where α, β are positive constants and p is a positive integer . The GRLW equation was ﬁrst put forward by Peregrine  and Benjamin et al.  as a model for small-amplitude long waves on the surface of water in a channel. Many authors [–] have recently stud- ied models for long waves in nonlinear dispersive systems. When p = , (.) is usually called the RLW equation. When p = , (.) is called a modiﬁed regularizedlong-wave (MRLW) equation. Various numerical techniques have been developed to solve the equa- tion. These partly include the ﬁnite diﬀerence method, ﬁnite element methods, the least squares method, and a collocation method with quadratic B-splines, cubic B-splines and septic splines; we refer to [–], and references therein.
where δ and µ are positive parameters, is known as regularizedlongwave (RLW) equa- tion. The equation was ﬁrst introduced by Peregrine  to describe the development of an undular bore. This equation is one of the most improtant equations of the nonlinear dispersive waves having many applications in diﬀerent areas, including ion-acoustic and magneto hydrodynamic waves in plasma, the transverse waves in shallow water, phonon packets in non-linear crystals, pressure waves in liquid-gas bubble mixtures and rotating ﬂow down a tube. Benjamin et al.  also introduced a mathematical theory of the equa- tion. Bona and Pryant  have discussed the existence and uniqueness of the equation. There are few analytical solutions available in the literature. Thus, the numerical solu- tions of the RLW equation have been subject of many papers. Various numerical studies including ﬁnite diﬀerence [4-7], ﬁnite element [8-21] and pseudo-spectral method have been reported recently. A special property of the equation is the fact that the solutions
We study the initial-boundary problem of dissipative symmetric regularizedlongwave equations with damping term by finite di ﬀ erence method. A linear three-level implicit finite di ﬀ erence scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite diﬀerence scheme is of second-order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify that the method is accurate and e ﬃ cient.
 T. S. El-Danaf, M. A. Ramadan and F. E. I. AbdAlaal, “The Use of Adomian Decomposition Method for Solv- ing the RegularizedLong-Wave Equation,” Chaos, Soli- tons & Fractals, Vol. 26, No. 3, 2005, pp. 747-757. doi:10.1016/j.chaos.2005.02.012
the power and elegance of the present method, we compared our result with the exact travelling wave solution of the symmetric regularizedlong-wave equation with quadratic nonlinearity. These results show that for weakly nonlinear case the solutions for both approaches coincide with each other. The present method is seen to be fairly simple as compared to the renormalization method of Kodama and Taniuti  and the multiple scale expansion method of Kraenkel et al .
This paper investigates the solitary wave solutions of the (2+1)-dimensional regularizedlong-wave (2DRLW) equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas and (2+1) dimensional Davey-Stewartson (DS) equation which is governing the dynamics of weakly nonlinear modulation of a lattice wave packet in a multidimensional lattice. By using extended mapping method technique, we have shown that the 2DRLW-2DDS equations can be reduced to the elliptic-like equation. Then, the extended mapping method is used to obtain a series of solutions including the single and the combined non degenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear partial differential equations (NLPDEs). Keywords: Exact Solitary Solutions; Extended Mapping Method; Two Dimension RegularizedLongWave
lution at diﬀerent times for both cases are drawn in Figure and Figure for the RLW equation, and Figure and Figure for the MRLW equation, respectively. Furthermore, the quantities for three conservative laws are presented in Table . As seen from these results, an initial date with Maxwellian disturbance will resolve into a sequence of solitary waves in the stable range ordered by amplitude with the larger waves in the front, followed by a dispersive tail. Clearly, the presented results are consistent with earlier work on this topic in [, ].
This superiority arises because, unlike the KdV equation, the dispersion relation associ- ated with the linearized RLW equation yields the frequency that is bounded for large wave numbers . But they have found an analytical solution of the RLW equation under the restricted initial and boundary conditions. So, various numerical techniques have been introduced to solve the equation. These include the ﬁnite diﬀerence [–], ﬁnite element [–], Fourier pseudo-spectral  methods and the meshfree method . One of the special properties of the equation is that the solutions may exhibit solitons whose mag- nitudes, shapes and velocities are not changed after the collision. The RLW equation is a special case of the generalized longwave (GRLW) equation having the form
where Q() and E() are two positive constants which relate to the initial condition. Existence and uniqueness of the solution of the RLW equation are given in . Its analytical solution was found  under restricted initial and boundary conditions, and, therefore, it became interesting from a numerical point of view. Some numeri- cal methods for the solution of the RLW equation such as variational iteration method [, ], ﬁnite-diﬀerence method [–], Fourier pseudospectral method , ﬁnite element method [–], collocation method  and adomian decomposition method  have been introduced in many works. In , Li and Vu-Quoc pointed out that ‘in some ar- eas, the ability to preserve some invariant properties of the original diﬀerential equa- tion is a criterion to judge the success of a numerical simulation.’ Meanwhile, Zhang et al.  thought that the conservative diﬀerence schemes perform better than the non- conservative ones, and the non-conservative diﬀerence schemes may easily show non- linear ‘blow-up.’ Hence, constructing a conservative diﬀerence scheme for the numerical solution of the nonlinear partial diﬀerential equation is quite signiﬁcant. In this paper, coupled with the Richardson extrapolation, a two-level nonlinear Crank-Nicolson ﬁnite diﬀerence scheme for problems (.)-(.), which has the accuracy of O(τ + h ) without
It is known the conservative scheme is better than the nonconservative ones. Zhang et al. 1 point out that the nonconservative scheme may easily show nonlinear blow up. In 2 Li and Vu-Quoc said “. . . in some areas, the ability to preserve some invariant properties of the original diﬀerential equation is a criterion to judge the success of a numerical simulation”. In 3–11 , some conservative finite diﬀerence schemes were used for a system of the generalized nonlinear Schr ¨odinger equations, RegularizedlongwaveRLW equations, Sine-Gordon equation, Klein-Gordon equation, Zakharov equations, Rosenau equation, respectively. Numerical results of all the schemes are very good. Hence, we propose a new conservative diﬀerence scheme for the general Rosenau-RLW equation, which simulates conservative laws 1.4 and 1.5 at the same time. The outline of the paper is as follows. In Section 2, a nonlinear diﬀerence scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, some numerical experiments are shown.
Due to its properties, the KdV equation was the source of many applications and results in a large area of nonlinear physics . Certain theoretical physics phenomena in the quantum mechanics domain are explained by means of a KdV model. It is used in fluid dynamics, aero- dynamics, and continuum mechanics as a model for shock wave formation, solitons, turbulence, boundary layer behavior, and mass transport. The alternative equation of the non-linear dis- persive waves to the more usual KdV-equation, modelled to govern a large number of physical phenomena such as shallow waters and plasma waves, is the Regularised LongWave (RLW) equation u˙t+ u˙x+u u˙x- u˙xxt=0, The regular- ized longwave (RLW) equation belongs to a class of the nonlinear evolution equations which pro- vide good models for predicting a variety of phys- ical phenomena. Solitary waves are wave pack- ets or pulses which propagate in nonlinear media. Due to dynamical balance between the non-linear and dispersive effects these waves retain a sta- ble waveform. The regularizedlongwave (RLW) equation was originally introduced to describe the behavior of the undular bore . It has also been derived from the study of water waves and ion acoustic plasma waves. It was first proposed by
The exact solutions of some conformable time fractional PDEs are pre- sented explicitly. The modified Kudryashov method is applied to construct the solutions to the conformable time fractional RegularizedLongWave- Burgers (RLW-Burgers, potential Korteweg-de Vries (KdV) and clannish random walker’s parabolic (CRWP) equations. Initially, the predicted so- lution in the finite series of a rational form of an exponential function is substituted to the ODE generated from the conformable time fractional PDE by using wave transformation. The coefficients used in the finite series are determined by solving the algebraic system derived from the coefficients of the powers of the predicted solution.
In this work, our main purpose is to develop of a suﬃciently robust, accurate and eﬃcient numerical scheme for the solution of the regularizedlongwave (RLW) equation, an important partial diﬀerential equation with quadratic nonlinearity, describing a large number of physical phenomena. The crucial idea is based on the discretization of the RLW equation with the aid of a combination of the discontinuous Galerkin method for the space semi-discretization and the backward diﬀerence formula for the time discretization. Furthermore, a suitable linearization preserves a linear algebraic problem at each time level. We present error analysis of the proposed scheme for the case of nonsymmetric discretization of the dispersive term. The appended numerical experiments conﬁrm theoretical results and investigate the conservative properties of the RLW equation related to mass, momentum and energy. Both procedures illustrate the potency of the scheme consequently.
Multi-decadal wind-wave hindcasts have become a common tool in supporting the assessment of wave climate variability and change, such as its extremes, trends or seasonal and inter- annual to decadal variability (e.g. WASA-Group, 1998; Sterl et al., 1998; Cox and Swail, 2001; Weisse and Günther, 2007; Dodet et al., 2010). Data from wind-wave hindcasts are also frequently used in practically oriented applications such as in navigation, shipbuilding, offshore design or strategic plan- ning of logistics for the operation of future offshore wind farms (Weisse et al., 2009, 2015). Further applications com- prise studies such as evaluating the impact of waves on sea salt emissions (Neumann et al., 2016) or the evaluation of the potential success or failure of different response strategies to oil pollution (Schwichtenberg et al., 2016). For all these dif- ferent types of applications long, homogeneous and consis- tent wind-wave data are needed to derive robust estimates of wind-wave related parameters specific to the problem. Often such information is unavailable from in situ or satellite data
A BSTRACT . In this paper, the coupled dispersive (2+1)-dimensional longwave equa- tion is studied. The traveling wave hypothesis yields complexiton solutions. Subse- quently, the wave equation is studied with power law nonlinearity where the ansatz method is applied to yield solitary wave solutions. The constraint conditions for the existence of solitons naturally fall out of the derivation of the soliton solution.
for source separation is found as 2. From these experiments it was shown that Regularized EMML algorithm outperforms the Regu- larized ISRA,IS-NMF and SNMF algorithms for NMF-based sin- gle channel speech and music separation when complexity of the mixture increases. But the computation time of the algorithm is comparatively smaller for SNMF, so mixture with only two under- lying sources SNMF outperforms the other three algorithms. Even though all the NMF algorithms are itself easy to implement and compute, makes NMF good for BSS method.
In this paper we compare research into several methods to both reduce the noise in the ESF estimation and to accurately model usable PSFs for the acquisition system. In Section 2 we review PSF measurement techniques and describe how a super- resolution ESF, and eventually a PSF is computed from an ensemble of low-resolution measurements. In Section 3 we examine the theoretical PSF models that result from consideration of both geometric and diffraction optics. In Section 4 we describe improvements to the traditional super-resolution PSF measurement technique that involve:- (i) Compensation for non-uniform illumination within the light box used to produce the test images; (ii) A regularized numerical differentiation process to limit noise in the computed PSF; (iii) Models of the ESF that have been developed and used to compute PSFs that have then been compared with the theoretical models described in Section 2. Fitting the correct ESF model to the measured data is key to obtaining accurate PSFs for the system. Section 5 presents the experimental results from both focused and defocused systems. Specific 1-D results have been used to
Below canopy irradiance in terrestrial ecosystems are often associated with significant spatial heterogeneity. Therefore, monitoring of below-canopy PPFD (PPFD_ BC_IN) that allows a robust estimation of the mean PPFD_BC_IN for a given ecosystem, requires using a mo- bile automated systems (such as tram-systems) that allow a representative spatial sampling of PPFD_BC_IN with a limited number of sensors, or a rather large number of sensors spatially distributed at fixed-locations. Due to the cost and inherent problems of maintenance of mobile auto- mated systems for long-term measurements, we discard their systematic use at ICOS terrestrial ecosystem stations. Therefore, we only consider the use of spatially distrib- uted sensors at fixed-locations. This manuscript provides only basic guidelines and very minimal requirements for measurement of PPFD_BC_IN and should not be seen as a complete state-of-the-art protocol for PPFD_IN_BC measurements.
The analytical study of long-wave scattering in a canal with a rapidly varying cross- section is presented. It is assumed that waves propagate on a stationary current with a given flow rate. Due to the fixed flow rate, the current speed is different in the different sections of the canal, upstream and downstream. The scattering coefficients (the transmission and reflection coefficients) are calculated for all possible orientations of incident wave with respect to the background current (downstream and upstream propagation) and for all possible regimes of current (subcritical, transcritical, and supercritical). It is shown that in some cases negative energy waves can appear in the process of waves scattering. The conditions are found when the over-reflection and over-transmission phenomena occur. In particular, it is shown that a spontaneous wave generation can arise in a transcritical accelerating flow, when the background current enhances due to the canal narrowing. This resembles a spontaneous wave generation on the horizon of an evaporating black hole due to the Hawking effect.
(.). This means that in some limit, t → ±∞ or x → ±∞ or in both, this new solution will approach the original plane-wave solution, up to some phase shift. It is shown that this solution, given by (.), represents a kind of homoclinic solution and meanwhile contains a periodic wave cosξ whose amplitude periodically oscillates with the evolution of time. So this solution represented by (p, q, b) of (.) is a homoclinic breather solution. The trajectory of these solutions is deﬁned explicitly by