Progressive Wave Solutions Research Articles

New Traveling Wave Solutions to the Simplified Modified Camassa–Holm Equation and the Landau-Ginsburg-Higgs Equation

Researchers are interested in the (1+1)-dimensional Camassa-Holm and Landau-Ginzburg-Higgs equations as they allow for the study of unidirectional wave propagation in shallow waters with a flat seabed, as well as nonlinear media exhibiting dispersion systems and superconductivity. This work has effectively developed exact wave solutions to the stated models, which may have significant consequences for characterising the nonlinear dynamical behaviour related to the phenomena. The extended -expansion technique is employed to procure a diverse array of progressive wave solutions characterized by hyperbolic, trigonometric, and rational functions. The solutions are shown as 3D profiles with a variety of shapes, including kink, singular kink, periodic, singular periodic, etc. The physical significance of the solutions is discussed by these plots, and the approach used in this study is considered efficient and capable of finding analytical solutions for the nonlinear models. Dhaka Univ. J. Sci. 72(1): 7-13, 2024 (January)

New investigation of the analytical behaviors for some nonlinear PDEs in mathematical physics and modern engineering

Numerous complex situations arising from science and engineering are modeled by integral-differential equations (IDEs). Examples of these situations include mathematical modeling of infectious diseases, epidemics, circuit analysis, voltage drop equations, computational neuroscience, intergalactic modeling, and many more. To explain the mathematical structure and the behavior of various phenomena in nature, two equations with simple mathematical structure namely: the (1 + 1)-dimensional integro-differential Ito equation (IDIE) and the (2 + 1)-dimensional integro-differential Sawada-Kotera equation (IDSKE) are considered. In this analytical investigation, we have conducted an in-depth enquiry to trace the distinctive closed form progressive wave solutions of these proposed models using the (G′G′+G+A)-expansion method. The graphical forms of the acquired solutions have been displayed in the bell-shaped soliton, singular periodic, and anti-kink shape soliton solutions after the values for the free parameters were specified. The two non-algebraic solutions, named, exponential and trigonometric function solutions, have emerged by applying our proposed method. Based on the general findings of our investigation for different parametric values, the attained wave solutions revealed can keep a vital role for natural balances and can be engaged in modern physical applications. The parameters have a visible effect on the wave amplitude and the swiftness of the traveling wave. The executed outcomes are innovative and have monumental applications in the present research, especially in the fields of theoretical physics, astrophysics, plasma physics, particle physics, theory of relativity, advance quantum mechanics, string theory, and geotechnical engineering.

Evolution of cylindrical/spherical shock formation in a dusty plasma with nonadiabatic dust charge variation

Cylindrical/spherical dust acoustic shock waves are studied in an unmagnetized plasma consisting of Maxwellian electrons, superthermal ions under the effect of nonadiabatic dust charge fluctuation. The dissipation is introduced in the system via nonadiabaticity, a new mechanism for the formation of shock waves. Using standard reductive perturbation method, nonplanar equation namely Korteweg–de Vries Burgers (KdVB) equation is derived and solution is obtained using the Weighted residual method. By use of weighted residual method, a progressive wave solution to nonplanar modified KdV Burgers equation is presented. It is shown that dust charge fluctuation produces dissipation which is responsible for shock formation. The solution shows that as one goes towards the axis of the cylinder or center of the sphere, the wave attenuates faster and the effects are shown in some 2D figures. We have studied various parametric influences on such shock structure and also showed how the gradual variations of these parameter affect the generation and structure of the shocks in their respective domain. Our results should be applicable to the formation of nonlinear DIA shock wave structures in regions in which dust grain is embedded in a Kappa distributed plasma, such as Saturn's magnetosphere.

Effect of non adiabatic dust charge fluctuation on nonplanar dust acoustic waves in superthermal polarized plasma

Some properties of cylindrical/spherical dust acoustic solitary waves have been studied in collisionless dusty plasma with variable charges under the influence of polarization force by reductive perturbation method. It is shown that the collisionless damping due to dust charge fluctuation causes the dust acoustic wave propagation to be described by the nonplanar damped Korteweg de Vries equation. By use of weighted residual method, a progressive wave solution to nonplanar damped KdV equation is presented for the first time. It is shown that dust charge fluctuation produce damping which is responsible for wave attenuation. The solution shows that as one goes towards the axis of the cylinder or centre of the sphere, the wave attenuates faster and the effects are shown in some 2D figures. Within a small amplitude limit, the influence of dust charge fluctuation, polarisation power, superthermality and ion temperature on the time evolution of dust acoustic solitons is also demonstrated. In laboratory experiments and space regions, findings obtained here may shed light on the salient characteristics of DA waves.

Collisionless damping of nonplanar dust acoustic waves due to dust charge fluctuation in nonextensive polarized plasma

A rigorous theoretical investigation has been made of the cylindrical/spherical dust acoustic waves in collisionless dusty plasma with variable charges under the influence of polarization force by reductive perturbation method. It is shown that the collisionless damping due to dust charge fluctuation causes the dust acoustic wave propagation to be described by the nonplanar damped Korteweg–de Vries equation. By use of weighted residual method, a progressive wave solution to nonplanar damped KdV equation is presented for the first time. It is shown that dust charge fluctuation produce damping which is responsible for wave attenuation. The solution shows that as one goes towards the axis of the cylinder or centre of the sphere, the wave attenuates faster and the effects are shown in some 2D figures. Within a small amplitude limit, the influence of dust charge fluctuation, polarisation power, nonextensivity and ion temperature on the time evolution of dust acoustic solitons is also demonstrated. In laboratory experiments and space regions, findings obtained here may shed light on the salient characteristics of DA waves.

Modeling of Thin Plate Flexural Vibrations by Partition of Unity Finite Element Method

This paper presents a conforming thin plate bending element based on the Partition of Unity Finite Element Method (PUFEM) for the simulation of steady-state forced vibration. The issue of ensuring the continuity of displacement and slope between elements is addressed by the use of cubic Hermite-type Partition of Unity (PU) functions. With appropriate PU functions, the PUFEM allows the incorporation of the special enrichment functions into the finite elements to better cope with plate oscillations in a broad frequency band. The enrichment strategies consist of the sum of a power series up to a given order and a combination of progressive flexural wave solutions with polynomials. The applicability and the effectiveness of the PUFEM plate elements are first verified via the structural frequency response. Investigation is then carried out to analyze the role of polynomial enrichment orders and enriched plane wave distributions for achieving good computational performance in terms of accuracy and data reduction. Numerical results show that the PUFEM with high-order polynomials and hybrid wave-polynomial combinations can provide highly accurate prediction results by using reduced degrees of freedom and improved rate of convergence, as compared with the classical FEM.

Analytical approximate solutions for nonplanar Burgers equations by weighted residual method

In this work, analytical approximate progressive wave solutions for the generalized form of the nonplanar KdV-Burgers (KdV-B) and mKdV-Burgers (mKdV-B) equations are presented and the results are discussed. Motivated with the exact solutions of the planar KdV-B and mKdV-B equations, the weighted residual method is applied to propose analytical approximate solutions for the generalized form of the nonplanar KdV-B and mKdV-B equations. The structure of the KdV-B equation assumes a solitary wave type of solution, whereas the mKdV-B equation assumes a shock wave type of solution. The analytical approximate progressive wave solutions for the cylindrical(spherical) KdV-B and mKdV-B equations are obtained as some special cases and compared with numerical solutions and the results are depicted on 2D and 3D figures. The results revealed that both solutions are in good agreement. The advantage of the present method is that it is rather simple as compared to the inverse scattering method and gives the same results with the perturbative inverse scattering technique. Moreover, the present analytical solutions allow readers to carry out physical parametric studies on the behavior of the solution. In addition to the present solutions are defined overall the problem domain not only over the grid points, as well as the solution calculation has less CPU time-consuming and round-off error.

Different Types of Progressive Wave Solutions via the 2D-Chiral Nonlinear Schrödinger Equation

A versatile integration gadget namely the protracted (or extended) Fan sub-equation (PFS-E) technique is devoted to retrieving a variety of solutions for different models in many fields of the sciences. This essay treatises the dynamics of progressive wave solutions via the 2D-chiral nonlinear Schrodinger (2D-CNLS) equation. The acquired solutions comprise of dark optical solitons, periodic solitons, singular dark (bright) solitons, and singular periodic solutions. By emulating the results gained in this work with other literature it can be noticed that the PFS-E method is an auspicious technique for finding solutions to other similar problems. Furthermore, it revealed some new types of solutions that will help us better understand the dynamic behaviors of the 2D-CNLS model.

On the evolution of finite and small amplitude waves in non-ideal gas with dust particles

The present work concerns with the study of progressive wave solution for the waves of finite and moderately small amplitude in the mixture of the non-ideal gas and dust particles governed by quasilinear hyperbolic system of PDEs. Using the progressive wave method, we derive the transport equation which provides the conditions of the shock formation, and equation to determine the shock strength. Also, it is shown that how the presence of the parameter of non-idealness and mass fraction of dust particles influence the shock formation and shock strength for the planar and non planar cases. We also discuss the consequence of variation in the value of the non-ideal parameter and the mass fraction of small solid dust particles on the evolutionary process of the shock.

Stability of periodic progressive gravity wave solutions of the Whitham equation in the presence of vorticity

The modulational instability of two-dimensional nonlinear traveling-wave solutions of the Whitham equation in the presence of constant vorticity is considered. It is shown that vorticity has a significant effect on the growth rate of the perturbations and on the range of unstable wavenumbers. Waves with kh greater than a critical value, where k is the wavenumber of the solution and h is the fluid depth, are modulationally unstable. This critical value decreases as the vorticity increases. Additionally, it is found that waves with large enough amplitude are always unstable, regardless of wavelength, fluid depth, and strength of vorticity. Furthermore, these new results are in qualitative agreement with those obtained by considering fully nonlinear solutions of the water-wave equations.

Analytical solution for nonplanar waves in a plasma with q-nonextensive nonthermal velocity distribution: Weighted residual method

Abstract The basic nonlinear equations describing the dynamics of a two component plasma consisting of cold positive ions and electrons obeying hybrid q- nonextensive nonthermal velocity distribution are examined in the cylindrical(spherical) coordinates through the use of reductive perturbation method and the cylindrical(spherical) KdV and the modified KdV equations are obtained. An approximate analytical method for the progressive wave solution is presented for these evolution equation in the sense of weighted residual method. It is observed that both amplitudes and the wave speeds decrease with the time parameter τ. Since the wave profiles change with τ, the waves cannot be treated as solitons. It is further observed that the amplitudes of spherical waves are larger than those of the cylindrical waves; and the wave amplitudes of modified KdV equation are much larger than those of the KdV equation. The effects of physical parameters (α, q) on the wave characteristics are also discussed.

AN APPROXIMATE WAVE SOLUTION FOR PERTURBED KDV AND DISSIPATIVE NLS EQUATIONS: WEIGHTED RESIDUAL METHOD

In the present work, we modi ed the conventional "weighted residual method" to some nonlinear evolution equations and tried to obtain the approximate progressive wave solutions for these evolution equations. For the illustration of the method we studied the approximate progressive wave solutions for the perturbed KdV and the dissipative NLS equations. The results obtained here are in complete agreement with the solutions of inverse scattering method. The present solutions are even valid when the dissipative e ects are considerably large. The results obtained are encouraging and the method can be used to study the cylindrical and spherical evolution equations.

New exact solutions for coupled nonlinear system of ion sound and Langmuir waves

In this work, our main goal is to study a new wave behavior of the nonlinear system associated with ionic wave equations under the influence of the ponderomotive force caused by a nonlinear force experienced by a charged particle in an inhomogeneous oscillating electromagnetic field due to high-frequency field. Another variation in the modified exp-function method has been developed to find progressive wave solutions for the system of sound waveform equations under the pressure of Langmuir wave. The algorithm presented in this work offers new solutions for the complex redundant functions of the nonlinear system in relation to the considered model. Numerical simulations are provided to illustrate the effectiveness of proposed algorithm.

Analytical solutions of cylindrical and spherical dust ion-acoustic solitary waves

Abstract In the present work, employing the conventional reductive perturbation method to the field equations of an unmagnetized dusty plasma consisting of inertial ions, Boltzmann electrons and stationary dust particles in the nonplanar geometry we derived cylindrical(spherical) KdV and mKdV equations. Being aware of the fact that there exists no exact analytical solution for the progressive waves for such evolution equations we presented the exact analytical solution of a generalized form of such evolution equations in the planar geometry and used this solution to obtain an analytical approximate progressive wave solution to the generalized evolution equation. Then the progressive wave solutions for the cylindrical(spherical) KdV and m-KdV equations is obtained as some special cases. The analytical approximate solutions so obtained are compared with the numerical solutions of these equations. The results reveal that both solutions are in a good agreement. One advantage of present analytical approximate solution is that it allows readers to gain information, share understandings, or carry out a physical parametric study on the evolution solution behavior as well as the solution calculation has no CPU time-consuming or round off error.

Early history of nonlinear acoustics: Waveform distortion, disaster, and redemption

The first (although slightly incorrect) wave equations for finite-amplitude sound in lossless fluids were obtained independently by Euler in 1759 and Lagrange in 1760. Poisson (1808) provided the first major breakthrough with his exact solution for progressive waves of finite amplitude in a lossless gas. Although a far-reaching result, the progressive waveform distortion (and disastrous consequences) implied by his solution went unrecognized for 40 years. Challis (1848) showed that the Poisson solution is not single valued but did not understand why. Stokes (1848) provided the why. He saw that the Poisson waveform distorts as the wave travels, eventually threatening to become multivalued. He postulated that a discontinuity (shock) develops to avoid waveform overturning. He also saw that viscosity (not accounted for by Poisson) would prevent true discontinuities. Earnshaw (1860) and Riemann (1860) cleaned up plane waves in lossless gases. However, how to predict propagation after shocks form? Rankine (1870) and Hugoniot (1887, 1889) provided the first solutions for propagation when dissipation is included. These were the first steps toward redemption. The curtain rang down on this era of nonlinear acoustics with the excellent papers in 1910 by Rayleigh and Taylor on steady shocks in a thermoviscous fluid.

Transient flexural gravity waves in two-layer fluid

In the present manuscript, flexural gravity wave generation due to an initial disturbance in the case of two-layer fluid of finite depth in the presence of current is analyzed in three dimensions. From the plane progressive wave solution, effect of wave and current direction and the combined effect of current and compressive force on flexural gravity wave motion are discussed. Using Laplace and Fourier transform, the solution of the transient wave structure interaction problem is obtained and integral forms of velocity potentials, structural deflection and interface elevations are derived. Assuming the initial disturbances is on the plate covered surface asymptotic solutions for structural deflection and interface elevations are obtained using method of stationary phase for large time and space. The study reveals that the effect of opposing current and critical compressive force in the presence of current has significant impact on the plate deflection.

Modulation of cylindrical (spherical) waves in a plasma with vortex electron distribution

In the present work, employing cylindrically (spherically) symmetric field equations of a plasma consisting of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution and stationary ions, we studied the amplitude modulation of electron-acoustic waves. Due to the physical nature of the problem under investigation, the nonlinearity of the field equations is of order (3/2), which causes considerable difficulty in the analysis of modulation problems. To solve this difficulty, we expanded this nonlinear term into the Fourier cosine series of the phase function and obtained the modified cylindrical (spherical) nonlinear Schrödinger (NLS) equation. A consistent analysis for the modulational instability is presented and a criterion between the time parameter τ and the wave number K is established. In addition, motivated with the solitonic solution of modified NLS equation for planar case and utilizing the “weighted residual method,” we proposed a harmonic wave of variable frequency with progressive wave amplitude to the evolution equation. It is found that the modified cylindrical (spherical) NLS equation assumes an envelope type of progressive wave solution in the sense weighted residual. The numerical results reveal that the amplitude of spherical wave is much larger than that of the cylindrical wave and that both amplitudes decrease with increasing time parameter τ. It is further observed that the wave profiles get distorted with progressing time.

Cylindrical and spherical solitary waves in an electron-acoustic plasma with vortex electron distribution

We consider the nonlinear propagation of electron-acoustic waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions. The basic nonlinear equations of the above described plasma are re-examined in the cylindrical (spherical) coordinates by employing the reductive perturbation technique. The modified cylindrical (spherical) KdV equation with fractional power nonlinearity is obtained as the evolution equation. Due to the nature of nonlinearity, this evolution equation cannot be reduced to the conventional KdV equation. A new family of closed form analytical approximate solution to the evolution equation and a comparison with numerical solution are presented and the results are depicted in some 2D and 3D figures. The results reveal that both solutions are in good agreement and the method can be used to obtain a new progressive wave solution for such evolution equations. Moreover, the resulting closed form analytical solution allows us to carry out a parametric study to investigate the effect of the physical parameters on the solution behavior of the modified cylindrical (spherical) KdV equation.

A one-dimensional nonlinear problem of thermoelasticity in extended thermodynamics

Abstract We solve a nonlinear, one-dimensional initial boundary-value problem of thermoelasticity in generalized thermodynamics. A Cattaneo-type evolution equation for the heat flux is used, which differs from the one used extensively in the literature. The hyperbolic nature of the associated linear system is clarified through a study of the characteristic curves. Progressive wave solutions with two finite speeds are noted. A numerical treatment is presented for the nonlinear system using a three-step, quasi-linearization, iterative finite-difference scheme for which the linear system of equations is the initial step in the iteration. The obtained results are discussed in detail. They clearly show the hyperbolic nature of the system, and may be of interest in investigating thermoelastic materials, not only at low temperatures, but also during high temperature processes involving rapid changes in temperature as in laser treatment of surfaces.

The method of lines solution of the Forced Korteweg-de Vries-Burgers equation

In this paper, the application of the method of lines (MOL) to the Forced Korteweg-de Vries-Burgers equation with variable coefficient (FKdVB) is presented. The MOL is a powerful technique for solving partial differential equations by typically using finite-difference approximations for the spatial derivatives and ordinary differential equations (ODEs) for the time derivative. The MOL approach of the FKdVB equation leads to a system of ODEs. The solution of the system of ODEs is obtained by applying the Fourth-Order Runge-Kutta (RK4) method. The numerical solution obtained is then compared with its progressive wave solution in order to show the accuracy of the MOL method.