In the present work, we examine the dynamics of a model for oscillons in one-dimensional space-time field theories with a cubic nonlinearity. We utilize a reduction of the model to first and third harmonics, which leads to a reduced partial differential equation (PDE) system whose steady states are candidates for the original PDE oscillons. We analyse the steady states of this model and their stability, via tools such as index theory. We develop suitable functionals needed for the study of such stationary states, as well as an analogue of the famous Vakhitov–Kolokolov criterion for a quantity whose change of monotonicity reflects a change of stability. Then, we test the relevant predictions, over the full range of oscillon frequencies, through systematic numerical computations of both the reduced model, its steady states and stability, and also of the original PDE model, identifying its time-periodic oscillon solution. Our results yield some significant connections with previous studies, but also some key new insights, both on the reduced system and the dynamics of the original system.

We investigate the dynamical role of internal vibrational modes in the self-dual Abelian-Higgs model, focusing on how Derrick-type breathing excitations modify vortex dynamics and scattering processes. While unexcited BPS vortices follow geodesic motion leading to the well-known $90^\circ$ scattering, initially excited vortices experience vibration-induced forces arising from the spectral flow of internal modes as the inter-vortex distance changes. This mechanism intensifies the resonant energy transfer between translational and vibrational channels, producing non-adiabatic dynamics, superelastic collisions and fractal structures in scattering diagrams.

A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi–Pasta–Ulam lattice with a general cubic interaction force. Explicit periodic travelling wave solutions in terms of Jacobi elliptic functions are classified, and their solitary-wave, kink and trigonometric limits are obtained. The Whitham modulation equations describing slow modulations of periodic travelling wave solutions are derived using an averaged variational principle. The convexity (strict hyperbolicity, genuine nonlinearity) of the resulting hydrodynamic-type equations is examined numerically in general and analytically in the solitary-wave and harmonic limits. In particular, the loss of hyperbolicity and the formation of complex conjugate characteristic velocities is shown to lead to modulational instability of periodic travelling waves. The onset of modulational instability is verified by numerical computations of linearized spectra for periodic travelling waves and initial value problems that also reveal additional short-wavelength instabilities.

We study the modulational stability problem for the travelling periodic waves (called Stokes waves) in an infinitely deep fluid by using pseudo-differential operators in conformal variables. We derive the criteria and the normal forms for four bifurcations which are repeated recurrently when the steepness of the Stokes wave is increased towards the highest wave with the peaked profile. The four bifurcations are observed in the following order: (a) new figure-8 bands appearing at each extremal point of speed, (b) degeneration of figure-8 bands resulting in vertical slopes, (c) new circular bands around the origin appearing at each period-doubling bifurcation and (d) reconnection of figure-$\infty$ bands at each extremal point of energy. Our work uses the analytic theory of Stokes waves developed previously for Babenko’s equation. The novelty of our work is the analytic extension of the modulational stability problem for singular pseudo-differential operators in terms of the Floquet parameter. The derivation of the normal form uses some structural assumptions which are known to be true for the Stokes waves. For the first and second bifurcation cycles, we compute numerically with a higher-order accuracy the actual values of wave steepness for which the structural assumptions are satisfied and the numerical coefficients of the normal forms to show the excellent agreement between the normal form theory and the numerical approximations of the spectral bands.

We construct a nonlinear-wave analogue of the celebrated inverted, Kapitza, pendulum. We consider bright soliton solutions of the one-dimensional nonlinear Schrödinger (NLS) equation under the influence of a repelling, when stationary, hump (or ‘hill’) potential and a confining harmonic trap. Using the variational approximation methodology, we reduce the soliton dynamics to an effective Newton-type model on the soliton position and study its behaviour when the hump potential is made to vibrate vertically. We corroborate that the effective model is able to qualitatively and quantitatively describe the statics, stability and dynamical properties of the original NLS bright soliton. Using the effective model, we find the stability regions as a function of the hump parameters (height and vertical shaking frequency), which correspond to the stability tongues of the Mathieu equation. We show that these theoretically predicted regions accurately describe the corresponding, numerically computed, Floquet stability regions of the original soliton on top of the vertically vibrating hill. Inside these stability regions, we are able to fully stabilize the soliton on top of the hill as a result of the vertical shaking in a manner akin to the inverted, Kapitza, pendulum.

A (2+1)-dimensional hyperbolic system of four quasi-linear partial differential equations is derived that describes the modulations of lump solutions of the Kadomtsev–Petviashvili I (KPI) equation in the presence of a mean field. The system is then shown to satisfy the necessary conditions for integrability of hydrodynamic chains. Moreover, a suitable reduction of the resulting modulation system is applied to study the interactions between lumps and a rarefaction wave for the mean field. Precise conditions are derived that describe how the lump parameters change as a result of the interaction, and which in particular determine whether the lump is transmitted through or trapped inside the rarefaction wave. The theoretical predictions are compared to direct numerical simulations of the KPI equation, showing excellent agreement.