Probability Distribution Characteristics of Strong Nonlinear Waves Under Typhoon Conditions in the Northern South China Sea

The generation and propagation mechanism of strong nonlinear waves in the South China Sea is an essential research area. In this study, the third-generation wave model WAVEWATCH Ⅲ is employed to simulate wave fields under extreme sea states. The model, integrating the ST6 source term, is validated against observed data, demonstrating its credibility. The spatial distribution of the occurrence probability of strong nonlinear waves during typhoons is shown, and the waves in the straits and the northeastern part of the South China Sea show strong nonlinear characteristics. The high-order spectral model HOS-ocean is employed to simulate the random wave surface series beneath five different platform areas. The waves during the typhoon exhibit strong nonlinear characteristics, and freak waves exist. The space-varying probability model is established to describe the short-term probability distribution of nonlinear wave series. The exceedance probability distributions of the wave surface beneath different platform areas are compared and analyzed. The results show that with an increase in the platform area, the probability of a strong nonlinear wave beneath the platform increases.

MULTIPLE SOLUTIONS FOR HAMILTONIAN SYSTEMS WITH PERIODIC NONLINEARITY AND STRONG RESONANCE

I. Introduction In this paper we are looking for solutions of the following Hamiltonian system of second order: where x= （x1, x2） and V satisfies （V. 1） V: R×R2→R is a C1-function, 1-periodic In t, （V.2） V is periodic in x1 with the period T>0, （V. 3） V→O, Vx→O as |x2|→∞, uniformly in （t, x1）.

Bifurcation and chaos of airfoil with multiple strong nonlinearities

The bifurcation and chaos phenomena of two-dimensional airfoils with multiple strong nonlinearities are investigated. First, the strongly nonlinear square and cubic plunging and pitching stiffness terms are considered in the airfoil motion equations, and the fourth-order Runge-Kutta simulation method is used to obtain the numerical solutions to the equations. Then, a post-processing program is developed to calculate the physical parameters such as the amplitude and the frequency based on the discrete numerical solutions. With these parameters, the transition of the airfoil motion from balance, period, and period-doubling bifurcations to chaos is emphatically analyzed. Finally, the critical points of the period-doubling bifurcations and chaos are predicted using the Feigenbaum constant and the first two bifurcation critical values. It is shown that the numerical simulation method with post-processing and the prediction procedure are capable of simulating and predicting the bifurcation and chaos of airfoils with multiple strong nonlinearities.

Primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method

A homotopy analysis method（HAM）is presented for the primary resonance of multiple degree-of-freedom systems with strong non-linearity excited by harmonic forces.The validity of the HAM is independent of the existence of small parameters in the considered equation.The HAM provides a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter.Two examples are presented to show that the HAM solutions agree well with the results of the modified Linstedt-Poincar＇e method and the incremental harmonic balance method.

IMPROVED L-P METHOD FOR SOLVING STRONGLY NONLINEAR PROBLEMS

Using the improved L-P method, the authors seek to salve a class of problems of square strongly nonlinear free oscillations and of strongly nonlinear nonoscillations. Their first-order approximate solutions which has high accuracy are obtained. The method of this paper is different from the known L-P methods.

NONLINEAR TIME TRANSFORMATION METHOD FOR STRONG NONLINEAR OSCILLATION SYSTEMS

In this paper,a nonlinear time transformation method is presented for the analysis of strong nonlinear oscillation systems.This method can be used to study the limit cycle behavior of the autonomous systems and to analyze the forced vibration of a strong nonlinear system.

A high order energy preserving scheme for the strongly coupled nonlinear Schr¨odinger system

A high order energy preserving scheme for a strongly coupled nonlinear Schrōdinger system is roposed by using the average vector field method. The high order energy preserving scheme is applied to simulate the soliton evolution of the strongly coupled Schrōdinger system. Numerical results show that the high order energy preserving scheme can well simulate the soliton evolution, moreover, it preserves the discrete energy of the strongly coupled nonlinear Schrōdinger system exactly.

Generalized Strongly Nonlinear Quasi-Complementarity Problems

Using the algorithm in this paper, we prove the existence of solutions to the gene-ralized strongly nonlinear quasi-complementarity problems and the convergence of theiterative sequences generated by the algorithm. Our results improve and extend thecorresponding results of Noor and Chang-Huang. Moreover, a more general iterativealgorithm for finding the approximate solution of generalized strongly nonlinear quasi-complementarity problems is also given. It is shown that the approximate solution ob-tained by the iterative scheme converges to the exact solution of this quasi-com-plementarity problem.

High-codimensional static bifurcations of strongly nonlinear oscillator

The static bifurcation of the parametrically excited strongly nonlinear oscillator is studied. We consider the averaged equations of a system subject to Duffing-van der Pol and quintic strong nonlinearity by introducing the undetermined fundamental frequency into the computation in the complex normal form. To discuss the static bifurcation, the bifurcation problem is described as a 3-codimensional unfolding with Z2 symmetry on the basis of singularity theory. The transition set and bifurcation diagrams for the singularity are presented, while the stability of the zero solution is studied by using the eigenvalues in various parameter regions.

The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant

In this paper, the extended Pade approximant is used to construct the homoclinic and the heteroclinic trajectories in nonlinear dynamical systems that are asymmetric at origin. Meanwhile, the conservative system, the autonomous system, and the nonautonomous system equations with quadratic and cubic nonlinearities are considered. The disturbance parameter ~ is not limited to being small. The ranges of the values of the linear and the nonlinear term parameters, which are variables, can be determined when the boundary values are satisfied. New conditions for the potentiality and the convergence are posed to make it possible to solve the boundary-value problems formulated for the orbitals and to evaluate the initial amplitude values.

INCREMENTAL HARMONIC BALANCE METHOD FOR AIRFOIL FLUTTER WITH MULTIPLE STRONG NONLINEARITIES

The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.

A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.

Propagation properties and radiation force of circular Airy Gaussian vortex beams in strongly nonlocal nonlinear medium

We study the abruptly autofocusing and autodefocusing properties of the circular Airy Gaussian vortex(CAi GV)beams in strongly nonlocal nonlinear medium for the first time through numerical simulations.The magnitude of topological charges and the position of the vortex could change not only the light spot pattern but also the intensity contrast.Meanwhile,we can change the position of the autofocusing and autodefocusing planes by changing the parameter of the incident beam.Furthermore,we can control the peak intensity contrast through choosing properly the truncation factor.As for the radiation force,we study the gradient and the scattering forces of CAi GV beams on Rayleigh dielectric sphere.Our analyses demonstrate that the radiation force can be enhanced by choosing proper parameters of CAi GV beams.

A NEW METHOD FOR THE PERIODIC SOLUTION OF STRONGLY NONLINEAR SYSTEMS

A new method for the periodic solution of strongly nonlinear system is given. By using this method, the existance and stability of the periodic solution can be decided, and the approximate expression of the periodic solution can also be found.

PROXIMAL POINT ALGORITHM WITH ERRORS FOR GENERALIZED STRONGLY NONLINEARQUASIVARIATIONAL INCLUSIONS

In this paper, a class of generalized strongly nonlinear quasivariational inclusions are studied. By using the properties of the resolvent operator associated with a maximal monotone; mapping in Hilbert space, an existence theorem of solutions for generalized strongly nonlinear quasivariational inclusion is established and a new proximal point algorithm with errors is suggested for finding approximate solutions which strongly converge to the exact solution of the generalized strongly, nonlinear quasivariational inclusion. As special cases, some known results in this field are also discussed.

The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term

We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear damping σ and ?, we make assumptions (H1) - (H4). Under of the proper assume, the main results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.

Strongly nonlinear long interfacial waves in uniform basic flows and their dispersion relationship

In this paper, long interfacial waves of finite amplitude in uniform basic flows are considered with the assumption that the aspect ratio between wavelength and water depth is small. A new model is derived using the velocities at arbitrary distances from the still water level as the velocity variables instead of the commonly used depth-averaged velocities. This significantly improves the dispersion properties and makes them applicable to a wider range of water depths. Since its derivation requires no assumption on wave amplitude, the model thus can be used to describe waves with arbitrary amplitude.

THE MLP METHOD FOR SUBHARMONIC AND ULTRA-HARMONIC RESONANCE SOLUTIONS OF STRONGLY NONLINEAR SYSTEMS

A new parameter transformation alpha = alpha (epsilon, n omega (0)/m, omega (l)) was defir2ed for extending the applicable range of the modified Lindstedt-Poincare method. It is suitable for determining subharmonic and ultraharmonic resonance solutions of strongly nonlinear systems. The 1/3 subharmonic and 3 ultraharmonic resonance solutions of the Duffing equation and the 1/2 subharmonic resonance solution of the Van der Pol-Mathieu equation were studied. These examples show approximate solutions are in good agreement with numerical solutions.

A Class of Cauchy Problems for Porous Medium Equations with Strongly Nonlinear Sources and Convections

In this paper we study existence of solutions of a class of Cauchy problems for porous medium equations with strongly nonlinear sources or absorptions and convections when the initial trace is a Radon measure μ on RN.

Propagation of optical vortex solitons due to the Gouy phase in strongly nonlocal nonlinear media

In this paper, we present a study on the propagation of the symmetrical optical vortices formed by two collinear Laguerre-Gauss solitons in strongly nonlocal nonlinear media. The optical vortices, which move along the beam axis as the light propagates, result in a rotation of the beam＇s transverse profile. This physical reason of the rotation is the Gouy phase acquired by the component beams.