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In this paper we derive a higher-order KdV equation (HKdV) as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a `flat surface' assumption are expressed in a general form as a system of two coupled equations through Dirichlet-Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long-wave, and small-amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of similar type: KdV5, Kaup-Kuperschmidt equation, Sawada-Kotera equation, Camassa-Holm and Degasperis-Procesi equations., Comment: 29 pages, 2 figures. Dedicated to the memory of Prof. David J. Kaup (1939-2022)

Based on a novel treatment of near resonances, we introduce a new approximation for the rotating stratified Boussinesq system on three-dimensional tori with arbitrary aspect ratios. The rotation and stratification parameters are arbitrary and not equal. We obtain global existence for the proposed nonlinear system for arbitrarily large initial data. This system is sufficiently accurate, with an important feature of coupling effects between slow and fast modes. The key to global existence is a sharp counting of the relevant number of nonlinear interactions. An additional regularity advantage arises from a careful examination of some mixed type interaction coefficients. In a wider context, the significance of our near resonant approach is a delicate balance between the inclusion of more interacting modes and the improvement of regularity properties, compared to the well-studied singular limit approach based on exact resonance.

We formalise the concept of near resonance for the rotating Navier-Stokes equations, based on which we propose a novel way to approximate the original PDE. The spatial domain is a three-dimensional flat torus of arbitrary aspect ratios. We prove that the family of proposed PDEs are globally well-posed for any rotation rate and initial datum of any size in any $H^s$ space with $s\ge0$. Such approximations retain much more 3-mode interactions, thus more accurate, than the conventional exact resonance approach. Our approach is free from any limiting argument that requires physical parameters to tend to zero or infinity, and is free from any small divisor argument (so estimates depend smoothly on the torus' aspect ratios). The key estimate hinges on counting of integer solutions of Diophantine inequalities rather than Diophantine equations. Using a range of novel ideas, we handle rigorously and optimally challenges arising from the non-trivial irrational functions in these inequalities. The main results and ingredients of the proofs can form part of the mathematical foundation of a non-asymptotic approach to nonlinear oscillatory dynamics in real-world applications., Comment: 44 pages, 2 figures

We examine skyrmionic field configurations on a spherical ferromagnet with large normal anisotropy. Exploiting variational concepts of angular momentum we find a new family of localized solutions to the Landau-Lifshitz equation that are topologically distinct from the ground state and not equivariant. Significantly, we observe an emergent spin-orbit coupling on the level of magnetization dynamics in a simple system without individual rotational invariance in spin and coordinate space.

We establish global existence, uniqueness, regularity and long-time asymptotics of strong solutions to the half-harmonic heat flow and dissipative Landau-Lifshitz equation, valid for initial data that is small in the homogeneous Sobolev norm $\dot{H}^{\frac n 2}$ for space dimensions $n \le 3$.