A narrow-banded spectrum of gravity waves propagating in one horizontal direction 4 at the surface of water with finite depth may be governed by a nonlinear Schrödinger equation 5 (NLSE). If the depth is non-uniform, the coefficients in the NLSE are variable and an additional 6 linear term due to conservation of wave action flux is present, which may act as a dissipative term 7 that causes the ...