We investigate exact plane-fronted gravitational-wave ($pp$-wave) solutions within the framework of shift-symmetric quadratic-order higher-order scalar-tensor (HOST) theories. These solutions represent fully nonlinear radiative spacetimes that extend beyond the linearized approximation. We demonstrate that, under the algebraic conditions on the coupling functions, the gravitational field equations reduce to a two-dimensional Laplace equation for the wave profile, recovering the structural form of vacuum general relativity. By adopting a scalar field ansatz that depends linearly on transverse coordinates and arbitrarily on the retarded null coordinate, we maintain a constant kinetic term of the scalar field. This configuration allows for a “stealth $pp$-wave” solution, where a nontrivial scalar field profile coexists with the gravitational wave without backreacting on the spacetime geometry. We further show that these stealth configurations are fully compatible with the degeneracy conditions of class-Ia degenerate HOST (DHOST) theories and satisfy current observational constraints. Finally, we examine the behavior of these solutions under disformal transformations, revealing that while the Brinkmann form is preserved, the stealth property is generically lost due to the mixing of scalar and tensor degrees of freedom. These results establish the robustness of $pp$-wave solutions in viable DHOST frameworks and highlight their utility for probing nonlinear effects in modified gravity.

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• Alternative gravity theories
• General relativity