Solitary Waves Research Articles

Nonlinear long waves in shallow water for normalized Boussinesq equations

In this work, we investigate a partial differential equation (PDE) derived from a Boussinesq system of two equations that describe the wave motion in shallow water when the wavelength is small compared to wave amplitude. The Lie point symmetries of the PDE are utilized to reduce it to an ordinary differential equation (ODE) the travelling wave solutions of which are obtained. These solutions are of trigonometric, hyperbolic and exponential types. Graphical representation of some solutions exhibiting solitary and cnoidal wave phenomena are presented.

Numerical Simulation for Solitary Waves of the Generalized Zakharov Equation Based on the Lattice Boltzmann Method

The generalized Zakharov equation is a widely used and crucial model in plasma physics, which helps to understand wave particle interactions and nonlinear wave propagation in plasma. The solitary wave solution of this equation provides insights into phenomena such as electron and ion acoustic waves, as well as magnetic field disturbances in plasma. The numerical simulation of solitary wave solutions to the generalized Zakharov equation is an interesting problem worth studying. This is crucial for plasma-based technology, as well as for understanding nonlinear wave propagation in plasma physics and other fields. In this study, a numerical investigation of the generalized Zakharov equation using the lattice Boltzmann method has been conducted. The lattice Boltzmann method is a new modeling and simulating method at the mesoscale. A lattice Boltzmann model was constructed by performing Taylor expansion, Chapman–Enskog expansion, and time multiscale expansion on the lattice Boltzmann equation. By defining the moments of the equilibrium distribution function appropriately, the macroscopic equation can be restored. Furthermore, the numerical experiments for the equation are carried out with the parameter lattice size m=100, time step Δt=0.001, and space step size Δx=0.4. The solitary wave solution of the equation is numerically simulated. Numerical results under different parameter values are compared, and the convergence and effectiveness of the model are numerically verified. It is obtained that the model is convergent in time and space, and the convergence orders are all 2.24881. The effectiveness of our model was also verified by comparing the numerical results of different numerical methods. The lattice Boltzmann method demonstrates advantages in both accuracy and CPU time. The results indicate that the lattice Boltzmann method is a good tool for computing the generalized Zakharov equation.

Lump, periodic, travelling, semi-analytical solutions and stability analysis for the Ito integro-differential equation arising in shallow water waves

Nonlinear evolution equations are typically used to model and analyse the majority of natural processes. These equations are extremely important for comprehending many intricate situations involving wave propagation, including heat transfer, fluid dynamics, optical fibres, electrodynamics, physics, chemistry, biology, condensed matter physics, ocean engineering, and many other nonlinear science fields. In this study, the anticipation of a bilinear Kdv model known as the (1+1)-dimensional integro-differential Ito equation is considered which depicts the oceanic shallow water wave dynamics. Firstly, we determine the bilinear form of the model and then we develop lump waves and analyse their interactions with periodic waves using symbolic computation approaches. Later, the interactions between single, double-kink and lump waves solutions, as well as interactions between periodic, lump and soliton solutions is considered. Furthermore, the (G′/G2) expansion approach and the Adomian technique are implemented to acquire a variety of novel configurations for the governing dynamical model. Later, the corresponding absolute accuracy is evaluated through a comparison of several types of observed solutions. Additionally, the stability of the solitary travelling wave solutions to the specified model is also established. The 3D, 2D, and contour plots have been employed to demonstrate the characteristics of the nonlinear model and aid in determine the proper set of parameters. As a result, a collection of options for bright, dark, periodic, rational, and elliptic functions is established. To the best of our knowledge, the current work presents a novel case study that has not been previously studied in order to generate several new solutions to the integro-differential equation appearing in shallow water waves. The results show that the strategies that have been employed are more effective and capable than the traditional methods found in previous research.

Physical constructions of kink, anti-kink optical solitons and other solitary wave solutions for the generalized nonlinear Schrödinger equation with cubic–quintic nonlinearity

Physical constructions of kink, anti-kink optical solitons and other solitary wave solutions for the generalized nonlinear Schrödinger equation with cubic–quintic nonlinearity

A detailed analysis of the improved modified Korteweg-de Vries equation via the Jacobi elliptic function expansion method and the application of truncated M-fractional derivatives

Using the Jacobi elliptic function expansion method, which is improved by the novel use of truncated M-fractional derivatives, we thoroughly analyze the improved modified Korteweg-de Vries problem in this paper. Not only does this method provide new insight into the improved modified Korteweg-de Vries Equation, but it also represents a giant leap forward for nonlinear differential equations. Periodic, solitary, and trigonometric waves are among the answers found by us; all of them have important consequences for mathematical physics and engineering.The MATLAB-generated visual representations of our results are crucial because they provide aconcrete shape to the once intangible mathematical ideas. In addition to showing the solutions, these images highlight the waves' dynamic behaviors and stability under various conditions, providing a better understanding of the physical processes that the improved modified Korteweg-de Vries equation attempts to explain. The graphic representations in this study demonstrate how the equation can be used to simulate complicated systems in many scientific disciplines. The relevance of our findings goes beyond only theoretical studies. Our study lays the groundwork for more accurate predictive models in fields including quantum physics, optical fibers, and fluid dynamics by giving exact and concrete solutions to the improved modified Korteweg-de Vries equation. The combination of the Jacobi Elliptic Function Expansion Method with truncated M-fractional derivatives provides a robust framework for solving comparable complicated differential equations, which in turn opens up new research and development opportunities. Overall, this research both deepens our understanding of the Improved Modified Korteweg-de Vries equation and provides more evidence that theoretical mathematics can be useful in solving practical problems. Mathematical modeling and computational visualization have the potential to make substantial contributions to engineering and the sciences, and our results support a multidisciplinary approach to study.

Propagation of solitary waves in origami-inspired metamaterials

We propose a design strategy for creating origami-like mechanical metamaterials with diverse non-linear mechanical properties and capable of remote actuation. The proposed triangulated cylindrical origami (TCO)-inspired metamaterials enable the highly desirable strain-softening/hardening and snap-through behaviors via a multi-material and highly deformable hinge design. Moreover, we couple these novel non-linear mechanical properties of the TCO origami-inspired metamaterials with the transformative ability of hard-magnetic active materials, allowing for untethered shape- and property-actuation in the developed metamaterials. We develop a mathematical modeling framework for the proposed TCO origami-inspired metamaterials, building on approximating the highly deformable hinges as a combination of longitudinal and rotational springs. We validate the accuracy of the developed mathematical modeling approach by comparing the analytically predicted compressive response of a unit cell structure with the corresponding numerical and experimental results. Using the developed mathematical modeling framework, we investigate the magnetic field-induced large deformation and superimposed solitary wave propagation in the TCO origami-inspired metamaterial system. We show that the proposed metamaterial allows us to tune the key characteristics of the enabled non-linear solitary waves, including their characteristic width and amplitude. The proposed design strategy for readily manufacturable origami-inspired metamaterial systems paves a novel path for practical engineering applications. Our studies also underscore the potential of magneto-mechanical interaction in the design of reconfigurable metamaterial systems with superior non-linear mechanical and elastic wave properties.

Exploring plasma phenomena with the Nizhnik–Novikov–Veselov formula: Analyzing ion-acoustic waves, solitons, and shocks

This paper delves into the intricacies of the ([Formula: see text])-dimensional asymmetric Nizhnik–Novikov–Veselov ([Formula: see text]) model, a nonlinear partial differential equation governing ion-acoustic wave propagation in plasma. By employing advanced analytical and numerical approaches, the study explores innovative solitary wave solutions, particularly focusing on the dynamics of isochoric flow. Isochoric flow analysis is crucial for unraveling the complex behaviors exhibited by incompressible fluids like elastomers and bio-elastomers, which maintain a constant density. The derivation of the ([Formula: see text])-dimensional [Formula: see text] equation stems from fluid equations governing plasma dynamics. This model serves as a valuable tool for simulating experimental observations of plasma waves. The computational methodology applied in this research demonstrates a commendable level of precision and consistency, yielding novel solitary wave solutions previously unreported in the [Formula: see text] model. These results underscore the study’s importance and novelty. The outcomes not only contribute to our understanding of incompressible fluid dynamics, but also lay the groundwork for future investigations in this domain. The revealed solitary wave solutions have the potential to inform the development of more accurate models for predicting fluid dynamics, thereby advancing the field.

Phase trajectories and Chaos theory for dynamical demonstration and explicit propagating wave formation

This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions.

The Lie point symmetry criteria and formation of exact analytical solutions for Kairat-II equation: Paul-Painlevé approach

The renowned non-linear Kairat-II equation, which expresses the surface curves, is examined meticulously in this work. There has never been a study before that discussed conserved quantities, sensitivity analysis, Hamiltonian function, Lie symmetry invariance criteria, and invariant solutions of Kairat-II equation The Lie invariance criteria are taken into consideration by the symmetry generators. The suggested technique leads to in a four-dimensional Lie algebra, where the translation point symmetries in space and time correlate with the conservation of mass and the energy, respectively, and remaining point symmetries are dilation and scaling. Firstly, we use symmetry reduction of Lie subalgebras to obtain closed-form invariant solutions. In specific reduction cases, we transform the Kairat-II equation into a spectrum of non-linear ordinary differential equations, which have the benefit of being able to offer an extensive selection of closed-form solitary wave solutions. For this equation, the Cauchy problem cannot be solved by the inverse scattering transform, therefore, traveling wave exact solutions are generated using the analytical Paul-Painlevé approach. In both two and three dimensions, the graphical behavior of specific solutions is presented for specific quantities of physical factors of examined equation. Utilizing conservation laws multipliers, the study culminates by determining an extensive set of local conservation laws for the non-linear Kairat-II equation that are applicable to arbitrary constant coefficients. Instead of focusing on the physical aspects of conservation laws, this study computed the conserved quantities using a mathematical perspective that can be used to identify potential symmetries of partial differential equations, guiding readers toward the solutions of partial differential equations. The existence of the Hamiltonian function is presented. The model’s sensitivity to various starting conditions is highlighted by the sensitivity analysis.

Investigation of a Spatio-Temporal Fractal Fractional Coupled Hirota System

This article aims to examine the nonlinear excitations in a coupled Hirota system described by the fractal fractional order derivative. By using the Laplace transform with Adomian decomposition (LADM), the numerical solution for the considered system is derived. It has been shown that the suggested technique offers a systematic and effective method to solve complex nonlinear systems. Employing the Banach contraction theorem, it is confirmed that the LADM leads to a convergent solution. The numerical analysis of the solutions demonstrates the confinement of the carrier wave and the presence of confined wave packets. The dispersion nonlinear parameter reduction equally influences the wave amplitude and spatial width. The localized internal oscillations in the solitary waves decreased the wave collapsing effect at comparatively small dispersion. Furthermore, it is also shown that the amplitude of the solitary wave solution increases by reducing the fractal derivative. It is evident that decreasing the order α modifies the nature of the solitary wave solutions and marginally decreases the amplitude. The numerical and approximation solutions correspond effectively for specific values of time (t). However, when the fractal or fractional derivative is set to one by increasing time, the wave amplitude increases. The absolute error analysis between the obtained series solutions and the accurate solutions are also presented.

Experiments on droplet production from solitary waves impacting on a vertical wall

This work presents the results of an experimental investigation of solitary waves impacting on a vertical flat wall. The aim of the study was to correlate the wave characteristics and breaking stage before impact to the droplet production after the impact. The main characteristic of these experiments is that the wall height is lower than the wave height. The droplet sizes were obtained by combining 3D particle tracking velocimetry with an in-house code. Droplet diameters between 0.3–22 mm were measured at a distance of 35 cm and 68 cm from the wall. It was found that the breaking stage before the impact has a large influence on the mean size of the droplets produced, especially when there is air trapped between the wall and the wave at the impact moment. It was found that the mean droplet size decreases with the ratio of water jet velocity after impact and the wave speed. The resulting distributions of droplet sizes were compared to three different theoretical models used in other droplet production studies. Results were also compared to previous experimental studies of droplet formation from breaking waves to determine the best fit to the experimentally obtained distribution.

Higher order Galerkin finite element method for [formula omitted]-dimensional generalized Benjamin–Bona–Mahony–Burgers equation: A numerical investigation

In this article, we study solitary wave solutions of the generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equation using higher-order shape elements in the Galerkin finite element method (FEM). Higher-order elements in FEMs are known to produce better results in solution approximations; however, these elements have received fewer studies in the literature. As a result, for the finite element analysis of the gBBMB equation, we consider Lagrange quadratic shape functions. We employ the Galerkin finite element approximation to derive a priori error estimates for semi-discrete solutions. For fully discrete solutions, we adopt the Crank–Nicolson approach, and to handle nonlinearity, we utilize a predictor–corrector scheme with Crank–Nicolson extrapolation. Additionally, we perform a stability analysis for time using the energy method. In the space, O(h3) convergence in L2(Ω) norm and O(h2) convergence in H1(Ω) norm are observed. Furthermore, an optimal O(Δt2) convergence in the maximum norm for the temporal direction is also obtained. We test the theoretical results on a few numerical examples in one- and two-dimensional spaces, including the dispersion of a single solitary wave and the interaction of double and triple solitary waves. To demonstrate the efficiency and effectiveness of the present scheme, we compute L2 and L∞ normed errors, along with the mass, momentum, and energy invariants. The obtained results are compared with the existing literature findings both numerically and graphically. We find quadratic shape functions improve accuracy in mass, momentum, energy invariants and also give rise to a higher order of convergence for Galerkin approximations for the considered nonlinear problems.

Explicit solitary wave profiles and stability analysis of biomembranes and nerves

This paper examines the stability analysis and exact solitary wave solutions of the nonlinear partial differential equation known as the Heimburg model. The several types of solitary wave solutions, soliton solutions and Jacobi elliptic doubly periodic function solutions are explored by using the extended Sinh-Gordon equation expansion approach. These investigations exhibit the system’s astounding diversity of waveforms, highlighting its potential applications in nerves and biomembranes. By selecting some appropriate values for the parameters, 3D, 2D, and its corresponding contour graph are plotted to represent the physical relevance of some of the solutions. Additionally, the linearized stability of this system is analyzed. The suggested approach is the finest resource for the analytical investigation of any nonlinear issue that occurs in various scientific fields.

New exact solutions to the generalized shallow water wave equation

In this work, we study the generalized shallow water wave equation to obtain novel solitary wave solutions. The application of this nonlinear model can be found in tidal waves, weather simulations, tsunami prediction, river and irrigation flows. To obtain the new exact solutions of the considered model, we have applied a novel analytical technique namely [Formula: see text]-expansion method. Using the aforementioned method and computational software, we have obtained different kinds of periodic and singular solitary wave solutions of the generalized shallow water wave equation. The obtained solutions are exponential function and trigonometric function solutions. Using 2D and 3D plots of the wave solutions, the dynamic behaviors of the developed solutions are displayed. The retrieved solutions validated the effectiveness and robustness of the proposed technique.

Linear limit continuation: Theory and an application to two-dimensional Bose–Einstein condensates

We present a coherent and effective theoretical framework to systematically construct numerically exact nonlinear solitary waves from their respective linear limits. First, all possible linear degenerate sets are classified for a harmonic potential using lattice planes. For a generic linear degenerate set, distinct wave patterns are identified in the near-linear regime using a random searching algorithm by suitably mixing the linear degenerate states, followed by a numerical continuation in the chemical potential extending the waves into the Thomas–Fermi regime. The method is applied to the two-dimensional, one-component Bose–Einstein condensates, yielding a spectacular set of waveforms. Our method opens a remarkably large program, and many more solitary waves are expected. Finally, the method can be readily generalized to three dimensions, and also multi-component condensates, providing a highly powerful technique for investigating solitary waves in future works.

Painlevé Analysis and Kink-Type Solitary Waves of the Geophysical KdV Equation Involving a Source

Painlevé Analysis and Kink-Type Solitary Waves of the Geophysical KdV Equation Involving a Source

New solitary wave solutions to Biswas–Milovic and resonant nonlinear Schrödinger equations

New solitary wave solutions to Biswas–Milovic and resonant nonlinear Schrödinger equations

Dynamic catastrophe analysis of deepwater mooring platform/riser/wellhead coupled system under ISW

Mooring platform/riser/wellhead coupled system, as an important and large complex system during offshore drilling operations, is easily affected by internal solitary waves (ISW). The impact load of ISW may lead to failure cascade process and even serious accident consequences to the coupled system. This paper is aimed at proposing a dynamic catastrophic analysis model that combines mechanical quantitative analysis with process safety of mooring platform/riser/wellhead system under ISW. The proposed dynamic catastrophic model can consider limit state of each system used as the failure criterion, including the dragging/fracture of mooring system and the fracture failure of riser/wellhead/conductor system. A dynamic catastrophic analysis method was proposed and a simulation program was also developed based on MATLAB. A case study was presented to reveal the dynamic catastrophic evolution process and the susceptibility of coupled system to ISW. Results indicated that the coupled system would experience wellhead fracture, mooring line dragging or fracture, riser fracture, and flex joint failure under different ISW velocities. Through the quantitative analysis with different ISW strength, two catastrophic scenarios, namely, wellhead fracture-mooring line dragging and mooring line dragging-wellhead fracture-mooring line dragging, occurred.

Exploring solitary wave solutions to the simplified modified camassa-holm equation through a couple sophisticated analytical approaches

The simplified modified Camassa-Holm equation, a developed Korteweg-de Vries equation, is an integrable nonlinear dispersive water wave equation with potential applications in diverse fields such as shallow water waves, liquid drop patterning, water surface waves in channels, tsunamis in coastal regions, ocean engineering, etc. In this study, we derived solitary wave solutions to the described equation, employing two reliable approaches: the enhanced modified simple equation and the extended Kudryashov methods. The enhanced modified simple equation technique yields solutions with variable coefficients through a generalized wave transformation. On the other hand, the extended Kudryashov method exploits the solutions of the Riccati and Bernoulli differential equations. The determined results include diverse soliton solutions expressed in the form of trigonometric, hyperbolic, and rational functions. Notably, for specific parameter values, these solutions exhibit wave of distinct shapes, including kink, smooth kink, flat kink, periodic, singular periodic, one-sided kink, etc. To apparently explain the physical behavior and the impact of the fractional parameter in the solutions, we plotted three- and two-dimensional graphs. The obtained solutions have potential applications in oceanography, spanning offshore rigs, ocean gravity waves, and wave energy.

Numerical study of the Serre–Green–Naghdi equations in 2D *

A detailed numerical study of solutions to the Serre–Green–Naghdi (SGN) equations in 2D with vanishing curl of the velocity field is presented. The transverse stability of line solitary waves, 1D solitary waves being exact solutions of the 2D equations independent of the second variable, is established numerically. The study of localized initial data as well as crossing 1D solitary waves does not give an indication of existence of stable structures in SGN solutions localized in two spatial dimensions. For the numerical experiments, an approach based on a Fourier spectral method with a Krylov subspace technique is applied.