Unidirectional Wave Propagation Research Articles

Higher‐order integrable models for oceanic internal wave–current interactions

AbstractIn this paper, we derive a higher‐order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a “flat‐surface” assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long‐wave, and small‐amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations.

Diverse soliton wave profile analysis in ion-acoustic wave through an improved analytical approach

In engineering and applied sciences, several physical phenomena can be more precisely characterized by employing nonlinear fractional partial differential equations. The primary goal of this research is to examine the traveling wave solution in closed form for the nonlinear acoustic wave propagation model known as the time fractional simplified modified Camassa–Holm equation, which is used to explain the unidirectional propagation of shallow-water waves with non-hydrostatic pressure and explains the dispersion properties of numerous phenomena like fluid flow, control theory, liquid drop patterning in plasma, acoustics, fusion, and fission processes, etc. The utmost potential approach, namely the new auxiliary equation technique, is applied for analyzing the time nonlinear fractional simplified modified Camassa-Holm equation in the logic of the newest established truncated M-fractional derivative. The fractional partial differential equations have been transformed to the ordinary differential equation using the complex wave transformation in the sense of truncated M-fractional derivative. A variety of soliton solutions, including anti-kink, single soliton, anti-bell, bell, kink, multiple soliton, double soliton, singular-kink, compacton shape, periodic shape, and so many, are displayed in the diagram of 3D and contour plots by taking into account a number of various parameters. It is essential to point out that all derived outcomes are directly compared to the original solutions to certify their exactness. Results show that the used scheme is capable, simple, and straightforward and can be useful to a variety of complex phenomena. The acquired results are unique for the model equation and could be applied to the analysis of several nonlinear study fields.

Orbital stability of smooth solitons for the modified Camassa-Holm equation

The modified Camassa-Holm equation with cubic nonlinearity is completely integrable and is considered a model for the unidirectional propagation of shallow-water waves. The localized smooth-wave solution exists uniquely, up to translation, within a certain range of the linear dispersive parameter. By constructing conserved H1 and L1 quantities in terms of the momentum variable m, this study demonstrates that the smooth soliton, when regarded as a solution of the initial-value problem for the modified Camassa-Holm equation, is orbitally stable to perturbations in the Sobolev space H3. Furthermore, the global well-posedness of the solution is established for certain initial data in Hs with s≥3.

Observation of dynamic non-Hermitian skin effects

Non-Hermitian physics has emerged as a new paradigm that profoundly changes our understanding of non-equilibrium systems, introducing novel concepts such as exceptional points, spectral topology, and non-Hermitian skin effects (NHSEs). Most existing studies focus on non-Hermitian eigenstates, whereas dynamic properties have been discussed only recently, and the dynamic NHSEs are not yet confirmed in experiments. Here, we report the experimental observation of non-Hermitian skin dynamics using tunable one-dimensional nonreciprocal double-chain mechanical systems with glide-time symmetry. Remarkably, dynamic NHSEs are observed with various behaviors in different dynamic phases, which can be understood via the generalized Brillouin zone and the related concepts. Moreover, the observed dynamic NHSEs, amplifications, bulk unidirectional wave propagation, and boundary wave trapping provide promising ways to manipulate waves in a controllable and robust way. Our findings open a new pathway toward non-Hermitian dynamics, which will fertilize the study of non-equilibrium phases of matter.

Square-root topological insulator for a dual-band photonic waveguide

We show that square-root topological insulators can be utilized as a platform for dual-band topological waveguides in 2-dimensional photonic crystals. Valley states for each of two frequency band gaps in differently perturbed decorated honeycomb lattices have opposite directions of the phase rotations, revealing opportunities for dual-band unidirectional topological valley transport. The topological valley edge states exist at the boundary between differently perturbed photonic crystals in the two frequency bands. In addition, unidirectional, and robust propagation of electromagnetic waves along the boundary in the two frequency bands are numerically performed. Our work provides extensions of square-root topological insulators to robust, unidirectional, and dual-band photonic waveguides.

Solitary wave solutions of the time fractional Benjamin Bona Mahony Burger equation

The present study examines the approximate solutions of the time fractional Benjamin Bona Mahony Burger equation. This equation is critical for characterizing the dynamics of water waves and fluid acoustic gravity waves, as well as explaining the unidirectional propagation of long waves in nonlinear dispersive systems. This equation also describes cold plasma for hydromagnetic and audio waves in harmonic crystals. The natural transform decomposition method is used to obtain the analytical solution to the time fractional Benjamin Bona Mahony Burger equation. The proposed method uses the Caputo, Caputo Fabrizio, and Atangana Baleanu Caputo derivatives to describe the fractional derivative. We utilize a numerical example with appropriate initial conditions to assess the correctness of our findings. The results of the proposed method are compared to those of the exact solution and various existing techniques, such as the fractional homotopy analysis transform method and the homotopy perturbation transform technique. As a result, bell shaped solitons are discovered under the influence of hyperbolic functions. By comparing the outcomes with tables and graphs, the findings demonstrate the efficacy and effectiveness of the suggested approach.

Elastic wave spin and unidirectional routing in thin rod systems

The unusual topology of elastic waves in continuum systems has been recently uncovered. Prominently, the spin of Rayleigh-Lamb waves has been experimentally observed and used to achieve the unidirectional wave propagation. In this work, the elastic wave spin in thin rod systems is theoretically predicted and experimentally observed, for all the three basic wave modes, i.e., the zero-order flexural, longitudinal and torsional modes. By elaborately building up the subwavelength chiral sources, the spin-momentum locking has been achieved for these three modes within broad frequency range. Interestingly and surprisingly, the spin-momentum locking occurs for flexural and longitudinal modes in both circular and non-circular thin rods; the spin-momentum locking for torsional modes only exits in non-circular thin rods, due to the warping effect in non-circular cross-section rods. Our results open up the door for elastic wave spin in thin rod systems, and provide general tips for the research on basic modes in diverse thin rods, based on either the continuum media or the artificial metamaterials in future.

Exploring the dynamics of shallow water waves and nonlinear wave propagation in hyperelastic rods: Analytical insights into the Camassa–Holm equation

The objective of this paper is to examine the analytical properties of the nonlinear [Formula: see text]-dimensional Camassa–Holm equation ([Formula: see text]), a fundamental model within the domain of nonlinear evolution equations. The aforementioned equation serves as a valuable tool in elucidating the unidirectional propagation of shallow water waves over a level terrain, as well as in representing certain nonlinear wave phenomena seen in cylindrical hyperelastic rods. We use the Khater III ([Formula: see text]hat.III) and improved Kudryashov ([Formula: see text]ud) technique to provide accurate solutions, drawing inspiration from the intricate mathematical framework of the [Formula: see text] issue. He’s variational iteration ([Formula: see text]) technique is used as a numerical methodology to assess the correctness of the generated answers. This strategy reveals a notable concurrence between the analytical and numerical outcomes. This alignment guarantees the suitability of the acquired solutions within the framework of the studied model. The importance of this investigation lies in its ability to improve our comprehension of the intricate dynamics regulated by the [Formula: see text] equation and its connections with other nonlinear evolution equations that describe shallow water wave behaviors and nonlinear wave propagation in cylindrical hyperelastic rods. The results of the study demonstrate novel analytical approaches, expanding the range of potential solutions and offering valuable insights into the physical characteristics of the interconnected wave phenomena. This research offers valuable insights and methodologies for addressing intricate mathematical models in shallow water wave theory and studying nonlinear waves in hyperelastic materials, therefore making substantial advances to the subject of nonlinear partial differential equations.

Unidirectional propagation of water waves near ancient Luoyang Bridge

Unidirectional propagation of water waves near ancient Luoyang Bridge

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)

Backscattering-free edge states below all bands in two-dimensional auxetic media

Unidirectional and backscattering-free propagation of sound waves is of fundamental interest in physics and highly sought-after in engineering. Current strategies utilize topologically protected chiral edge modes in bandgaps or complex mechanisms involving active constituents or nonlinearity. Here, we propose a new class of passive, linear, one-way edge states based on spin-momentum locking of Rayleigh waves in two-dimensional media in the limit of vanishing bulk modulus, which provides 100% unidirectional and backscattering-free edge propagation immune to any edge roughness at a broad range of frequencies instead of residing in gaps between bulk bands. We further show that such modes are characterized by a new topological winding number that is analogous to discrete angular momentum eigenvalues in quantum mechanics. These passive and backscattering-free edge waves have the potential to enable a new class of phononic devices in the form of lattices or continua that work in previously inaccessible frequency ranges.

Numerical Solution of Time-fractional BBMB Equation via LRPSM

The time-fractional Benjamin-Bona-Mahony-Burger (BBMB) differential equation plays an important role in explaining the unidirectional propagation of long waves in definite nonlinear differential systems. This work presents an analytical numerical solution of considered equation by Laplace transform with residual power series method (LRPSM) which is a generalized Taylor series together with Laplace transform and the residual error function. Using the proposed approach, the series solution of this equation is obtained. The analytical numerical solution shows that the LRPSM is a reliable and powerful method for solving the time fractional differential equations (FDEs) with less number of terms. The obtained results of numerical solutions are also compared with the exact solution and presented as absolute errors at different time levels.

ON THE SOLITARY WAVE SOLUTIONS OF DIFFERENT VERSIONS OF FRACTIONAL 3D- WAZWAZ -BENJAMIN-BONA-MAHONY EQUATIONS

Nonlinear fractional Wazwaz -Benjamin-Bona-Mahony (WBBM) equations play an important role in physics. The equations form an important model for studying the approximately unidirectional propagation of small amplitude long waves in certain nonlinear distribution systems as an alternative to Kortweg and de Vries (KDV). In this study, the fractional 3D-WBBM equations are solved by using the Improved Bernoulli Sub-Equation Function (IBSEF) method. 3D, 2D and contour plots are given to show the physical properties of the solutions. The main aim of this method is to clarify obvious the exact solutions to the equations. Moreover, the effectiveness of the method is demonstrated by the findings presented in this paper.

Forcing the silence of the Lamb waves: Uni-directional propagation in structured gyro-elastic strips and networks

In this paper, we propose an innovative design of an elastic network, which is capable of channelling the energy supplied by an external source towards any of its endpoints, that can be chosen arbitrarily and in advance. This system, named Mechanical Switching Network (MSN), consists of an interconnected array of branches, each of which is represented by a lattice strip endowed with gyroscopic spinners. The latter make the system non-reciprocal and, hence, are responsible for the preferential directionality exhibited by the network. We formulate and solve the forced problem for the gyro-elastic strip in the analytical form and compare the derived solutions with the results of independent finite element simulations, showing an excellent agreement. Additionally, we carry out a parametric analysis to evaluate the influence of the main parameters of the system on the uni-directional wave propagation of Lamb waves. We envisage that the proposed model can have important implications in many engineering applications, where control and tunability of guided waves play a key role.

The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

Plenty of accurate, explicit solitary unidirectional wave solutions of the nonlinear Gilson–Pickering model

This paper proposes abundant accurate wave solutions that represent the plasma wave propagation based on crystal lattice theory and physicochemical characterization. The Gilson–Pickering ([Formula: see text]) model is analytically and numerically solved by two recent techniques. This model is a basic unidirectional wave propagation model that describes the prorogation of waves in crystal lattice theory and plasma physics. The investigated model has a deep connection with some nonlinear evolution equations under specific values of its parameters, such as the Fuchssteiner–Fokas–Camassa–Holm ([Formula: see text]) equation, the Rosenau–Hyman ([Formula: see text]) equation, and the Fornberg–Whitham ([Formula: see text]) equation. The Sardar sub-equation and He’s variational iteration techniques are employed to construct novel and accurate solitary wave solutions of the handled model. The obtained solutions are explained through some distinct graphs in contour, three-, two-dimensional. The research value is explained by comparing our results with some recently published studies. The employed methods show their simple, direct, effectiveness and their ability for handling many nonlinear evolution equations.

Numerical simulation of time fractional Benjamin-Bona-Mahony-Burger equation describing propagation of long waves on the water surface

The unidirectional propagation of long waves in a certain nonlinear dispersive system is explained in the Benjamin-Bona-Mahony-Burger equation. The motivation of this manuscript is to examine the Benjamin-Bona-Mahony-Burger equation numerically and analytically by employing reproducing kernel functions. The implementation of reproducing kernel Hilbert space method produces an analytical solution in the form of a series representation, and after truncating the infinite series, an approximate solution is attained. With the support of numerical examples, the accuracy of this method has been shown through graphs and error tables. The convergence analysis of the introduced scheme has been explained by some useful theorems and lemmas. Finally, it has been observed that the approximate solution consistently converges to the exact solution with less computational overhead. The proposed method would be very effective in solving various fractional partial differential equation models.

Galilei-invariant and energy-preserving extensions of Benjamin–Bona–Mahony-type equations

The classical Benjamin–Bona–Mahony equation (BBM equation) models unidirectional propagation of long gravity surface waves of small amplitude. Unlike many other water wave models, it lacks the Galilean invariance, which is an essential property of physical systems. It is shown that by an addition of a higher asymptotic order nonlinear term, this deficiency can be corrected, giving rise to a new Galilei invariant Benjamin–Bona–Mahony equation (iBBM equation). Moreover, further additional higher-order terms can be chosen in a way that the augmented model preserves the energy conservation property along with Hamiltonian and Lagrangian structures. The resulting equation is referred to as energy-preserving Benjamin–Bona–Mahony equation (eBBM).It is shown that both the classical BBM equation and the energy-preserving eBBM equations belong to a one-parameter (α) family that shares essentially the same local and nonlocal symmetries, conservation laws, Hamiltonian, and Lagrangian structures, with the BBM and eBBM equations corresponding to parameter values α=0 and α=1, respectively. Symmetry and conservation law classifications reveal a special case α=1/3, which is shown to correspond to a rescaled version of the celebrated integrable Camassa–Holm (CH) equation. Local symmetries and conservation laws are computed, and numerical solution behaviour is compared for the three BBM-type modes and the CH-equivalent eBBM1/3 model.

On H 2-solutions for a Camassa-Holm type equation

Abstract Camassa-Holm type equations arise as models for the unidirectional propagation of shallow water waves over a flat bottom. They also describe finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods. Under appropriate assumption on the initial data, on the time T T , and on the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem.

An improved method for unidirectional mechanical wave propagation in a metamaterial beam

An improved method for unidirectional mechanical wave propagation in a metamaterial beam