Order Nonlinear Wave Research Articles

Traveling Wave Solutions of Nonlinear Wave Equation Based on Functional Analysis and Multiple Variables as Applied to IoT

Abstract In applied mathematics, physics, and mechanics, many mathematical models are established in the form of nonlinear wave equations. Therefore, a method for studying traveling wave solutions of nonlinear wave equations based on functional analysis is proposed. The boundary value stability of the nonlinear second order nonlinear wave equation is modified by using functional analysis of multiple variables. In this paper, a new integral term is introduced into the organization mapping functional, the concept and construction of the nonlinear wave equation are analyzed, and its continuity is proved. According to the growth order of the traveling wave solution of nonlinear wave equation, the method of finding the traveling wave solution and its stability are studied.

Research on p order nonlinear half wave Schrödinger equations

The objective of our research is to scrutinize the presence and singularity of the comprehensive potent solution for the half-wave Schrödinger equation characterized by order. Take into account the nonlinear half-wave Schrödinger equations represented by order: . We apply the Brezis-Gallouet style inequality to attain a logarithmic form of regulation. These rationales will be pivotal in validating our primary theorem within Global Well-Posedness. When discussing global Solutions, we infer the solutions for Schrödinger equations within the realm when . Specifically, when , we leverage the Strichartz estimations alongside inequalities following the Brezis-Gallouët pattern to ascertain the well-posedness of Schrödinger equations within the framework. To elucidate the global well-posedness of Schrödinger equations in the energy subspace , we undertake a traditional tactic to craft the frail solution within the realm, succeeded by the application of a rationale found in existing academic literature to verify the singularity of the frail solution. The coherent progression of the frail solution is derived from the sustained conservation of both mass and energy elements. An analogous strategy is likewise harnessed to affirm the global well-posedness of Schrödinger equations in the sphere when .

Dynamic Analysis of an Articulated Offshore Tower Using Stokes Fifth Order Nonlinear Wave Theory

In offshore engineering, a compliant offshore structure accommodates dynamic forces by flexibility rather than rigidly withstanding load effects. As an alternative to conventional fixed platforms for loading/mooring in deep waters, articulated offshore towers are highly appealing because of their reduced structural weight. This study presents the results of a probabilistic random wave with and without current force on a 500 m high articulated tower with two hinges. By using the Lagrangian approach, governing nonlinear equations of motion are derived. The motion equation has been resolved in the time domain with the Houbolt approach. Based on Stokes fifth order nonlinear wave theory, wave forces have been assessed. Hydrodynamic forces are determined using a modified Morison equation with stretching modifications developed by Chakrabarti. The study examined two marine states, i.e., 10 m/10 s and 15 m/15 s. The current intensity of 2 m/sec has been taken into account. The salient statistical parameters such as maximum, minimum, mean, and standard deviation are presented under both the ocean environments. The proposed model is feasible for both the sea states. When comparing sea states, those with a greater wave height and period and those with a smaller wave height and period have more energy in their PSDs for all responses.

Shock waves with irrotational Rankine-Hugoniot conditions

Shock wave stability for isentropic irrotational flow is studied for Euler system but with shock front conditions corresponding to the second order nonlinear wave equation. It is shown that the usual Lax’ shock condition still guarantees the uniform linear stability and therefore the existence of the shock waves solution.

Nonlinear analytical solution for radiation stress of higher-order Stokes waves on a flat bottom

The radiation stress of water surface waves plays an important role in many aspects of marine hydrodynamics. The nonlinear effects of highly nonlinear waves on the radiation stress were investigated on a flat bottom under finite water depth and deep water conditions. Analytical solutions for radiation stress were derived based on different order nonlinear Stokes wave theory, respectively. The results indicate that the nonlinear radiation stresses are significantly greater than those based on the linear wave, and they increase as the nonlinearity strengthens. Through an inter-comparison of the expressions of nonlinear radiation stress of various orders of Stokes waves, it turned out that the results based on the second-order Stokes wave overestimated the radiation stress, while the expressions based on Fenton's fifth-order Stokes wave were more reasonable in reflecting the nonlinear effect of waves. In the case of extreme wave steepness, the fifth-order Stokes wave radiation stresses Sxx5 was approximately 17% larger than that of the linear wave. Finally, using the expressions of the fifth-order Stokes wave radiation stress, bound infragravity waves excited by the bichromatic wave groups formed by the superposition of two fifth-order Stokes waves were obtained.

Bifurcation of traveling waves in a liquid film with broken time-reversal symmetry

According to the analysis of the lubrication theory, the surface wave will develop in the region of low Reynolds number, and the presence of an odd viscosity. Fourth order nonlinear wave equations (generalized Kuramoto-Sivashinsky equation) that appear in the theory of thin liquid films are introduced. Analyses of linear stability for sinusoidal waves in various regimes are investigated. The Hopf bifurcation points for the traveling waves are investigated. Surface wave stability will arise more quickly if an odd viscosity is present.

Long-time behavior for fourth order nonlinear wave equations with dissipative and dispersive terms

The initial boundary value problem for a class of fourth-order nonlinear wave equations with dissipative terms and dispersive terms is considered in this paper. We first establish the global existence and the exponential decay of the solution. Next, based on Condition (C) and semigroup theory arguments, the existence of the global attractor is derived.

HIGH ORDER NONLINEAR WAVE INTERACTIONS FROM DEEP TO FINITE WATER DEPTH WITH BOTTOM TOPOGRAPHY CHANGE

In recent years, freak wave/rouge wave has become an important problem in science and engineering. Modulational instability is considered to be an important factor leading to freak wave in the wave evolution of deep water, and Janssen (2003) defined Benjamin-Feir index (BFI) to reflect it. Mori and Janssen (2006) gave the occurrence probability of freak waves based on a weakly non-Gaussian theory, and distribution of wave height is determined by skewness and kurtosis of surface elevation to a considerable extent in deep water. According to observational record, freak wave has not only been found in deep water in the ocean, but also been observed in shallow water and coastal areas. In the process of water wave entering continental shelf, water depth is changing with mild slope after a long distance propagation. This study focus on investigating how water depth affect skewness and kurtosis in the high order nonlinear wave evolution from deep water to finite water depth in two-dimension.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/a8LiJvXWRrw

Global strong solution of fourth order nonlinear wave equation

In this paper, we consider a class of fourth order nonlinear wave equation in a bounded domain with smooth boundary, and discuss the existence and uniqueness of the global strong solutions by gradually smoothing method.

Interconnections among nonlinear field equations

Galileons are related to other implicitly integrable equations such as the first order nonlinear wave equation and the universal field equation, which is a Lagrangian for the Galileon.

Asymptotic behavior and finite time blow up for damped fourth order nonlinear evolution equation

This paper investigates the initial boundary value problem for a class of fourth order nonlinear damped wave equations modeling longitudinal motion of an elasto-plastic bar. By applying a suitable potential well-convexity method, we derive the global existence, asymptotic behavior and finite time blow up for the considered problem with more generalized nonlinear functions at subcritical initial energy level. Further for arbitrarily positive initial energy we give some sufficient conditions ensuring finite time blow up.

Group classification of third order nonlinear wave equations chain’s

The paper presents the results of point symmetrys Lie algebras for third-order nonlinear wave equations calculating linked into a chain by Bäcklund transformations. Calculations are carried out by using Lie-Ovsyannikov method of group analysis. The basic algebras of point symmetries of the indicated equations are found, all possible cases of their extension are revealed, and the commutator tables of algebras found are calculated.

On Integrability of Christou’s Sixth Order Solitary Wave Equations

We examine the integrability in terms of Painlevè analysis for several models of higher order nonlinear solitary wave equations which were recently derived by Christou. Our results point out that these equations do not possess Painlevè property and fail the Painlevè test for some special values of the coefficients; and that indicates a non-integrability criteria of the equations by means of the Painlevè integrability.

Integrable nonlocal asymptotic reductions of physically significant nonlinear equations

Quasi-monochromatic complex reductions of a number of physically important equations are obtained. Starting from the cubic nonlinear Klein–Gordon (NLKG), the Korteweg–de Vries (KdV) and water wave equations, it is shown that the leading order asymptotic approximation can be transformed to the well-known integrable AKNS system (Ablowitz et al 1974 Stud. Appl. Math. 53 249) associated with second order (in space) nonlinear wave equations. This in turn establishes, for the first time, an important physical connection between the recently discovered nonlocal integrable reductions of the AKNS system and physically interesting equations. Reductions include the parity-time, reverse space-time and reverse time nonlocal nonlinear Schrödinger equations.

Interaction behavior between solitons and (2+1)-dimensional CDGKS waves

Starting from the Darboux transformation (DT) related nonlocal symmetry of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation, the original symmetry is localized by introducing four field quantities. On the basis of the local symmetry, some novel group invariant solutions are derived by utilizing the classical symmetry reduction method. The result shows that the essential and unique role of the DT is to add an additional soliton to the fifth order nonlinear wave, which is the basic reduction of the (2+1)-dimensional CDGKS equation. As an illustration, the interactions between solitons and fifth order nonlinear waves expressed by Jacobi elliptic functions and incomplete elliptic integrals of the third kind are exhibited, and some of their dynamical properties are analyzed.

Global solutions and finite time blow-up for fourth order nonlinear damped wave equation

In this paper, we study the initial boundary value problem and global well-posedness for a class of fourth order wave equations with a nonlinear damping term and a nonlinear source term, which was introduced to describe the dynamics of a suspension bridge. The global existence, decay estimate, and blow-up of solution at both subcritical (E(0) < d) and critical (E(0) = d) initial energy levels are obtained. Moreover, we prove the blow-up in finite time of solution at the supercritical initial energy level (E(0) > 0).

Asymptotic Calculation of the Wave Trough Exceedance Probabilities in A Nonlinear Sea

This paper concerns the calculation of the wave trough exceedance probabilities in a nonlinear sea. The calculations have been carried out by incorporating a second order nonlinear wave model into an asymptotic method. This is a new approach for the calculation of the wave trough exceedance probabilities, and, as all of the calculations are performed in the probability domain, avoids the need for long time-domain simulations. The proposed asymptotic method has been applied to calculate the wave trough depth exceedance probabilities of a sea state with the surface elevation data measured at the coast of Yura in the Japan Sea. It is demonstrated that the proposed new method can offer better predictions than the theoretical Rayleigh wave trough depth distribution model. The calculated results by using the proposed new method have been further compared with those obtained by using the Arhan and Plaisted nonlinear distribution model and the Toffoli et al.’s wave trough depth distribution model, and its accuracy has been once again substantiated. The research findings obtained from this study demonstrate that the proposed asymptotic method can be readily utilized in the process of designing various kinds of ocean engineering structures.

Higher order nonlinear dust-acoustic waves in a dusty plasma with dust of opposite polarity

The nonlinear propagation of dust-acoustic (DA) waves in an unmagnetized dusty plasma (containing positive and negatively charged inertial dust grains and nonextensive distributed electrons and ions) have been investigated theoretically. Three different types of nonlinear dynamical equations, namely, Korteweg-de Vries (K-dV), modified K-dV (mK-dV) and Gardner equations have been derived by employing the reductive perturbation method. The basic features (viz., polarity, amplitude, width, etc.) of solitary waves have been also analyzed numerically. It has been observed that the effects of nonextensivity of electrons and ions, mass and charge of positive and negative dust, and different dusty plasma parameters have significantly modified the fundamental properties of the solitary waves.

The initial-boundary value problems for a class of sixth order nonlinear wave equation

This paper considers the initial boundary value problem of solutions for a class of sixth order 1-D nonlinear wave equations. We discuss the probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and non-global existence of solutions at three different initial energy levels, i.e., sub-critical level, critical level and sup-critical level.

Analysis of Structural Second Order Effects on a Floating Concrete Platform for FOWT's

The irruption of the floating wind energy in the offshore engineering is changing the paradigm of the structural analysis for those new structures, which are significantly smaller and more slender than those previously existing from the O&G industry and in which the most important load is applied around 100m above the mean sea level. Those characteristics require the structural analysis to include the dynamic effects over the deformation of the structure, integrating the instantaneous deformation of the structure into the nonlinear dynamic finite element analysis (FEA) computation.A numerical model which couples the hydrodynamic and aerodynamic effects in a time domain structural analysis, has been developed at the UPC. The code integrates the forces exerted by the waves and currents as well as the aerodynamic loads, including the wind turbine, and the mooring system. For the hydrodynamic loads, Morison's equation in combination with the Stoke's 5th order nonlinear wave theory are used to compute the resulting forces. The mooring system is treated in a quasi-static way and the aerodynamic loads are computed with FAST. The structure is discretized with one-dimensional beam elements, in which the formulation considers small strains as well as large motions.The model computes a dynamic time domain nonlinear FEA for FOWT's, integrating all the effects of the external forces and the structural stiffness to obtain the displacements at each point of the structure at each time step. With this approach, the dynamic interaction between the wind turbine and the structure, as well as the effects on the internal forces are implicitly considered in the formulation. A comparison of the structure motions and internal forces for a concrete spar concept is presented assuming a nonlinear rigid body approach and a nonlinear dynamic time-domain FEA.