Long Wave Equation Research Articles

On similarity solutions to (2+1)-dispersive long-wave equations

This work is devoted to get a new family of analytical solutions of the (2+1)-coupled dispersive long wave equations propagating in an infinitely long channel with constant depth, and can be observed in an open sea or in wide channels. The solutions are obtained by using the invariance property of the similarity transformations method via one-parameter Lie group theory. The repeated use of the similarity transformations method can transform the system of PDEs into system of ODEs. Under adequate restrictions, the reduced system of ODEs is solved. Numerical simulation is performed to describe the solutions in a physically meaningful way. The profiles of the solutions are simulated by taking an appropriate choice of functions and constants involved therein. In each animation, a frame for dominated behavior is captured. They exhibit elastic multisolitons, single soliton, doubly solitons, stationary, kink and parabolic nature. The results are significant since these have confirmed some of the established results of S. Kumar et al. (2020) and K. Sharma et al. (2020). Some of their solutions can be deduced from the results derived in this work. Other results in the existing literature are different from those in this work.

Droplet motion on chemically heterogeneous substrates with mass transfer. I. Two-dimensional dynamics

We consider the dynamics of thin two-dimensional viscous droplets on chemically heterogeneous surfaces moving under the combined effects of slip, mass transfer and capillarity. The resulting long-wave evolution equation for the droplet thickness is treated analytically via the method of matched asymptotic expansions in the limit of slow mass transfer rates, quasi-static dynamics and vanishingly small slip lengths to deduce a lower-dimensional system of integrodifferential equations for the two moving fronts. We demonstrate that the predictions of the deduced system agree excellently with simulations of the full model for a number of representative cases within the domain of applicability of the analysis. Specifically, we focus on situations where the mass of the drop changes periodically to highlight a number of interesting features of the dynamics, which include stick-slip, hysteresis-like effects, as well as the possibility for the droplet to alternate between the constant-radius and constant-angle stages which have been previously reported in related works. These features of the dynamics are further scrutinized by investigating how the bifurcation structure of droplet equilibria evolves as the mass of the droplet varies.

New trigonometric B-spline approximation for numerical investigation of the regularized long-wave equation

Abstract The objective of this work is to propose a collocation technique based on new cubic trigonometric B-spline (NCTB-spline) functions to approximate the regularized long-wave (RLW) equation. This equation is used for modelling numerous problems occurring in applied sciences. The NCTB-spline collocation method is used to integrate the spatial derivatives. We use the Rubin–Graves linearization technique to linearize the non-linear term. The accuracy and efficiency of the technique are examined by employing it on three important numerical examples which have three invariants of motion viz. mass, momentum, and energy. It is observed that the error norms of the present method are less than the error norms of the methods available in the literature. The numerical values of these invariants have also been approximated, which remain conserved during the program run which shows that the propagation of the solitary wave is represented perfectly. The propagation of one and two solitary waves and undulations of waves are depicted graphically. The stability analysis shows that the method is unconditionally stable.

Long-wave equation for a confined ferrofluid interface: periodic interfacial waves as dissipative solitons

We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted non-uniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports travelling waves, governed by a novel modified Kuramoto–Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equation allows for the existence of dissipative solitons. These permanent travelling waves’ propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The travelling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the travelling waves. Interestingly, multi-periodic waves, which are a non-integrable analogue of the double cnoidal wave, are also found to propagate under the model long-wave equation. These multi-periodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stable periodic states identified.

Classes of new analytical soliton solutions to some nonlinear evolution equations

Undoubtedly, all applied sciences are closely corresponding to differential equations, especially partial differential equations. This contribution aims to examine the efficiency of a modified version of the generalized exponential rational function method in solving some well-known nonlinear evolution equations. The investigated models in this article include the Zakharov–Kuznetsov, the cubic Boussinesq, and the modified regularized long-wave equation. In the structure of these solutions, various familiar elementary functions, including exponential, trigonometric, and hyperbolic functions are used. This important feature makes it easy to use these solutions in real applications. Numerical behaviors corresponding to the obtained solutions have been demonstrated through some three-dimensional diagrams in the article. This technique can be also adopted in determining wave soliton solution to other equations with partial derivatives.

The solution of the long-wave equation for various nonlinear depth and breadth profiles in the power-law form

Long waves bring many important challenges in the ocean and coastal engineering, including but are not limited to harbor resonance and run-up. Therefore, understanding and modeling their dynamics is crucially important. Although their dynamics over various types of geometries are well-studied in the literature, the study of the geometries with power-law variations remains an open problem in this setting. With this motivation, in this paper, we derive the exact analytical solutions of the long-wave equation over nonlinear depth and breadth profiles having power-law forms given by h(x)=c1xa and b(x)=c2xc, where the parameters c1,c2,a,c are some constants. We show that for these types of power-law forms of depth and breadth profiles, the long-wave equation admits solutions in terms of Bessel functions and Cauchy–Euler series. We also derive the seiching periods and resonance conditions for these forms of depth and breadth variations. Our results can be used to investigate the long-wave dynamics and their envelope characteristics over equilibrium beach profiles, the effects of nonlinear harbor entrances and angled nonlinear seawall breadth variations in the power-law forms on these dynamics, and the effects of reconstruction, geomorphological changes, sedimentation, and dredging to harbor resonance, to the shift in resonance periods and to the seiching characteristics in lakes and barrages.

Investigation of adequate closed form travelling wave solution to the space-time fractional non-linear evolution equations

This work aims to construct exact solutions for the space-time fractional (2 + 1)- dimensional dispersive longwave (DLW) equation and approximate long water wave equation (ALW) utilizing the two-variable (G′/G,1/G)-expansion method and the modified Riemann–Liouville fractional derivative. The recommended equations play a significant role to describe the travel of the shallow water wave. The fractional complex transform is used to convert fractional differential equations into ordinary differential equations. Several wave solutions have been successfully achieved using the proposed approach and the symbolic computer Maple package. The Maple package program was used to set up and validate all of the computations in this investigation. By choosing particular values of the embedded parameters, we produce multiple periodic solutions, periodic wave solutions, single soliton solutions, kink wave solutions, and more forms of soliton solutions. The achieved solutions might be useful to comprehend nonlinear phenomena. It is worth noting that the implemented method for solving nonlinear fractional partial differential equations (NLFPDEs) is efficient, and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering.

Kink-Type Solution for One Modification of the Regularized Long-Wave Equation

Kink-Type Solution for One Modification of the Regularized Long-Wave Equation

Comments on the paper ”A conservative linear difference scheme for the 2D regularized long-wave equation”, by Xiaofeng Wang, Weizhong Dai and Shuangbing Guo [Applied Mathematics and Computation, 342 (2019) 55-70

In the above article, a second-order convergence in the maximum norm of the two-dimensional regularized long-wave equation is proved. However, due to a wrong inequality in Lemma 3.3 used in X. Wang et al.[1], there are some fundamental errors in this paper, in particular the proof of Theorem 3.3 and the convergence Theorem are no longer correct. The present brief paper contains clarifying comments.

The cubic B-spline least-squares finite-element method for the numerical solutions of regularized long-wave equation

A new least-squares method based on cubic B-spline functions is presented to solve regularized long-wave (RLW) equation numerically. An algebraic system of equations is obtained by employing the least-squares method based on cubic B-spline approximation for spatial discretization and the Crank–Nicolson technique for time discretization for integrating the RLW equation. The two test problems are implemented to show suitability of the proposed scheme. New results have been compared with those of some previous studies available in the literature.

On equatorial wave–current interactions

We use the Hamiltonian structure of equatorial wave-current interactions in the case of non-zero constant vorticity in the near-surface homogeneous layer, studying the dispersion relation for linear long waves outrunning the currents (downstream waves). We also investigate the behavior of the lifespan of the solution of the evolution equation for nonlinear long internal waves with respect to vorticity values.

Solitary wave patterns and conservation laws of fourth-order nonlinear symmetric regularized long-wave equation arising in plasma

In this paper, we study a fourth-order nonlinear symmetric regularized long-wave equation which arises in several physical applications including ion sound waves in plasma. This model also describes nonlinear ion-acoustic and space-charge waves. Lie group analysis and the tanh-coth method is utilized for the construction of solitary wave solutions to the equation. Based on each symmetry, we reduce the considered partial differential equation into different ordinary differential equations. The obtained equations are solved and new solitary wave patterns are reported. To understand the physical interpretation, some solitary wave structures are exhibited graphically. Also, by using the multiplier approach, we constructed the local conservation laws of the considered equation.

Droplets on the liquid substrate: Thermocapillary oscillatory instability

We investigate the influence of heating on the shape and dynamics of a droplet on a liquid substrate. The problem is studied numerically in the framework of longwave amplitude equations. The effect of gravity is taken into account. We show that the thermocapillary stresses produce a significant deformation of the droplet's interfaces. Heating the droplet from above can create an oscillatory instability generating periodic and quasiperiodic oscillations, change of droplet shape, and droplet decomposition. Heating from below can lead to the substrate layer's rupture due to its monotonic instability.

Traveling fronts in dissipative granular chains and nonlinear lattices

We consider an infinite chain of particles with nonlinear elastic and dissipative nearest neighbors interactions. Assuming the existence of a traveling front between uniformly compressed (or stretched) states, we obtain jump conditions relating the wave speed and limiting particle velocities to the relative displacements at infinity. Using this result, we characterise compression fronts in chains of touching beads, for viscoelastic contact laws that include a nonlinear elastic force (generalised Hertz contact) and viscous dissipation. We compute compression fronts numerically for the generalised Kuwabara–Kono model in which the viscous contact force is proportional to the derivative of the elastic force, without precompression of the chain. Steady fronts are obtained both as the end result of the compression of one end of the chain and using a shooting method which provides numerically exact traveling waves. Depending on the magnitudes of contact damping and strain applied downstream, we obtain either overdamped (monotonic) or underdamped (oscillatory) compression fronts. To explain this transition and approximate the front profiles, we consider a continuum limit valid when the exponent of the contact nonlinearity is close to (and above) unity. Using multiscale expansions, we formally derive two different amplitude equations for long waves, a Burgers equation with logarithmic nonlinearity, and a logarithmic Korteweg–de Vries (KdV)–Burgers equation for small contact damping. Both models possess traveling front solutions that are in good agreement with the front profiles computed numerically in the granular chain. The analysis of the logarithmic KdV–Burgers equation allows one to approximate the critical damping corresponding to the transition from underdamped to overdamped fronts.

The Sloshing and Diapycnal Meridional Overturning Circulations in the Indian Ocean

AbstractThe meridional overturning circulation (MOC) seasonality in the Indian Ocean is investigated with the ocean state estimate product ECCO v4r3. The vertical movements of water parcels are predominantly due to the heaving of the isopycnals all over the basin except off the western coast. Aided by the linear propagation equation of long baroclinic Rossby waves, the driving factor determining the strength of the seasonal MOC in the Indian Ocean is identified as the zonally integrated Ekman pumping anomaly, rather than the Ekman transport concluded in earlier studies. A new concept of sloshing MOC is proposed, and its difference with the classic Eulerian MOC leads to the so-called diapycnal MOC. The striking resemblance of the Eulerian and sloshing MOCs implies the seasonal variation of the Eulerian MOC in the Indian Ocean is a sloshing mode. The shallow overturning cells manifest themselves in the diapycnal MOC as the most remarkable structure. New perspectives on the upwelling branch of the shallow overturn in the Indian Ocean are offered based on diapycnal vertical velocity. The discrepancy among the observation-based estimates on the bottom inflow across 32°S of the basin is interpreted with the seasonal sloshing mode. Consequently, the “missing mixing” in the deep Indian Ocean is attributed to the overestimated diapycnal volume fluxes. Decomposition of meridional heat transport (MHT) into sloshing and diapycnal components clearly shows the dominant mechanism of MHT in the Indian Ocean in various seasons.

Lie Symmetry Analysis and Explicit Solutions for the Time-Fractional Regularized Long-Wave Equation

This paper systematically investigates the Lie group analysis method of the time-fractional regularized long-wave (RLW) equation with Riemann–Liouville fractional derivative. The vector fields and similarity reductions of the time-fractional (RLW) equation are obtained. It is shown that the governing equation can be transformed into a fractional ordinary differential equation with a new independent variable, where the fractional derivatives are in Erdèlyi–Kober sense. Furthermore, the explicit analytic solutions of the time-fractional (RLW) equation are obtained using the power series expansion method. Finally, some graphical features were presented to give a visual interpretation of the solutions.

Corrigendum to “A conservative linear difference scheme for the 2D regularized long-wave equation” [Appl. Math. Comput. 342 (2019) 55–70

The authors found a minor incorrect citation (Lemma 3.3) in their above titled paper which should be corrected in this Corrigendum.

Conservative Difference Scheme of Solitary Wave Solutions of the Generalized Regularized Long-Wave Equation

Conservative difference scheme for the nonlinear dispersive generalized regularized long-wave (GRLW) equation is proposed. Existence of its difference solutions has been shown. It is proved by the discrete energy method that the difference scheme is uniquely solvable, unconditionally stable and the convergence is of second-order in the maximum norm. The particular case known as the modified regularized long-wave (MRLW) equation is also discussed numerically in details. Furthemore, three invariants of motion are evaluated to determine the conservation properties of the problem. Interaction of two and three solitary waves are shown. Some numerical examples are given in order to validate the theoretical results.

Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves

<p style='text-indent:20px;'>This paper aimed at obtaining the traveling-wave solution of the nonlinear time fractional regularized long-wave equation. In this approach, firstly, the time fractional derivative is accomplished using a finite difference with convergence order <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{O}(\delta t^{2-\alpha}) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ 0 < \alpha< 1 $\end{document}</tex-math></inline-formula> and the nonlinear term is linearized by a linearization technique. Then, the spatial terms are approximated with the help of the radial basis function (RBF). Numerical stability of the method is analyzed by applying the Von-Neumann linear stability analysis. Three invariant quantities corresponding to mass, momentum and energy are evaluated for further validation. Numerical results demonstrate the accuracy and validity of the proposed method.</p>

Traveling wave solutions for space-time fractional Cahn Hilliard equation and space-time fractional symmetric regularized long-wave equation

Over the last few years, a direct method known as the new auxiliary method has been shown to solve non-linear partial differential equations effectively where this method can give more exact solutions compared to the traditional direct methods. Due to these promising results, efforts are now being made to extend this method in solving the more complicated non-linear fractional partial differential equations. The new auxiliary method is a very powerful, felicitous, and effective method to get exact and traveling wave solutions of partial differential equations. Therefore, we apply the new auxiliary method to solve for the traveling wave solutions of the space-time fractional Cahn-Hilliard equation and the space-time fractional symmetric regularized long-wave equation. Furthermore, it is considered as more general method for solving that time of equations because it is shown that more exact solutions for the traveling wave on these equations will be obtained with the application of the new auxiliary method.