct the depth function using only zeros of one eigenfunction. Finally, we investigate 7 the eigenvalue gap under the

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... restriction of symmetric single-well depth functions. We show the gap attains its minimum 8 when the depth function is constant. 9 Key words: inverse nodal problem; Tikhonov regularization method; eigenvalue gap; p -Laplacian has attracted considerable attention and arises in various fields, such as non-Newtonian fluids and nonlinear 13 diffusion problems. The quantity p is a characteristic of the fluid medium.