Dimensional Wave Equation Research Articles

Novel nonlinear wave equation: Regulated rogue waves and accelerated soliton solutions

A new exactly solvable (1+1)-dimensional complex nonlinear wave equation exhibiting rich analytic properties has been introduced. A rogue wave (RW), localized in space–time like Peregrine RW solution, though richer due to the presence of free parameters is discovered. This freedom allows to regulate amplitude and width of the RW as needed. The proposed equation allows also an intriguing topology changing accelerated dark soliton solution in spite of constant coefficients in the equation.

Inelastic dromions, rogue waves and lumps of (2+1) dimensional long dispersive wave equation

In this paper, we revisit the (2+1) dimensional long dispersive wave equation employing the truncated Painlevé approach. We then generate the solutions in the closed form in terms of lower dimensional arbitrary functions of space and time. By suitably harnessing the arbitrary functions present in the solution, we then construct localized solutions such as dromions, lumps and rogue waves. We have also explicitly brought out the generality of the localized solutions compared to the localized solutions generated earlier. The collisional dynamics of dromions, lumps and rogue waves is then explored.

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A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation

In this paper, we present an approach for seeking exact solutions with coefficient function forms of conformable fractional partial differential equations. By a combination of an under-determined fractional transformation and the Jacobi elliptic equation, exact solutions with coefficient function forms can be obtained for fractional partial differential equations. The innovation point of the present approach lies in two aspects. One is the fractional transformation, which involve the traveling wave transformations used by many articles as special cases. The other is that more general exact solutions with coefficient function forms can be found, and traveling wave solutions with constants coefficients are only special cases of our results. As of applications, we apply this method to the space-time fractional (2+1)-dimensional dispersive long wave equations and the time fractional Bogoyavlenskii equations. As a result, some exact solutions with coefficient function forms for the two equations are successfully found.

The Application of Scilab Software in Frequency Mode Simulation on the Circular Membrane

Scilab software is utilized for solving physics cases in two dimensional wave equations. This study aims to obtain a frequency mode simulation on the circular membrane so that it can display the wave pattern of each frequency mode. The method is to compare the simulation results with the normal mode circular membrane in each mode in to the theoretical calculation. The simulation has been done by utilizing the physics concept in a wave equation involving cos θ. The function of the wave equation is , assuming that r is the radius of the circle membrane. The value of the vibration frequency mode in theoretical are f<sub>01</sub> = (200.6 ± 2.0) Hz, f<sub>11</sub> = (319.7 ± 3.1) Hz, f<sub>21</sub> = (428.5 ± 4.2) Hz. By using the Scilab program application, the results obtained show that the results frequency mode simulation f<sub>01</sub>, f<sub>11</sub>, dan f<sub>21</sub> are similar to the normal mode in theoretical calculation. Thus, the simulation can be used to visualize waves of two dimensions in vibration and wave learning.

Oblique closed form solutions of some important fractional evolution equations via the modified Kudryashov method arising in physical problems

The paper deals with the obliquely propagating wave solutions of fractional nonlinear evolution equations (NLEEs) arising in science and engineering. The conformable time fractional (2 + 1)-dimensional extended Zakharov-Kuzetsov equation (EZKE), coupled space-time fractional (2 + 1)-dimensional dispersive long wave equation (DLWE) and space-time fractional (2 + 1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation are considered to investigate such physical phenomena. The modified Kudryashov method along with the properties of conformable and modified Riemann-Liouville derivatives is employed to construct the oblique wave solutions of the considered equations. The obtained results may be useful for better understanding the nature of internal oblique propagating wave dynamics in ocean engineering.

The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space

The Boussinesq abcd system is a 4-parameter set of equations posed in Rt×Rx, originally derived by Bona, Chen and Saut [11,12] as first order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation [17]. Among many particular regimes, depending each of them in terms of the value of the parameters (a,b,c,d) present in the equations, the generic regime is characterized by the setting b,d>0 and a,c<0. If additionally b=d, the abcd system is Hamiltonian. The equations in this regime are globally well-posed in the energy space H1×H1, provided one works with small solutions [12]. In this paper, we investigate decay and the scattering problem in this regime, which is characterized as having (quadratic) long-range nonlinearities, very weak linear decay O(t−1/3) because of the one dimensional setting, and existence of non scattering solutions (solitary waves). We prove, among other results, that for a sufficiently dispersiveabcd systems (characterized only in terms of parameters a,b and c), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone |x|≤|t|. We prove this result by constructing three suitable virial functionals in the spirit of works [27,28], and more precisely [42] (valid for the simpler scalar “good Boussinesq” model), leading to global in time decay and control of all local H1×H1 terms. No parity nor extra decay assumptions are needed to prove decay, only small solutions in the energy space.

Dynamic behaviors of interaction solutions of (3+1)-dimensional Shallow Water wave equation

Rational solutions and interaction solutions of the (3+1)-dimensional Shallow Water wave equation are obtained depending on N-soliton solution by the Hirota bilinear method and long wave limit method. And breather, lump solutions and line rogue waves are obtained from two-soliton solution as well as their interaction solutions are derived from four-soliton solution. After the collision, breathers and lumps remain unchanged, but the line rogue wave has a difference. Furthermore, three-soliton solution is discussed, the interactions between some hybrid solutions, such as hybrid solution between the stripe soliton and breather, hybrid solution between stripe soliton and lump solution and so on, are demonstrated in detail. More interestingly, stripe soliton remain unchanged, but the amplitudes of the periodic line waves, lump solutions and rogue waves have a great change after the collision.

The Differential Transform Method for n1 - dimensional Equal Width Wave equation with damping term

The Differential Transform Method for n1 - dimensional Equal Width Wave equation with damping term

The refined impedance transform for 1D acoustic reflection data

The one dimensional wave equation serves as a basic model for imaging modalities such as seismic which utilize acoustic data reflected back from a layered medium. In 1955 Peterson et al described a single scattering approximation for the one dimensional wave equation that relates the reflection of Green’s function to acoustic impedance. The approximation is simple, fast to compute and has become a standard part of seismic theory. The present paper re-examines this classical approximation in light of new results concerning the (exact) measurement operator for reflection imaging of layered media, and shows that the classical approximation can be substantially improved. We derive an alternate formula, called the refined impedance transform, that retains the simplicity and speed of computation of the classical estimate, but which is qualitatively more accurate and applicable to a wider range of recorded data. The refined impedance transform can be applied to recorded data directly (without the need to deconvolve the source wavelet), and solves exactly the inverse problem of determining the value of acoustic impedance on the far side of an arbitrary slab of unknown structure. The results are illustrated with numerical examples.

Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation∂t2u=∂r2u+(d−1)r∂ru−(d−1)2r2sin⁡(2u). We construct for this equation a family of C∞ solutions which blow up in finite time via concentration of the universal profileu(r,t)∼Q(rλ(t)), where Q is the stationary solution of the equation and the speed is given by the quantized ratesλ(t)∼cu(T−t)ℓγ,ℓ∈N⁎,ℓ>γ=γ(d)∈(1,2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski [49] for the energy supercritical nonlinear Schrödinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.

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On the structure of conservation laws of (3+1)-dimensional wave equation

In this paper, a (3+1)-dimensional wave equation is studied from the point of view of Lie’s theory in partial differential equations including conservation laws. The symmetry operators are determined to find the reduced form of the considered equation. The non-local conservation theorems and multipliers approach are performed on the (3+1)-dimensional wave equation. We obtain conservation laws by using five methods, such as direct method, Noether’s method, extended Noether’s method, Ibragimov’s method; and finally we can derive infinitely many conservation laws from a known conservation law viewed as the last method. We also derive some exact solutions using some conservation laws Anco and Bluman (2002).

Moderate Deviations for a Stochastic Wave Equation in Dimension Three

In this paper, we prove a central limit theorem and establish a moderate deviation principle for a perturbed stochastic wave equation defined on $[0,T]\times \mathbb{R}^{3}$ . This equation is driven by a Gaussian noise, white in time and correlated in space. The weak convergence approach plays an important role.

Self-Excited Vibrations for Damped and Delayed 1-Dimensional Wave Equations

It is shown that time delay induces self-excited vibrations in one dimensional damped wave equations, via Hopf bifurcation. In order to obtain classical solutions and $$C^{\infty }$$ smoothness with respect to the amplitude parameter, we avoid the use of any abstract Hopf bifurcation theorem, and instead work directly in Sobolev spaces on the 2-torus. As a result, a “derivative loss” problem arises, which is overcome using the regularizing effects of the time delay and damping. Only the classical implicit function theorem is used. The direction of bifurcation is also obtained.

Periodic solutions for one dimensional wave equation with bounded nonlinearity

This paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients. The constant coefficient model corresponds to the classical wave equation, while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For finding the periodic solutions of variable coefficient wave equation, it is usually required that the coefficient u(x) satisfies ess infηu(x)>0 with ηu(x)=12u″u−14(u′u)2, which actually excludes the classical constant coefficient model. For the case ηu(x)=0, it is indicated to remain an open problem by Barbu and Pavel (1997) [6]. In this work, for the periods having the form T=2p−1q (p,q are positive integers) and some types of boundary value conditions, we find some fundamental properties for the wave operator with either constant or variable coefficients. Based on these properties, we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded. Such nonlinearity may cross multiple eigenvalues of the corresponding wave operator. In particular, we do not require the condition ess infηu(x)>0.

Chaotic behaviors of one dimensional wave equations with van der Pol nonlinear boundary conditions

For one-dimensional linear wave equations with van der Pol boundary conditions, there have existed several different ways in the literature to characterize the complexity of their solutions. In this paper, we give a new way to characterize the dynamical behaviors of the systems, including almost equicontinuity and Devaney chaos. In order to do this, we consider a one-dimensional system (I, f) and its functional envelope (L1(I,I),Hf) and show that Hf is weakly mixing if and only if f is; Hf is Devaney chaotic if f is weakly mixing; Hf is almost equicontinuous if f has an attracting 2-period orbit. Those abstract results have their own significance and can be applied to the linear wave equations with van der Pol boundary conditions.

Successive approximation method for solving (1+1)-dimensional dispersive long wave equations

Successive approximation method for solving (1+1)-dimensional dispersive long wave equations