1.Is there a Higgs wave (just like EM wave and GW)?
Quantum field theory for particle physics, posits for all elementary particles in the table,
a field. This is a mathematical field, represented by the free particle wavefunctions of the corresponding particle in the table. An electron field, a photon field, a gluon field, a Higgs field, ... at each space time point. It is based on the postulates of quantum mechanics, a mathematical extension and tool.
It is like a Lorentz invariant version of a special aether for each particle, where measured particles with their fixed mass and quantum numbers are defined by creation and annihilation operators at each space time point. Thus a particle is like a disturbance on its corresponding field. Because the field is in all space time, and the free particle wavefunction covers the whole space time, to get a localized particle, an electron, one has to use the concept of wavepackets, but that is an other story.
The QFT model is very good because it allows the Feynman diagram representation for interactions and decays, and simplifies calculations of real numbers to be compared with experiment.
So the Higgs boson is an excitation on all spacetime permeating Higgs field, ( as with other particles and fields). The difference lies in the vacuum expectation value, VEV, of the Higgs field versus all the other particles, which is the mechanism that the Higgs field gives masses to the particles in the table. The Higgs field has a 246GeV VEV, whereas all other particles in the table have a zero VEV
The electromagnetic wave is a superposition of real photons, and the gravitational wave is a superposition of real gravitons ( if they exist) and is an emergent classical level from the underlying quantum mechanical, and should not be confused.
- Is the Higgs wave (if exists) a herd of Higgs bosons, and is there any experiment where we try to detect Higgs waves?
No and no, as explained above. There is no Higgs wave , as the Higgs boson has a mass, in contrast to the photon and the graviton (if it exists). The Higgs boson is an excitation of the Higgs field, and its discovery is important because it validated the standard model of particle physics, which needs the Higgs field for the generation of masses.
Edit after comment by OP:
What makes me confused is when they say that an electron can behave like a wave. But the electron has rest mass. Is that just a mathematical description for the propagation of the electron that we call wave in that case? – Árpád Szendrei
Electromagnetic waves, and gravitational waves are classical models representing accurately what happens and is observed in experiments, down to sizes comensurate to $h$, i.e when sizes and velocities introduce the heisenberg uncertainty and quantum mechanical effects become dominant. These are an underlying level from which the classical has to emerge if our underlying theory is correct. By all the data and observations we have , the underlying level of all nature is quantum mechanical and the emergence from this of the classical fields, like the electric, etc can be demonstrated .
The quantum mechanical level is also modeled by wave equations., BUT, in contrast to the classical waves, it is not a wave of energy that is displayed in space time, but a probability wave . See my answer here on this. One needs many "events" with the same boundary conditions for the quantum mechanical equations , to build up the interference patterns predicted by the quantum mechanical wave equations.
Thus the electron "wave" behavior is a different manifestation than the classical electromagnetic or gravity waves, where there, it is the energy carried by the wave that is being transferred through a vacuum with velocity c, that is waving. Other waves, like acoustic and water waves, transfer the energy in a medium. A single electron can only be detected once at an (x,y,z,t) point and has a fixed four momentum vector.
So a Higgs boson, if it were long lived, would have left a point by its decay in a detector, and an accumulation of such points would show a wave nature, given similar boundary conditions as in the double slit experiments.
An electron beam can show collectively the patterns, thus in principle able to transfer energy in a medium, with wave properties, but this wave cannot be confused with the theoretical waves built up by zero mass gauge bosons macroscopically that do not need a medium to appear and transfer energy in spacetime.