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We investigate kinetic energy properties of an irrotational deep-water Stokes wave. By introducing a conformal hodograph transform, we perform complex analysis in the new variables, and show that the existence of the streamline time-period for deep-water Stokes waves. Relying on the conformal mappings, we prove some qualitative results about the kinetic energy such as the convexity property of integral means of kinetic energy, the streamline time-period being non-increasing and independent of initial data and the monotonicity and logarithmic convexity of the total kinetic energy over a streamline time-period. Moreover, taking advantage of the non-increasing nature of the streamline time-period, we show that the streamline time-period strictly larger than the wave period, which suggests the fact that there are no closed paths for all particles in a deep-water Stokes flow and the drift of any streamline is positive. These results allow us to quantify the observation that the kinetic energy and the streamline time-period are larger near the free surface and decrease with increasing the depth., Comment: arXiv admin note: text overlap with arXiv:2405.02554. arXiv admin note: text overlap with arXiv:2405.02554

In this paper, we investigate kinetic energy and streamline properties for an irrotational periodic geophysical traveling surface water waves propagating in equatorial oceanic regions. Relying on the methods from complex analysis, we prove the logarithmic convexity and monotonicity of specific flow variables. By means of conformal mappings, we derive some qualitative results for kinetic energy and streamline, such as streamline time-period being independent of any moment and any point on the streamline in steady flow, the concavity and monotonicity of total kinetic energy within the region between two streamlines and the convexity and monotonicity of total kinetic energy over a streamline time-period. Moreover, we present several results about irrotational equatorial wind waves, such as an upper bound of the minimum of streamline time-period, an upper bound of the maximum of area within the region between two streamlines. Taking advantage of the Bernoulli's law and the Schwarz reflection principle, we show that the extremum of the kinetic energy is attained on the free surface for irrotational equatorial wind waves.

In this paper we study wave breaking in the unidirectional non-local wave model describing the motion of a collision-free plasma in a magnetic field. By analyzing the monotonicity and continuity properties of a system of the Riccati-type differential inequalities involving the extremal slopes of flows, we show a new sufficient condition on the initial data to exhibit wave breaking. Moreover, the estimates of life span and wave breaking rate are derived.

In this paper, we investigate existence and stability of solitary waves to the rotation-Camassa-Holm equation which can be considered as a model in the shallow water for the long-crested waves propagating near the equator with effect of the Coriolis force due to the Earths rotation. We prove existence of solitary waves by performing a phase plane analysis. Moreover, utilizing the approach proposed by Grillakis-Shatah-Strauss, we prove stability of solitary waves.

In this paper, we consider the Kakutani-Matsuuchi model which describes the surface elevation of the water-waves under the effect of viscosity. We show wave breaking for the Kakutani-Matsuuchi model, namely, the solution remains bounded but its slope becomes unbounded in finite time, the slope of the initial data is sufficiently negative.