Is there a precise mathematical definition for a "wave"? I think not. Are there ongoing attempts to formulate one? Or is it accepted that "wave" should remain a mathematically ambiguous concept?

One could make the attempt: "A wave is any solution to the wave equation". However, although the phrase "the wave equation" is used to refer to a particular equation (https://en.wikipedia.org/wiki/Wave_equation) there are other partial differential equations whose solutions are also regarded as waves. If a "wave" is defined as a solution to some "wave equation", what is the definition of a "wave equation"?

According to http://www.scholarpedia.org/article/...onlinear_waves

Evaluating those definitions, the article says:Now, whilst most people have a general notion of what a wave is, based on their everyday experience, it is not easy to formulate a definition that will satisfy everyone engaged in or interested in this wide ranging subject. In fact, many technical works related to waves eschew a formal definition altogether and introduce the concept by a series of examples; for example, Physics of waves [Elm-69] and Hydrodynamics [Lam-93]. Nevertheless, it is useful to at least make an attempt and a selection of various definitions from normally authoritative sources is given below:

The variety of definitions given above, and their clearly differing degrees of clarity, confirm that 'wave' is indeed not an easy concept to define!