This review explains a practical way to study nonlinear wave equations by reducing them to simpler dynamical systems. The authors argue that writing down exact travelling-wave formulas is often not enough. To understand which waves exist, how they change with parameters, and whether they are robust, you need a geometric view of the equations in a reduced phase space.

The paper surveys methods that convert partial differential equations (PDEs) into ordinary differential equations (ODEs) on a special solution manifold. Typical reductions include travelling-wave transforms, Galilean or self-similar changes of variables, and symmetry-based (Lie) reductions. Once reduced, the finite-dimensional system can reveal equilibria, periodic orbits, and connecting orbits that correspond to different wave types.

The review stresses a direct correspondence between invariant objects in the reduced system and waveforms in the PDE. An equilibrium in the reduced system corresponds to a constant-amplitude state. A closed orbit corresponds to a periodic travelling wave. A homoclinic orbit — a trajectory that leaves and then returns to the same equilibrium — corresponds to a localized solitary wave. A heteroclinic connection between two different equilibria corresponds to a front or kink. The authors also emphasize that reduced trajectories describe the spatial shape of a profile, not the long-time temporal stability of that profile in the full PDE.

The framework helps with questions that matter in applications, such as how solutions change with parameters (bifurcation), whether a uniform background will break up (modulation instability, MI), and how solutions respond to perturbations. The review discusses many constructive tools that provide explicit waveforms for benchmarking and analysis, including inverse scattering, Hirota’s method, Darboux transforms, and function-expansion methods like tanh and elliptic expansions.