Modeling Nonlinear Dispersive Water Waves

An expository review is given on various theories of modeling weakly to strongly nonlinear, dispersive, time-evolving, three-dimensional gravity-capillary waves on a layer of water. It is based on a new model that allows the nonlinear and dispersive effects to operate to the same full extent as in the Euler equations. Its relationships with some existing models are discussed. Various interesting phenomena will be illustrated with applications of these models and with an exposition on the salient features of nonlinear waves in wave-wave interactions and the related processes of transport of mass and energy

On resolution to Wu’s conjecture on Cauchy function’s exterior singularities

This is a series of studies on Wu’s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy’s function f(z) (z = x + iy) and its integral J[f(z)] ≡ (2πi)^(−1) ∮_Cf(t)(t−z)^(−1)dt taken along the unit circle as contour C, inside which (the open domain D^+) f(z) is regular but has singularities distributed in open domain D^− outside C. Resolution is given to the inverse problem that the singularities of f(z) can be determined in analytical form in terms of the values f(t) of f(z) numerically prescribed on C (|t| = 1), as so enunciated by Wu’s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle, for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction, and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics, engineering and technology in analogy with resolving the inverse problem presented here

Stability of Nonlinear Waves Resonantly Sustained

This talk will examine the stability properties of representative cases of nonlinear dispersive waves generated and sustained at resonance of physical systems capable of supporting solitary waves. The criteria are sought for realizing the remarkable phenomenon of periodic production of upstream-radiating solitary waves by critical disturbances moving steadily in a layer of shallow water as modeled by the forced KdV equation. Of primary interest are the distinctive features of instabilities of a few typical steady basic flows, the salient new characteristics of the associated eigenvalue problems, the relevant nonlinear effects, and the resulting bifurcation diagrams

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f (z) and its integral, (2πi)J[f(z)] ≡ ∮f(t)dt(t−z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D^− outside C. (1) With f (z) assumed to be C^n (n < ∞-times continuously differentiable) ∀ z ∈ D^+ and in a neighborhood of C, f (z) and its derivatives f^(n)(z) are proved uniformly continuous in the closed domain D^+ = [D^+ + C]. (2) Cauchy’s integral formulas and their derivatives ∀z ∈ D^+ (or ∀z ∈ D^−) are proved to converge uniformly in D^+ (or in [D^ + C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[f (z)]) are shown extended to hold for the complement function F(z), defined to be C^n ∀z ∈ D^- and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four generalized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f(z) in D^− is elucidated by considering the direct problem exemplified with several typical singularities prescribed in D^−. (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical significances of these formulas are illustrated with applications to nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f(z) in domain D −, based on the continuous numerical value of f(z)∀z ∈ D+=[D++C], is presented for resolution as a conjecture

On theoretical modeling of aquatic and aerial animal locomotion

This chapter discusses the theoretical modeling of aquatic and aerial animal locomotion, several objectives that are focused on exploring how, why, and under what premises such high efficiency and low-energy cost can be achieved in these diverse modes of locomotion as a result of a long history of convergent evolution. The chapter takes an integral viewpoint from the foundation built by the pioneering leaders in the field, such as Herbert Wagner, Theodore von Karman, William R. Sears, and Sir James Lighthill, followed by other researchers through developing various theoretical and experimental methods used in studies on the subject. In subdividing the various classes of hydrodynamic theories and the underlying physical conceptions, it is seen that the generalized slender-body theory is readily capable of expounding the complex interaction between the swimming body and the vortex sheets shed from the appended fins, caudal fins, or lunate tails. Some important nonlinear effects are considered, and separate resort is made for mechanophysiological studies on energetics and hydromechanics of fish propulsion involving the biochemical and mechanical conversions of energy

Stability of Nonlinear Waves Resonantly Sustained

This talk will examine the stability properties of representative cases of nonlinear dispersive waves generated and sustained at resonance of physical systems capable of supporting solitary waves. The criteria are sought for realizing the remarkable phenomenon of periodic production of upstream-radiating solitary waves by critical disturbances moving steadily in a layer of shallow water as modeled by the forced KdV equation. Of primary interest are the distinctive features of instabilities of a few typical steady basic flows, the salient new characteristics of the associated eigenvalue problems, the relevant nonlinear effects, and the resulting bifurcation diagrams