Dispersive Wave Equations Research Articles

Solitary Wave Solutions of Some Coupled Nonlinear Evolution Equations

In this article, the modified simple equation (MSE) method has been executed to find the traveling wave solutions of the coupled (1+1)-dimensional Broer-Kaup (BK) equations and the dispersive long wave (DLW) equations. The efficiency of the method for finding exact solutions has been demonstrated. It has been shown that the method is direct, effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations. Keywords: MSE method; NLEE; BK equations; DLW equations; Solitary wave solutions. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i2.16671 J. Sci. Res. 6 (2), 273-284 (2014)

Blowup issues for a class of nonlinear dispersive wave equations

In this paper we consider the nonlinear dispersive wave equation on the real line, ut−utxx+[f(u)]x−[f(u)]xxx+[g(u)+f″(u)2ux2]x=0, that for appropriate choices of the functions f and g includes well known models, such as Dai's equation for the study of vibrations inside elastic rods or the Camassa–Holm equation modelling water wave propagation in shallow water. We establish a local-in-space blowup criterion (i.e., a criterion involving only the properties of the data u0 in a neighborhood of a single point) simplifying and extending earlier blowup criteria for this equation. Our arguments apply both to the finite and infinite energy cases, yielding the finite time blowup of strong solutions with possibly different behavior as x→+∞ and x→−∞.

The Investigation of Exact Solutions for the Appropriate Type of the Dispersive Long Wave Equation

Improved (G'/G)-expansion and first integral methods are used to construct exact solutions of the 2+1-dimensional Eckhaus-type extension of the dispersive long wave equation. The (G'/G)-expansion method is based on the assumptions that the travelling wave solutions can be expressed by a polynomial in (G'/G) and the first integral method is based on the theory of commutative algebra in which Division Theorem is of concern. It is worth mentioning that these methods are used for different systems and those two different systems can both be reduced to a system that will be mentioned in this paper. To recapitulate, this investigation has resulted in the exact solutions of the given systems.

Dynamics of dispersive long waves in fluids

We obtain many different variable separation solutions for (2+1)-dimensional variable coefficient dispersive long wave equation by means of five different methods, including the multilinear variable separation approach, the projective Ricatti equation method, the extended projective Ricatti equation method, the extended tanh-function method and the improved tanh-function method. However, by careful analysis, we find that variable separation solution obtained by the multilinear variable separation approach includes all variable separation solutions obtained by other four direct methods. Thus variable separation solution for (2+1)-dimensional variable coefficient dispersive long wave equation exists a uniform form. Based on this uniform variable separation solution, we discuss the completely elastic interaction between foldons, the non-completely elastic interaction between bell-like semi-foldon, peaked semi-foldon and foldon, and the completely non-elastic interaction between bell-like semi-foldon and peaked semi-foldon. These results are helpful to analyze more precisely nonlinear and dispersive long gravity waves traveling in two horizontal directions, such as the bubbles on (or under) a fluid surface and folded waves in various ocean waves.

KP solitons and total positivity for the Grassmannian

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional dispersive wave equation now known as the KP equation. It is well-known that one can use the Wronskian method to construct a soliton solution to the KP equation from each point of the real Grassmannian Gr_kn. More recently several authors have studied the regular solutions that one obtains in this way: these come from points of the totally non-negative part of the Grassmannian (Gr_kn)_{>= 0}. In this paper we exhibit a surprising connection between the theory of total positivity for the Grassmannian, and the structure of regular soliton solutions to the KP equation. By exploiting this connection, we obtain new insights into the structure of KP solitons, as well as new interpretations of the combinatorial objects indexing cells of (Gr_kn)_{>= 0}. In particular, we completely classify the spatial patterns of the soliton solutions coming from (Gr_2n)_{>0}, as well as those coming from (Gr_kn)_{>= 0} when the absolute value of the time parameter is sufficiently large. We also demonstrate an intriguing connection between soliton graphs for (Gr_kn)_{>0} and the cluster algebras of Fomin and Zelevinsky, and we use this connection to solve the inverse problem for generic KP solitons coming from (Gr_kn)_{>0}. Finally we construct all the soliton graphs for (Gr_2n)_{>0} using the triangulations of n-gon.

Exact soliton solutions for \((2+1)\)-dimensional dispersive long wave equation

In this paper, the first integral method is used to construct exact traveling wave solutions of $(2+1)-$ dimensional dispersive long wave equation. The first integral method is an efficient method for obtaining exact solutions some of nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones.

New exact solutions and complex wave excitations for the (2+1)-dimensional dispersive long wave equation

By the Riccati equation expansion method and a variable separation method, a series of variable separation solutions of the (2+1)-dimensional dispersive long wave equation is derived. According to the derived solitary wave solution, we obtain some nove complex wave localized structures and study the time evolutions of complex waves.

Analytical Solutions for the Elastic Circular Rod Nonlinear Wave, Boussinesq, and Dispersive Long Wave Equations

The solving processes of the homogeneous balance method, Jacobi elliptic function expansion method, fixed point method, and modified mapping method are introduced in this paper. By using four different methods, the exact solutions of nonlinear wave equation of a finite deformation elastic circular rod, Boussinesq equations and dispersive long wave equations are studied. In the discussion, the more physical specifications of these nonlinear equations, have been identified and the results indicated that these methods (especially the fixed point method) can be used to solve other similar nonlinear wave equations.

Symmetry Reduction and Explicit Solutions of the (2 + 1)-Dimensional DLW Equation

Utilizing the Clarkson-Kruskal direct method, the symmetry of the (2 + 1)-dimensional dispersive long wave equation is derived. From which, through solving the characteristic equations, four types of the explicit reduction solutions that related the hyperbolic tangent function are obtained. Finally, several soliton excitations are depicted from one of the solutions.

On Blow-Up Structures for a Generalized Periodic Nonlinearly Dispersive Wave Equation

The local well-posedness for a generalized periodic nonlinearly dispersive wave equation is established. Under suitable assumptions on initial valueu0, a precise blow-up scenario and several sufficient conditions about blow-up results to the equation are presented.

Bloch-Wave Homogenization on Large Time Scales and Dispersive Effective Wave Equations

We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in $\mathbb{R}^n$, $n\in\{1,2,3\}$. Standard homogenization theory provides, for the limit of a small periodicity length $\varepsilon>0$, an effective second order wave equation that describes solutions on time intervals $[0,T]$. In order to approximate solutions on large time intervals $[0,T\varepsilon^{-2}]$, one has to use a dispersive, higher order wave equation. In this work, we provide a well-posed, weakly dispersive effective equation and an estimate for errors between the solution of the original heterogeneous problem and the solution of the dispersive wave equation. We use Bloch-wave analysis to identify a family of relevant limit models and introduce an approach to select a well-posed effective model under symmetry assumptions on the periodic structure. The analytical results are confirmed and illustrated by numerical tests.

Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations

In this paper we study the global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, utt−Luxx=B(−|u|p−1u)xx,(p>1), where the nonlocality enters through two pseudo-differential operators L and B. We establish thresholds for global existence versus blow-up using the potential well method which relies essentially on the ideas suggested by Payne and Sattinger. Our results improve the global existence and blow-up results given in the literature for the present class of nonlocal nonlinear wave equations and cover those given for many well-known nonlinear dispersive wave equations such as the so-called double-dispersion equation and the traditional Boussinesq-type equations, as special cases.

A self-adjusting algorithm for solitary wave simulations

We introduce a practical algorithm to automate the simulations of solitary wave solutions of some nonlinear dispersive wave equations. The full discretization consists of a spatial discretization with a local basis and an invariant preserving time integrator. The algorithm includes a dynamic cleaning of dispersive tails and an automatic detection of the complete separation of the main pulses.

Exact solutions for fractional partial differential equations by a new fractional sub-equation method

In this paper, we propose a new fractional sub-equation method for finding exact solutions of fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative, which is the fractional version of the known (G′/G) method. To illustrate the validity of this method, we apply it to the space-time fractional Fokas equation, the space-time fractional -dimensional dispersive long wave equations and the space-time fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established. MSC:35Q51, 35Q53.

Decay kinematics in the aether: Pion decay, kaon decay, and superluminal Cherenkov radiation

Pion and kaon decay, π → μ + νμ, K → μ + νμ, into muons and GeV neutrinos is investigated with regard to a possible superluminal neutrino speed. Photonic Cherenkov emission by superluminal high-energy charges is shown to be forbidden by causality violation. A proper causality interpretation of decay processes outside the lightcone involving superluminal particles requires an absolute spacetime conception based on a distinguished frame of reference (aether frame). The universal reference frame is physically manifested as the rest frame of the cosmic microwave background (CMB) radiation. The propagation of particles and radiation modes in the CMB rest frame is determined by dispersive wave equations coupled to isotropic permeability tensors. The decay of ∼50 GeV pions and ∼85 GeV kaons generating the CERN neutrino beam to Gran Sasso (CNGS) is analyzed in this context. Causality constraints on the group velocity of the ∼17 GeV muon neutrinos produced in the decay are derived and compared to recent experimental bounds on the neutrino speed.

On the Galilean Invariance of Some Nonlinear Dispersive Wave Equations

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of these properties is the Galilean symmetry, that is, the invariance under Galilean transformations. In this paper, a mechanism to incorporate Galilean invariance in classical water wave models is proposed. The technique is applied to the Benajmin–Bona–Mahony (BBM) equation and the Peregrine (classical Boussinesq) system, leading to the corresponding Galilean invariant versions of these models. Some properties of the new equations are presented, with special emphasis on the computation and interaction of solitary wave solutions. A comparison with the Euler equations demonstrates the relevance of the Galilean invariance in the description of water waves.

Homotopy analysis and homotopy Padé methods for (1+1) and (2+1)‐dimensional dispersive long wave equations

PurposeThe purpose of this paper is to obtain analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations by the homotopy analysis and the homotopy Padé methods.Design/methodology/approachThe obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series.FindingsThe approximation solutions by [m,m] homotopy Padé technique is often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.Originality/valueIn this paper, analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations are obtained by the homotopy analysis and the homotopy Padé methods. The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m,m] homotopy Padé technique are often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.

Periodic traveling--wave solutions of nonlinear dispersive evolution equations

For a general class of nonlinear, dispersive wave equations, existence of periodic, traveling-wave solutions is studied. These travelingwaveforms are the analog of the classical cnoidal-wave solutions of the Korteweg-de Vriesequation. They are determined to bestable to perturbation of the same period. Their large wavelengthlimit is shown to be solitary waves.

A 2.5D boundary element formulation for modeling damped waves in arbitrary cross-section waveguides and cavities

A regularized 2.5D boundary element method (BEM) is proposed to predict the dispersion properties of damped stress guided waves in waveguides and cavities of arbitrary cross-section. The wave attenuation, induced by material damping, is introduced using linear viscoelastic constitutive relations and described in a spatial manner by the imaginary component of the axial wavenumber. The discretized dispersive wave equation results in a nonlinear eigenvalue problem, which is solved obtaining complex axial wavenumbers for a fixed frequency using a contour integral algorithm. Due to the singular characteristics and the multivalued feature of the wave equation, the requirement of holomorphicity inside the contour region over the complex wavenumber plane is fulfilled by the introduction of the Sommerfeld branch cuts and by the choice of the permissible Riemann sheets. A post processing analysis is developed for the extraction of the energy velocity of propagative guided waves. The reliability of the method is demonstrated by comparing the results obtained for a rail and a bar with square cross-section with those obtained from a 2.5D Finite Element formulation also known in literature as Semi Analytical Finite Element (SAFE) method. Next, to show the potential of the proposed numerical framework, dispersion properties are predicted for surface waves propagating along cylindrical cavities of arbitrary cross-section. It is demonstrated that the attenuation of surface waves approaches asymptotically the attenuation of Rayleigh waves.

Optimal distributed control of a nonlinear viscous dispersive wave equation for unidirectional propagation of a long wave

This paper is concerned with an optimal distributed control problem of a nonlinear viscous dispersive wave equation that approximately describes the unidirectional propagation of long waves. By the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the investigational system. The necessary condition for optimality is established for the controlled object in a fixed final horizon case and, subsequently, a remark on the applicability of the obtained results is made for the illustration.