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Path protection requires finding a working path and a protection path that are link disjoint. In this paper, we consider the dynamic lightpath protection problem in WDM mesh networks under the wavelength continuity constraint. Existing polynomial time algorithms can be applied to find a pair of link-disjoint lightpaths on a single wavelength; however, such algorithms fail if the working and protection lightpaths are on two different wavelengths. We prove the problem is NP-complete for both dedicated protection and shared protection. We develop an integer linear programming (ILP) formulation and heuristic solutions for the problem. Computer simulations are conducted to evaluate the performance of the heuristic algorithms.