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Hodograph equations for the n-dimensional Euler equations with the constant pressure and external force linear in velocity are presented. They provide us with solutions of the Euler in implicit form and information on existence or absence of gradient catastrophes. It is shown that in even dimensions the constructed solutions are periodic in time for particular subclasses of external forces. Several particular examples in one, two and three dimensions are considered, including the case of Coriolis external force., Comment: reference added 24 pages, 4 figures

Chynoweth and Sewell proposed a mathematical model for an atmospheric front based on the singularities of the Legendre transformation between different pairs of dual variables. Drawing inspiration from their work, we formalize the idea of a Chynoweth-Sewell front and illuminate its geometrical meaning from the viewpoint of Monge-Ampere geometry. This extends our previous work on the semigeostrophic model to dynamically-forming singularities. We use the notion of a Chynoweth-Sewell front to characterize the classical Eady problem and its known solutions in geometrical terms.

A Hamiltonian reduction approach is defined, studied, and finally used to derive asymptotic models of internal wave propagation in density stratified fluids in two-dimensional domains. Beginning with the general Hamiltonian formalism of Benjamin [1] for an ideal, stably stratified Euler fluid, the corresponding structure is systematically reduced to the setup of two homogeneous fluids under gravity, separated by an interface and confined between two infinite horizontal plates. A long-wave, small-amplitude asymptotics is then used to obtain a simplified model that encapsulates most of the known properties of the dynamics of such systems, such as bidirectional wave propagation and maximal amplitude travelling waves in the form of fronts. Further reductions, and in particular devising an asymptotic extension of Dirac's theory of Hamiltonian constraints, lead to the completely integrable evolution equations previously considered in the literature for limiting forms of the dynamics of stratified fluids. To assess the performance of the asymptotic models, special solutions are studied and compared with those of the parent equations., Comment: 29 pages, 4 figures

We study the (1+1) focussing nonlinear Schr\"{o}dinger equation for an initial condition with compactly-supported parabolic profile and phase depending quadratically on the spatial coordinate. In the absence of dispersion, using the natural class of self-similar solutions, we provide a criterion for blow-up in finite time, generalising a result by Talanov et al. In the presence of dispersion, we numerically show that the same criterion determines, even beyond the semi-classical regime, whether the solution relaxes or develops a high-order rogue wave, whose onset time is predicted by the corresponding dispersionless catastrophe time. The sign of the chirp appears to determine the prevailing scenario among two competing mechanisms for rogue wave formation. For negative values, the numerical simulations are suggestive of the dispersive regularisation of a gradient catastrophe described by Bertola and Tovbis for a different class of smooth, bell-shaped initial data. As the chirp becomes positive, the rogue wave seems to result from the interaction of counter-propagating dispersive dam break flows, as in the box problem recently studied by El, Khamis and Tovbis. As the chirp and amplitude of the initial profile are relatively easy to manipulate in optical devices and water tank wave generators, we expect our observation to be relevant for experiments in nonlinear optics and fluid dynamics., Comment: 17 pages, 7 figures, 1 table, typos corrected, references added, figures improved

Blowups of vorticity for the three- and two- dimensional homogeneous Euler equations are studied. Two regimes of approaching a blowup points, respectively, with variable or fixed time are analysed. It is shown that in the $n$-dimensional ($n=2,3$) generic case the blowups of degrees $1,..,n$ at the variable time regime and of degrees $1/2,..,(n+1)/(n+2)$ at the fixed time regime may exist. Particular situations when the vorticity blows while the direction of the vorticity vector is concentrated in one or two directions are realisable., Comment: 17 pages, 4 figures. Minor typos corrected

Telerobotics has shown promise in helping workers safely manipulate building components on construction sites; however, its primary reliance on visual feedback limits efficiency in situations with stiff contact or poor visibility. Reliable and economical haptic feedback could fill this perception gap and facilitate construction activities. Thus, we designed an audio-based haptic feedback system that measures the vibrations experienced by the robot tools and enables the operator to feel them in real time. Accurate haptic transmission was achieved by optimizing the positions of the system's off-the-shelf accelerometers and voice-coil actuators. A user study was conducted to evaluate how this naturalistic type of vibration feedback affects the operator's performance in telerobotic assembly. Thirty participants used a bimanual teleoperation system to build a structure under three randomly ordered haptic feedback conditions: no vibrations, one-axis vibrations, and three-axis vibrations. The results show that users took advantage of both kinds of haptic feedback after gaining some experience with the task, causing significantly lower vibrations and forces in the second trial. Subjective responses indicate that haptic feedback reduced the perceived task difficulty, task duration, and fatigue. These results demonstrate that providing this type of vibrotactile feedback on teleoperated construction robots would enhance user performance and experience., Comment: 10 pages, 7 figures in the main content, submitted to a journal

Blow-ups of derivatives and gradient catastrophes for the $n$-dimensional homogeneous Euler equation are discussed. It is shown that, in the case of generic initial data, the blow-ups exhibit a fine structure in accordance of the admissible ranks of certain matrix generated by the initial data. Blow-ups form a hierarchy composed by $n+1$ levels with the strongest singularity of derivatives given by $\partial u_i/\partial x_k \sim |\delta \mathbf{x}|^{-(n+1)/(n+2)}$ along certain critical directions. It is demonstrated that in the multi-dimensional case there are certain bounded linear superposition of blow-up derivatives. Particular results for the potential motion are presented too. Hodograph equations are basic tools of the analysis., Comment: 22 pages, 4 figures, 3 tables

Using Monge-Amp\`ere geometry, we study the singular structure of a class of nonlinear Monge-Amp\`ere equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Amp\`ere geometry by examining the role of an induced metric on Lagrangian submanifolds of the cotangent bundle. In particular, we show that the signature of the metric serves as a classification of the Monge-Amp\`ere equation, while singularities and elliptic-hyperbolic transitions are revealed by the degeneracies of the metric. The theory is illustrated by application to an example solution of the semigeostrophic equations., Comment: 22 pages, 5 figures

When creating 3D content, highly specialized skills are generally needed to design and generate models of objects and other assets by hand. We address this problem through high-quality 3D asset retrieval from multi-modal inputs, including 2D sketches, images and text. We use CLIP as it provides a bridge to higher-level latent features. We use these features to perform a multi-modality fusion to address the lack of artistic control that affects common data-driven approaches. Our approach allows for multi-modal conditional feature-driven retrieval through a 3D asset database, by utilizing a combination of input latent embeddings. We explore the effects of different combinations of feature embeddings across different input types and weighting methods.

Wave front propagation with non-trivial bottom topography is studied within the formalism of hyperbolic long wave models. Evolution of non-smooth initial data is examined, and in particular the splitting of singular points and their short time behaviour is described. In the opposite limit of longer times, the local analysis of wavefronts is used to estimate the gradient catastrophe formation and how this is influenced by the topography. The limiting cases when the free surface intersects the bottom boundary, belonging to the so-called "physical" and "non-physical" vacuum classes, are examined. Solutions expressed by power series in the spatial variable lead to a hierarchy of ordinary differential equations for the time-dependent series coefficients, which are shown to reveal basic differences between the two vacuum cases: for non-physical vacuums, the equations of the hierarchy are recursive and linear past the first two pairs, while for physical vacuums the hierarchy is non-recursive, fully coupled and nonlinear. The former case may admit solutions that are free of singularities for nonzero time intervals, while the latter is shown to develop non-standard velocity shocks instantaneously. We show that truncation to finite dimensional systems and polynomial solutions is in general only possible for the case of a quadratic bottom profile. In this case the system's evolution can reduce to, and is completely described by, a low dimensional dynamical system for the time-dependent coefficients. This system encapsulates all the nonlinear properties of the solution for general power series initial data, and in particular governs the loss of regularity in finite times at the dry point. For the special case of parabolic bottom topographies, an exact, self-similar solution class is introduced and studied to illustrate via closed form expressions the general results., Comment: 32 pages, 15 figures