Relativistic k-fields with massless soliton solutions in $$3+1$$ dimensions

In this work, the relativistic non-standard Lagrangian densities (k-fields) with massless solutions are generally introduced. Such solutions are not necessarily energetically stable. However, in 3+1 dimensions, we introduce a new k-field model that results in a single non-topological massless solitary wave solution. This special solution is energetically stable; that is, any arbitrary deformation above its background leads to an increase in the total energy. In other words, its energy is zero which is the least energy in all solutions. Hence, it can be called a massless soliton solution.