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The Whitham approach is a well-studied method to describe non-linearintegrable systems. Although approximate in nature, its results may predictrather accurately the time evolution of such systems in many situations giveninitial conditions. A similarly powerful approach has recently emerged that isapplicable to quantum integrable systems, namely the generalized hydrodynamicsapproach. This paper aims at showing that the Whitham approach is thesemiclassical limit of the generalized hydrodynamics approach by connecting thetwo formal methods explicitly on the example of the Lieb-Liniger model on thequantum side to the non-linear Schr\"{o}dinger equation on the classical sidein the $c\to0$ limit, $c$ being the interaction parameter. We show how quantumexpectation values may be computed in this limit based on the connectionestablished here which is mentioned above.

The Whitham Approach to the $c\to0$ limit of The Lieb-Liniger Model and Generalized Hydrodynamics