Nonlinear Wave Model Research Articles

Shape-morphing reduced-order models for nonlinear Schrödinger equations

We consider reduced-order modeling of nonlinear dispersive waves described by a class of nonlinear Schrodinger (NLS) equations. We compare two nonlinear reduced-order modeling methods: (i) The reduced Lagrangian approach which relies on the variational formulation of NLS and (ii) The recently developed method of reduced-order nonlinear solutions (RONS). First, we prove the surprising result that, although the two methods are seemingly quite different, they can be obtained from the real and imaginary parts of a single complex-valued master equation. Furthermore, for the NLS equation in a stationary frame, we show that the reduced Lagrangian method fails to predict the correct group velocity of the waves whereas RONS predicts the correct group velocity. Finally, for the modified NLS equation, where the reduced Lagrangian approach is inapplicable, the RONS reduced-order model accurately approximates the true solutions.

Data-driven soliton solutions and model parameters of nonlinear wave models via the conservation-law constrained neural network method

In the process of the deep learning, we integrate more integrable information of nonlinear wave models, such as the conservation law obtained from the integrable theory, into the neural network structure, and propose a conservation-law constrained neural network method with the flexible learning rate to predict solutions and parameters of nonlinear wave models. As some examples, we study real and complex typical nonlinear wave models, including nonlinear Schrödinger equation, Korteweg-de Vries and modified Korteweg-de Vries equations. Compared with the traditional physics-informed neural network method, this new method can more accurately predict solutions and parameters of some specific nonlinear wave models even when less information is needed, for example, in the absence of the boundary conditions. This provides a reference to further study solutions of nonlinear wave models by combining the deep learning and the integrable theory.

Analytical Investigation of Fractional-Order Korteweg–De-Vries-Type Equations under Atangana–Baleanu–Caputo Operator: Modeling Nonlinear Waves in a Plasma and Fluid

This article applies the homotopy perturbation transform technique to analyze fractional-order nonlinear fifth-order Korteweg–de-Vries-type (KdV-type)/Kawahara-type equations. This method combines the Zain Ul Abadin Zafar-transform (ZZ-T) and the homotopy perturbation technique (HPT) to show the validation and efficiency of this technique to investigate three examples. It is also shown that the fractional and integer-order solutions have closed contact with the exact result. The suggested technique is found to be reliable, efficient, and straightforward to use for many related models of engineering and several branches of science, such as modeling nonlinear waves in different plasma models.

Lie Symmetries, Closed-Form Solutions, and Various Dynamical Profiles of Solitons for the Variable Coefficient (2+1)-Dimensional KP Equations

This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with time-dependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and non-vanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are verified and validated by using back substitution to the original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these obtained solutions with some three-dimensional graphics via numerical simulation. The obtained variable coefficient solutions are more relevant and useful for understanding the dynamical structures of nonlinear KP equations and shallow water wave models.

A 2D Model for 3D Periodic Deep-Water Waves

The paper is devoted to further development of an accelerated method for simulation of the two-dimensional surface waves at infinite depth with the use of a two-dimensional model derived with simplifications of the three-dimensional equations for potential periodic deep-water waves. A 3D full wave model (FWM) is based on a numerical solution of a 3D Poisson equation written in the surface-fitted coordinates for a nonlinear component of the velocity potential. For sufficient vertical resolution used for the Poisson equation, the 3D model provides very high accuracy. The simplified model is based on the 2D Poisson equation written for a free surface. This exact equation contains both the first and second derivatives of the velocity potential, i.e., it is unclosed. The analysis of the accurate solutions for the 3D velocity potential obtained with the 3D model shows that those variables are linearly connected to each other. This property allows us to obtain a 2D equation for the first derivative of the velocity potential (i.e., vertical velocity on surface), which gives the closed 2D formulation for a 3D problem of two-dimensional waves. The previously developed scheme was not universal since the parameters of the closure scheme had to be adjusted to the specific setting. The current paper offers a new formulation of a closing scheme based on the integral parameters of wave field. The method of closing the equations, as well as the numerical parameters, were chosen on the basis of the multiple numerical experiments with the full nonlinear wave model (FWM) and selection of a suitable closing scheme. That is why the given model can be called Heuristic Wave Model (HWM). The connection between the first and second variables is not precise; hence, the method as a whole cannot be exact. However, the derived 2D model is able to reproduce different statistical characteristics of the 2D wave field with good accuracy. The main advantage of the model developed is its high performance exceeding that of 3D model by about two decimal orders.

A comparative study about the propagation of water waves with fractional operators

In the current research article, the truncated M-fractional derivative and Atangana–Baleanu derivative in Riemann–Liouville sense is practiced in the fractional modeling of the weakly non-linear shallow water wave equation. This paper provides some traveling wave transformations for the conversion of the partial fractional differential equation into an ordinary differential equation concerning fractional operators. In order to portray the wave propagation in weakly non-linear and dispersive media, the weakly non-linear shallow water wave model has been utilized. The proficient approach new extended direct algebraic scheme is used to find the solitonic structures. It is observed that the acquired solutions are purely new and have full novelty. The obtained results are carrying dark, dark singular, bright singular, and bright-dark singular solutions, which are obtained via Mathematica. The graphical comparison is also depicted in the perspective of operating fractional differential definitions with the selection of appropriate parametric values.

Scattering of Nonlinear Periodic (Cnoidal) Waves by a Partially Immersed Box-Type Breakwater

This paper presents a combined analytical and numerical (CAN) model to simulate the scattering of cnoidal waves by a fixed and partially immersed box-type breakwater. A set of Boussinesq equations are solved in the outer region using the finite-difference method to model the propagation of cnoidal waves and their subsequent reflection and transmission after encountering the breakwater. The two-dimensional (2D) velocity potential in the inner region beneath the body is derived analytically by solving the equations formulated from the orthogonality of eigenfunctions and the interfacial matching conditions. Experimental measurements on the wave profiles were carried out in a wave tank to verify the model solutions. Reflected and transmitted wave elevations obtained from the present CAN model match closely with the measured data. Additionally, the calculated horizontal and vertical forces on the body using the developed CAN model are in reasonable agreement with those from a potential 2D flow-based fully nonlinear wave model (FNWM). The method and proposed CAN model, if applied to a simple parametric investigation, can provide the expected trends in terms of applied forces, wave reflection, and transmission.

Wave propagation behavior in nonlinear media and resonant nonlinear interactions

The nonlinear Landau–Ginsberg–Higgs model, which depicts wave propagation in nonlinear media with a scattering system, classifies wave velocities, and effectively generates real events, is examined in this article. We use the recently enhanced couple of rising procedures to extract the important, applicable, and further general solitary wave solutions to the formerly stated nonlinear wave model via the complex traveling wave transformation. The solitons are constructed in terms of hyperbolic, exponential, rational, and trigonometric functions as well as their integration. The physical implications of the extracted wave solutions for the specific values of the resultant parameters are illustrated graphically, and the internal structure of the connected physical phenomena is analyzed using the Wolfram Mathematica program. This study shows that the method utilized is effective and may be used to find appropriate closed-form solitary solitons to a variety of nonlinear evolution equations (NLEEs).

Efficient time second-order SCQ formula combined with a mixed element method for a nonlinear time fractional wave model

<abstract><p>In this article, a kind of nonlinear wave model with the Caputo fractional derivative is solved by an efficient algorithm, which is formulated by combining a time second-order shifted convolution quadrature (SCQ) formula in time and a mixed element method in space. The stability of numerical scheme is derived, and an optimal error result for unknown functions which include an original function and two auxiliary functions are proven. Further, the numerical tests are conducted to confirm the theoretical results.</p></abstract>

Strong and weak interactions of rational vector rogue waves and solitons to anyn-component nonlinear Schrödinger system with higher-order effects

The higher-order effects play an important role in the wave propagations of ultrashort (e.g. subpicosecond or femtosecond) light pulses in optical fibres. In this paper, we investigate anyn-component fourth-order nonlinear Schrödinger (n-FONLS) system with non-zero backgrounds containing then-Hirota equation and then-Lakshmanan–Porsezian–Daniel equation. Based on the loop group theory, we find the multi-parameter family of novel rational vector rogue waves (RVRWs) of then-FONLS equation starting from the plane-wave solutions. Moreover, we exhibit the weak and strong interactions of some representative RVRW structures. In particular, we also find that the W-shaped rational vector dark and bright solitons of then-FONLS equation as the second- and fourth-order dispersion coefficients satisfy some relation. Furthermore, we find the higher-order RVRWs of then-FONLS equation. These obtained rational solutions will be useful in the study of RVRW phenomena of multi-component nonlinear wave models in nonlinear optics, deep ocean and Bose–Einstein condensates.

Two-dimensional self-similarity transformation theory and line rogue waves excitation

A two-dimensional self-similarity transformation theory is established, and the focusing (parabolic) (2 + 1)-dimensional NLS equation is taken as the model. The two-dimensional self-similarity transformation is proposed for converting the focusing (2 + 1)-dimensional NLS equation into the focusing (1 + 1) dimensional NLS equations, and the excitation of its novel line-rogue waves is further investigated. It is found that the spatial coherent structures induced by the Akhmediev breathers (AB) and Kuznetsov-Ma solitons (KMS) also have the short-lived characteristics which are possessed by the line-rogue waves induced by the Peregrine solitons, and the other higher-order rogue waves and the multi-rogue waves of the (1 + 1) dimensional NLS equations. This is completely different from the evolution characteristics of spatially coherent structures induced by bright solitons (including multi-solitons and lump solutions), with their shapes and amplitudes kept unchanged. The diagram shows the evolution characteristics of all kinds of resulting line rogue waves. The new excitation mechanism of line rogue waves revealed contributes to the new understanding of the coherent structure of high-dimensional nonlinear wave models.

Spatial self-similar transformation and novel line rogue waves in the Fokas system

We first propose a two-dimensional spatial self-similar transformation of the Fokas system, which is translated into the (1+1)-dimensional nonlinear Schrödinger equation, and is construct its abundant line rogue waves excitation. It found that the line rogue waves induced by the Akhmediev breathers and Kuznetsov-Ma solitons also have the short life characteristics possessed for the line rogue waves induced by the Peregrine solitons and other higher-order rogue waves and multi-rogue waves of the (1+1) dimensional standard NLS equations. This is completely different from the evolution characteristics of line soliton induced by bright solitons and the multi-solitons, which keeping their shape and amplitude unchanged. The diagram shows the evolution characteristics of the resulting all kinds of line rogue waves. The new excitation mechanism of line rogue waves for the Fokas system revealed contributes to the new understanding of the localized coherent structure of high-dimensional nonlinear wave models.

Modelling the Impact of Climate Change on Coastal Flooding: Implications for Coastal Structures Design

In the present work, the impact of climate change on coastal flooding is investigated through a set of interoperable models developed by the authors, following a modular modelling approach and adapting the modelling sequence to two separate objectives with respect to inundation over large-scale areas and coastal protection structures’ design. The modelling toolbox used includes a large-scale wave propagation model, a storm-induced circulation model, and an advanced nearshore wave propagation model based on the higher order Boussinesq-type equations, all of which are presented in detail. Model capabilities are validated and applications are made for projected scenarios of climate change-induced wave and storm surge events, simulating coastal flooding over the low-lying areas of a semi-enclosed bay and testing the effects of different structures on a typical sandy beach (both in northern Greece). This work is among the few in relevant literature that incorporate a fully non-linear wave model to a modelling system aimed at representing coastal flooding. Results highlight the capabilities of the presented modelling approach and set the basis for a comprehensive evaluation of the use of advanced modelling tools for the design of coastal protection and adaptation measures against future climatic pressures.

Experimental and Numerical Simulation of Cross-Shore Morphological Processes in a Nourished Beach

Spyrou, D. and Karambas, T., 2021. Experimental and numerical simulation of cross-shore morphological processes in a nourished beach. Journal of Coastal Research, 37(5), 1012–1024. Coconut Creek (Florida), ISSN 0749-0208. This paper presents the results of an experimental and numerical research carried out to reproduce the cross-shore evolution of nourished sandy beaches. An extensive two-dimensional laboratory investigation was performed to study the cross-shore profile evolution after the application of a beach nourishment method. A 1:20 scale physical model was carried out in the Laboratory of the Department of Hydraulics and Environmental Engineering, Aristotle University of Thessaloniki. The analysis considered only cross-shore sediment transport. Long-shore sediment transport is considered negligible. Consequently, the obtained results are related to the short-term response of the cross-shore profile. Ten different wave combinations were generated in an attempt to reproduce erosive conditions. Experimental results are given in terms of wave parameters, bed slope, and berm height of the nourished beach. An advanced nonlinear wave model was developed to simulate cross-shore morphodynamical processes. The comparison between the experimental and numerical beach profile indicated good agreement. Also, comparisons between Dean's analytical method and the experimental results were made.

Nonlinear wave transitions and their mechanisms of (2+1)-dimensional Sawada–Kotera equation

The transitions and mechanisms of nonlinear waves in the (2+1)-dimensional Sawada–Kotera (2DSK) equation are studied by means of characteristic line and phase shift analysis, and the dynamic behavior of various nonlinear transformed waves is analyzed. Firstly, we obtain the N-soliton solution based on the Hirota bilinear method, from which the breath wave solution is constructed by changing the parameters into a complex form in pairs, and lump solution is obtained via the long wave limit method. Then the mechanism of the breath wave solution transformation is studied by characteristic line analysis, we present the types of transformed nonlinear waves, including quasi-anti-dark soliton, M-shaped soliton, W-shaped soliton, multi-peak soliton, and quasi-periodic wave soliton, and the distribution diagrams of these nonlinear waves on the (α,β) plane is rendered. We further reveal the gradient properties of the transformed wave. In addition, the transformed wave is decomposed into a solitary wave and a periodic wave component, and the formation mechanism, locality, and oscillation properties of the nonlinear transformed wave are explained through the nonlinear superposition. Furthermore, we demonstrate that the geometric properties of the characteristic lines vary with time essentially resulting in the time-varying properties of nonlinear waves, which have never been found in (1+1)-dimensional systems. Based on high-order nonlinear waves, the state transitions of the mixed solution and the second-order breath wave solution are investigated. We show several collision models of nonlinear waves, and reveal that the phase shift difference between the solitary and the periodic wave component leads to the deformable collision of the transformed wave. Such phase shift is due to time evolution and wave interaction. Finally, the dynamic process of nonlinear wave collision under the combined action of time and collision is presented.

Asimmetriya i ekstsess poverkhnostnykh voln v pribrezhnoy zone Chernogo morya

Purpose. The aim of the study is to analyze variability of the statistical moments characterizing deviations of the sea surface elevation distributions from the Gaussian one. Methods and Results. Field studies of the sea waves’ characteristics were carried out from the stationary oceanographic platform located in the Black Sea near the Southern coast of Crimea. The data obtained both in summer and winter, were used. The statistical moments were calculated separately for wind waves and swell. The measurements were performed in a wide range of meteorological conditions and wave parameters (wind speed varied from 0 to 26 m/s, wave age – from 0 to 5.2 and steepness – from 0.005 to 0.095). For wind waves, the coefficients of skewness correlation with the waves’ steepness and age were equal to 0.46 and 0.38. The kurtosis correlation coefficients with these parameters were small (0.09 and 0.07), but with the confidence level 99.8% – significant. For swell, the correlation coefficients were 1.5 – 2.0 times lower. Conclusions. The statistical moments of the sea surface elevations of the third and higher orders are the indicators of the wave field nonlinearity, which should be taken into account when solving a wide range of the applied and fundamental problems. The deviations of the surface elevation distributions from the Gaussian one are not described unambiguously by the waves’ steepness and age. At the fixed values of these parameters, a large scatter in the values of the surface elevations’ asymmetry and kurtosis is observed. This imposes significant limitations on the possibility of applying the nonlinear wave models based on the wave profile expansion by small parameter (steepness) degrees, in engineering calculations.

Skewness and Kurtosis of the Surface Wave in the Coastal Zone of the Black Sea

Purpose. The aim of the study is to analyze variability of the statistical moments characterizing deviations of the sea surface elevation distributions from the Gaussian. Methods and Results. Field studies of the sea waves’ characteristics were carried out from the stationary oceanographic platform located in the Black Sea near the Southern coast of Crimea. The data obtained both in summer and winter, were used. The statistical moments were calculated separately for wind waves and swell. The measurements were performed in a wide range of meteorological conditions and wave parameters (wind speed varied from 0 to 26 m/s, wave age – from 0 to 5.2 and steepness – from 0.005 to 0.095). For wind waves, the coefficients of skewness correlation with the waves’ steepness and age were equal to 0.46 and 0.38. The kurtosis correlation coefficients with these parameters were small (0.09 and 0.07), but with the confidence level 99.8% – significant. For swell, the correlation coefficients were 1.5 – 2.0 times lower. Conclusions. The statistical moments of the sea surface elevations of the third and higher orders are the indicators of the wave field nonlinearity, which should be taken into account when solving a wide range of the applied and fundamental problems. The deviations of the surface elevation distributions from the Gaussian one are not described unambiguously by the steepness and wave age. At the fixed values of these parameters, a large scatter in the skewness and kurtosis of the surface elevations is observed. This imposes significant limitations on the possibility of applying the nonlinear wave models based on the wave profile expansion by small parameter (steepness) degrees, in engineering calculations.

Explicit wave phenomena to the couple type fractional order nonlinear evolution equations

We utilize the fractional modified Riemann-Liouville derivative in the sense to develop careful arrangements of space–time fractional coupled Boussinesq equation which emerges in genuine applications, for instance, nonlinear framework waves iron sound waves in plasma and in vibrations in nonlinear string and space–time fractional-coupled Boussinesq Burger equation that emerges in the investigation of liquids stream in a dynamic framework and depicts engendering of shallow-water waves. A decent comprehension of its solutions is exceptionally useful for beachfront and engineers to apply the nonlinear water wave model to the harbor and seaside plans. A summed-up partial complex transformation is correctly used to change this equation to a standard differential equation thus, many precise logical arrangements are acquired with all the free parameters. At this point, the traveling wave arrangements are articulated by hyperbolic functions, trigonometric functions, and rational functions, if these free parameters are considered as specific values. We obtain kink wave solution, periodic solutions, singular kink type solution, and anti-kink type solutions which are shown in 3D and contour plots. The presentation of the method is dependable and important and gives even more new broad accurate arrangements.

Resonant line wave soliton solutions and interaction solutions for (2+1)-dimensional nonlinear wave equation

Based on the N-soliton solutions, the resonant line wave soliton and interaction solutions are derived through some constraints in the (2+1)-dimensional nonlinear wave equation. General resonant line wave soliton solutions are firstly presented and their changing routes are illustrated. Then, several interaction solutions including a nonlinear superposition of resonant line wave soliton with breather wave, a hybrid between resonant line wave soliton and lump wave are constructed via long-wave limit method and module resonant mechanism. The characteristics and properties of these interaction solutions are discussed analytically and graphically. Localized wave and interaction solutions of the nonlinear wave models have a great impact on oceanography and physics. The results may be useful in investigating the physical phenomena in shallow water waves and nonlinear optics.

Nonlinear Wave Model for Transport Phenomena in Media with Non-local Effects

The purpose of the work is to establish and substantiate the prerequisites under which the modified Whitham equation can play the role of a transfer equation in physical-chemical systems and technological processes.The contribution of this work lie in the fact that the perturbed Whitham equation describing propagation of nonlinear waves in media with non-locality has been derived using the nonlinear relaxation function. It is shown that while deriving the equations of the Whitham type, the presence of the spatial non-locality of the medium can play a fundamental role. The paper also considers the issues of physical interpretation of the model under study. The principal presence of non-local effects in studied systems can be substantiated and obtained as a result of the presence of domains with a complexly organized spatial structure as well as in the presence of heat and mass sources. The new model submitted in the work can find application in methods for calculating the intensity of heat and mass transfer in complexly structured heterogeneous systems.