Nonlinear Dispersive Model Research Articles

Analytical and numerical studies on dynamic of complexiton solution for complex nonlinear dispersive model

In this paper, we apply the basic one‐way propagator in order to introduce complex regularized long wave (RLW) equation. The complex RLW can be transformed into a system of nonlinear equations. By adopting a consequence nonlinear system, we can derive an energy conservation law of a complex regularized long wave equation. We then investigate how the Ansatz method may be applied to find a class of solitary wave solutions. Simultaneously, a numerical scheme for solving the model is implemented using a finite difference method based on the energy‐preserving Crank–Nicolson/Adams–Bashforth technique. It is worth mentioning that the obtained system is nonlinear. However, by using the present algorithm, we are able to linearize this system and solve it because of the implicit nature of the system of equations. An a priori estimate of the numerical solutions is derived to obtain a convergence and stability analysis; this yields second‐order accuracy in both time and space. Additionally, some numerical experiments verify computational efficiency. The results indicate that this method is an excellent way to preserve energy conservation, providing second‐order accuracy both in time and space with a maximum norm. In addition, we use the proposed scheme to study the effects of dispersive parameters when proceeding with an initial complex Gaussian condition.

Evolution of plume geometry, dilution and reactive mixing in porous media under highly transient flow fields at the surface water-groundwater interface

Highly transient boundary conditions affect mixing of dissolved solutes in groundwater. An example of these transient boundary conditions occurs at the surface water-groundwater interface, where the water level in rivers can change rapidly due to the operation of hydropower plants, leading to a regime known as hydropeaking. Inspired by this phenomenon, this work studies at laboratory scale the effects of fluctuating surface water bodies on solute transport in aquifers. We performed flow-through experiments at two different flow velocities and under steady and transient flow conditions where a conservative tracer was injected in the system and its concentration measured with optical imaging methods. The experimental results were quantitatively interpreted with numerical simulations implementing a non-linear velocity-dependent dispersive transport model. We estimated plume dilution by computing the dilution index and its evolution as well as two key geometrical metrics of the transient plumes: the perimeter and the area. We further investigated reactive mixing and mixing enhancement considering mixing-controlled bimolecular reactions using different critical mixing ratios. In general, highly transient boundary conditions lead to a larger area, perimeter and plume dilution and the results show greater relative enhancement for the scenarios with low groundwater flow velocity. A linear relationship was observed between the evolution of the area and the dilution index of the plumes for the transient flow scenarios investigated. Considering reactive transport and mixing-limited reactions at the surface water-groundwater interface, we identified different dilution and reaction dominated regimes, characterized, respectively, by increasing and decreasing plume entropies at different mixing ratios of the reactants. Furthermore, reactive mixing was enhanced by transient flows leading to a faster degradation of contaminant plumes compared to corresponding steady flow conditions.

Numerical modeling of the long surface wave impact on a partially immersed structure in a coastal zone: Solitary waves over a flat slope

This paper describes the numerical simulation of the solitary wave impact on a partially immersed and fixed structure located over a flat coastal slope. This topic is related to the need for assessment of the possible impact of long waves, such as tsunamis, on partially immersed structures in coastal waters. Numerical algorithms on a movable grid adapting to the motion of the shore point are developed for a fully nonlinear dispersive model and a dispersionless shallow water model. Their validation is carried out by comparing the obtained solutions with the data from laboratory experiments and with the results obtained using a fully nonlinear potential flow model. The study shows that the difference between the maximum wave impact on the body at the foot of the slope and near the shore can be up to 6 times. In many cases, the maximum horizontal component of the wave force occurs under the influence of the wave reflected from the shore, indicating the need to consider the influence of the shore-reflected wave when assessing the impact of long waves on structures located in coastal waters. Furthermore, the need to use runup algorithms instead of reflective boundary conditions (vertical wall) has been identified for gentler slopes, where the differences in the wave impact for these two configurations can be 2–3 times.

Phase and Amplitude Characteristics of Nonlinear Dispersion Models Improved Precision

Phase and Amplitude Characteristics of Nonlinear Dispersion Models Improved Precision

Shallow water modeling of wave–structure interaction over irregular bottom

This paper describes the results from a numerical simulation of long surface waves interaction with fixed and partially immersed structure located over irregular bottom. The topic is connected with risk of critical wave impact on structures placed in the coastal waters of seas and oceans. The wave–structure interaction is simulated within the framework of a hierarchy of mathematical models, including the model of potential flows of an ideal fluid, a fully nonlinear dispersive and dispersionless shallow water models. The numerical algorithm for the dispersive model is described in detail. The obtained results show that rectangular cutout in the structure bottom reduces the amplitude of the reflected wave and increases that of the transmitted one; the basin bottom irregularity plays the most important role when located directly under the body. The protrusion behind the body may slightly decrease the wave force, and it significantly increases the force at other locations. Generally, the results obtained by the dispersive model correspond well with those of potential model, while the dispersionless model often significantly overestimates the wave force and amplitude of the reflected wave.

A NOVEL ANALYTICAL APPROACH TO FRACTAL NONLINEAR DISPERSIVE MODEL WITH VARIABLE COEFFICIENTS

In this paper, the fractal nonlinear dispersive model (FNDM) with variable coefficients is investigated by variational perspective, and its fractal variational principle is constructed by using the He’s fractal semi-inverse method. Based on the fractal variational principle, a novel analytical algorithm is successfully established to solve the fractal model. Two examples are given to illustrate the efficiency and accuracy of the proposed algorithm.

Assorted soliton solutions to the nonlinear dispersive wave models in inhomogeneous media

The Sharma-Tasso-Olver and Klein–Gordon equations are significant models to interpret plasma physics, relativistic physics, quantum mechanics, nonlinear optics, etc. In the present article, an advanced generalized approach, namely the generalized (G′/G)-expansion approach is scheduled to unfold certain wave solutions to the nonlinear evolution equations earlier described. The soliton solutions exposed are estimated in the form of hyperbolic, rational and trigonometric functions. The physical implications of the ascertained solutions are summed up by setting specific values of the integrated constraints and delineating the profiles to decode the physical phenomena. The numerical simulation of the solutions extracted is standard kink, rogue wave, bright-dark soliton, periodic soliton, breather type soliton, compaction, singular kink soliton, etc. This study affirms that the introduced scheme is robust, efficient in searching for nonlinear evolution equations, computer algebra compatible, and capable of finding further inclusive wave solutions.

Computational investigation on a nonlinear dispersion model with the weak non-local nonlinearity in quantum mechanics

In the presence of nonlinear dispersion, we examine certain solitary wave solutions of a dimensionless nonlinear Schrödinger (DLNLS) equation with the parabolic law of nonlinearity. We use a unique computational (generalized rational (GRat)) approach to build a variety of solutions with varying shapes. These answers show how the nonlinear interaction between Langmuir waves and electrons causes the parabolic law nonlinearity. For a better visual description of the examined model, these interactions are shown using several graphs in three-, two-dimensional, and polar plots. These representations of the obtained solutions are considered as numerical simulaions to explain the investigated model’s characterizations, the pulse waves’ behaviour and the interaction between complex short wave and real long wave envelope.

New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography

This paper presents a study of nonlinear waves in shallow water. The Korteweg-de Vries (KdV) equation has a canonical version based on oceanography theory, the shallow water waves in the oceans, and the internal ion-acoustic waves in plasma. Indeed, the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method (HPTM) to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness. This approach is connected with the fuzzy generalized integral transform and HPTM. Besides that, two novel results for fuzzy generalized integral transformation concerning fuzzy partial gH-derivatives are presented. Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method. Furthermore, 2D and 3D simulations depict the comparison analysis between two fractional derivative operators (Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense) under generalized gH-differentiability. The projected method (GHPTM) demonstrates a diverse spectrum of applications for dealing with nonlinear wave equations in scientific domains. The current work, as a novel use of GHPTM, demonstrates some key differences from existing similar methods.

Плановая модель волновой гидродинамики с дисперсионным соотношением повышенной точности. II. Четвертый, шестой и восьмой порядки

Построена полностью нелинейная слабо дисперсионная модель волновой гидродинамики четвертого порядка длинноволновой аппроксимации. За скорость в модели взята усредненная по глубине горизонтальная составляющая скорости трехмерного течения. Учтена подвижность дна. Выполненная модификация модели обеспечивает шестой и восьмой порядки точности аппроксимации дисперсионного соотношения трехмерной модели потенциальных течений. In the numerical simulation of medium-length surface waves in the framework of nonlinear dispersive (NLD) models, an increased accuracy of reproducing the characteristics of the simulated processes is required. A number of works (Kirby (2016), e.g.) describe approaches to improve the known NLD-models. In particular, NLD-models of the fourth order of the long-wave approximation have been proposed and, based on a comparison of numerical results with experimental data, their high accuracy has been demonstrated (Ataie-Ashtiani and Najafi-Jilani (2007); Zhou and Teng (2010)). In these new models, the horizontal component of the velocity vector of the threedimensional (FNPF-) model of potential flows at a certain surface located between the bottom and the free boundary was chosen as the velocity vector. The result was a very cumbersome form of equations. In addition, the laws of conservation of mass and momentum do not hold for these models. The main result of this work is the derivation of a two-parameter fully nonlinear weakly dispersive (mSGN4) model of the fourth order of the long-wave approximation, which is a generalization of the well-known Serre-Green-Naghdi (SGN) second order model. In the derivation, the velocity averaged over the thickness of the liquid layer was used. The assumption about the potentiality of the three-dimensional flow was used only at the stage of closing the model. The movement of the bottom is taken into account. For the derived model, the law of conservation of mass is satisfied, and the law of conservation of total momentum is satisfied in the case of a horizontal stationary bottom. The equations of the mSGN4-model are invariant under the Galilean transformation and are presented in a compact form similar to the equations of gas dynamics. The dispersion relation of the mSGN4-model has the fourth order of accuracy in the long wave region and satisfactorily approximates the dispersion relation of the FNPF-model in the short wave region. Moreover, with a special choice of the values of the model parameters, an increased accuracy of approximating the dispersion relation of the FNPF-model at long waves (sixth or eighth order) is achieved. Analysis of the deviations of the values of the phase velocity of the mSGN4 model from the values of the “reference” speed of the FNPF model in the entire wavelength range showed that the most preferable is the mSGN4 model with the parameter values corresponding to the Pad’e approximant (2,4).

High-order Accurate Schemes for Maxwell-Bloch Equations with Material Interfaces

Efficient and high-order accurate FDTD schemes are developed for nonlinear dispersive models in active multi-level media with material interfaces in 2D and 3D arbitrary geometry. The considered nonlinear systems are multi-level generalizations of 2-level optical Maxwell-Bloch equations. Arbitrary numbers of atomic levels and macroscopic polarization vectors can be accounted for in the multi-level atomic (MLA) system. The MLA models are suitable for modeling active media with various properties, such as lasing, saturable absorption, reverse saturable absorption, etc. Composite overlapping grids are employed to handle complex geometry with material interfaces, using locally conforming curvilinear grids for curved boundaries and interfaces and Cartesian grids for the rest. The developed high-order schemes allow compact stencils in time integration, and efficient point-wise update of the numerical solutions.

Multiphysics modelling of electroporation under uni- or bipolar nanosecond pulse sequences

A nonlinear dispersive multiphysics model of single-cell electroporation was proposed in this paper. The time-domain Debye model was utilised to describe the membrane dispersion while the dynamic pore radius function was deployed to modify the plasma membrane conductivity. The dynamic spatial distributions of the ion concentration were dominated by the Nernst-Planck function. First, a single nanosecond pulsed electric field was applied to verify our model and to explore the effects of dispersion and dynamic pore radius on the redistribution of the electric field. The dispersive membrane was found to increase the transmembrane potential, expedite the electroporation process, and weaken the membrane permeability; however, adding the dynamic pore radius function had the opposite effect on transmembrane potential and membrane permeability. The responses of the cells exposed to unipolar and bipolar nanosecond pulse sequences were subsequently simulated. During the application of unipolar pulse sequences, the pore radius and perforation area showed a step-like accumulation, and significant increases in the perforation area and intracellular ion concentration were observed with higher frequency pulse sequences and wider subpulse intervals. The bipolar cancellation effect was also observed in terms of membrane permeability and pore radius.

Nonlinear dispersive equations: classical and new frameworks

The purpose of this manuscript associated to the Golden Jubilee of the IME-USP is to present selected material from the author’s scientific contribution dealing with nonlinear phenomena of dispersive type. That study has been a source for modern research in the dynamic of traveling wave solutions of different type and it has been disseminated through publication of scientific articles and/or books. Here, we provide the reader with information about current research in stability theory for nonlinear dispersive equations and possible developments. Also, I hope it inspires future developments in this important and fascinating subject. In this manuscript we consider the following topics: stability theory of solitary waves and the applicability of the concentration–compactness principle, the existence and orbital (in)stability of periodic traveling wave solutions for nonlinear dispersive models, nonlinear Schrodinger and Korteweg–de Vries models on star-shaped metric graphs. The use of tools of the theory of spaces of Hilbert, the spectral theory for unbounded self-adjoint operators, Sturm–Liouville’s theory, variational methods, analytic perturbation theory of operators, and the extension theory of symmetric operators are pieces fundamental in our study. The methods presented in this manuscript have prospect for the study of the dynamic of solutions for nonlinear evolution equations around of different traveling waves profiles which may appear in non-standard environments such as star-shaped metric graphs.

Study on the dynamics of a nonlinear dispersion model in both 1D and 2D based on the fourth-order compact conservative difference scheme

In this paper, the nonlinear dispersion wave model in both 1D and 2D is studied by the compact finite difference method, which is called the generalized Rosenau–RLW equation. A fourth-order compact three-level and linearized difference scheme that maintains the original conservative properties of equation is proposed. The discrete mass conservation and discrete energy conservation of compact difference scheme are obtained. The solvability of numerical scheme is obtained. By using the discrete energy method, convergence and unconditional stability can also be obtained without relying on the grid ratio, and the optimal error estimates in the L∞ norm are fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. The scheme is conservative so can be used for long time computation. Numerical experiment results show that the theory is accurate and the method is efficient and reliable. Finally, the new numerical scheme is used to study the nonlinear dynamic of 1D generalized Rosenau–RLW equation and the wave interference of 2D generalized Rosenau–RLW equation, focusing on the influence of nonlinear convection term on the wave and the conservation of wave propagation.

Nonlinear dispersive cell model for microdosimetry of nanosecond pulsed electric fields

For applications based on nanosecond pulsed electric fields (nsPEFs), the underlying transmembrane potential (TMP) distribution on the plasma membrane is influenced by electroporation (EP) of the plasma membrane and dielectric dispersion (DP) of all cell compartments which is important for predicting the bioelectric effects. In this study, the temporal and spatial distribution of TMP on the plasma membrane induced by nsPEFs of various pulse durations (3 ns, 5 ns unipolar, 5 ns bipolar, and 10 ns) is investigated with the inclusion of both DP and EP. Based on the double-shelled dielectric spherical cell model, the Debye equation describing DP is transformed into the time-domain form with the introduction of polarization vector, and then we obtain the time course of TMP by solving the combination of Laplace equation and time-domain Debye equation. Next, the asymptotic version of the Smoluchowski equation is included to characterize the EP of plasma membrane in order to observe more profound electroporation effects with larger pore density and electroporated areas in consideration of both DP and EP. Through the simulation, it is clearer to understand the relationship between the applied nsPEFs and the induced bioelectric effects.

Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model

Before going further with fractional derivative which is constructed by Rabotnov exponential kernel, there exist many questions that are not addressed. In this paper, we try to recapitulate all the fundamental calculus, which we can obtain with this new fractional operator. The problems in this paper are to determine the solutions of the fractional differential equations where the second members are constant functions, polynomial functions, exponential functions, trigonometric functions, or Mittag-Leffler functions. For all the fractional differential equations, the obtained solutions are represented graphically. The Laplace transform of the fractional derivative with Rabotnov exponential kernel is the primary tool in the investigations. Finally, we give the fundamental solution to the nonlinear time-fractional modified Degasperis–Procesi equation by considering the fractional operator with Rabotnov exponential kernel.

Mechanical Solving a Few Fractional Partial Differential Equations and Discussing the Effects of the Fractional Order

With the help of Maple, the precise traveling wave solutions of three fractal-order model equations related to water waves, including hyperbolic solutions, trigonometric solutions, and rational solutions, are obtained by using function expansion method. An isolated wave solution is selected from the solution of each nonlinear dispersive wave model equation, and the influence of fractional order change on these isolated wave solutions is discussed. The results show that the fractional derivatives can modulate the waveform, local periodicity, and structure of the isolated solutions of the three model equations. We also point out the construction rules of the auxiliary equations of the extended (G′/G)-expansion method. In the “The Explanation and Discussion” section, a more generalized auxiliary equation is used to further emphasize the rules, which has certain reference value for the construction of the new auxiliary equations. The solutions of fractional-order nonlinear partial differential equations can be enriched by selecting other solvable equations as auxiliary equations.

New nonlinear theory for a piston-type wavemaker: The classical Boussinesq equations

In this study, we present a new nonlinear theory for a moving boundary wavemaker of piston-type based on a nonlinear dispersive shallow water model, where the classical Boussinesq equations are employed as a starting point. The new theory is inherently different from the traditional wavemaker theories, such as the usual theories employed for solving the Laplace equation equipped with the free surface boundary conditions by using the perturbation approach. To verify the wavemaker theory proposed in this study, the ratio of the wave height relative to the stroke characterizing the performance of the wavemaker was observed and compared with numerical, experimental, and Havelock's theoretical results, thereby confirming that the results obtained with the proposed theory were in significant agreement. Furthermore, a comparison of the solitary wave generated by the proposed theory and the known exact solution showed that they were in good agreement.

Numerical simulation of ship-borne waves using a 2DH post-Boussinesq model

This work presents results on the simulation of the generation and propagation of ship-borne waves, using an advanced nonlinear dispersive wave model based on the higher order Boussinesq-type equations. The model includes a single frequency dispersion term, expressed through convolution integrals; it is adapted to represent ship-borne waves by adopting the approach for wave generation by a moving local pressure disturbance, achieved through adding a respective pressure term in its governing equations. The model is tested against an analytical solution for the calculation of the angles of ship wakes and laboratory experiments of waves produced by a high-speed ship in a channel. Results compare well to data from both the aforementioned studies, confirming the model's capabilities and highlighting its accuracy in the representation of ship-borne waves; the model's applicability to practical coastal engineering applications is also investigated, through a set of numerical experiments of waves generated at sea and in a harbour.

High order ADER-DG schemes for the simulation of linear seismic waves induced by nonlinear dispersive free-surface water waves

In this paper, we propose a unified and high order accurate fully-discrete one-step ADER Discontinuous Galerkin method for the simulation of linear seismic waves in the sea bottom that are generated by the propagation of free surface water waves. In particular, a hyperbolic reformulation of the Serre-Green-Naghdi model for nonlinear dispersive free surface flows is coupled with a first order velocity-stress formulation for linear elastic wave propagation in the sea bottom. To this end, Cartesian non-conforming meshes are defined in the solid and fluid domains and the coupling is achieved by an appropriate time-dependent pressure boundary condition in the three-dimensional domain for the elastic wave propagation, where the pressure is a combination of hydrostatic and non-hydrostatic pressure in the water column above the sea bottom. The use of a first order hyperbolic reformulation of the nonlinear dispersive free surface flow model leads to a straightforward coupling with the linear seismic wave equations, which are also written in first order hyperbolic form. It furthermore allows the use of explicit time integrators with a rather generous CFL-type time step restriction associated with the dispersive water waves, compared to numerical schemes applied to classical dispersive models that contain higher order derivatives and typically require implicit solvers. Since the two systems that describe the seismic waves and the free surface water waves are written in the same form of a first order hyperbolic system they can also be efficiently solved in a unique numerical framework. In this paper we choose the family of arbitrary high order accurate discontinuous Galerkin finite element schemes, which have already shown to be suitable for the numerical simulation of wave propagation problems. The developed methodology is carefully assessed by first considering several benchmarks for each system separately, i.e. in the framework of linear elasticity and non-hydrostatic free surface flows, showing a good agreement with exact and numerical reference solutions. Finally, also coupled test cases are addressed. Throughout this paper we assume the elastic deformations in the solid to be sufficiently small so that their influence on the free surface water waves can be neglected.