Nonlinear Dispersive Waves Research Articles

Evolution of high frequency waves in shoaling and breaking wave spectra

Mathematical derivation and numerical verification of a wave transformation model in the frequency domain are discussed. The model is a fully dispersive nonlinear wave model and is derived based on the boundary value problem. Transforming the problem in the frequency domain and using multiple scale analysis in space and perturbation theory, the model is expanded up to second order in wave steepness. This fully dispersive nonlinear wave model is a set of evolution equations which explicitly contains quadratic near-resonant interactions. The comparison between the presented model, the existing fully dispersive model, and a nearshore model with different sets of laboratory and field data shows that the presented model provides significant improvements particularly at higher frequency.

Error estimates for Galerkin finite element methods for the Camassa–Holm equation

We consider the Camassa–Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order $$L^{2}$$ -error estimates for the semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge–Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the ‘peakon’ type.

Tsunami Squares modeling of landslide generated impulsive waves and its application to the 1792 Unzen-Mayuyama mega-slide in Japan

The frequent occurrence of landslide tsunami has aroused great social concern, hence an efficient approach for modeling the phenomena is required urgently. Herein, we present and supplement a meshless numerical model named ‘Tsunami Squares’ (TS). TS simulates both the landslide and its generated wave using an updated numerical scheme that has enhanced computational efficiency. TS also provides an artful way of approximating linear dispersive wave propagation and non-linear wave inundation that makes it quite capable of simulating real-scale events. A well-designed test that compares TS output with an analytical formula for a tsunami propagation calculation provides firm validation of the approach. We then applied the ‘Tsunami Squares’ to simulate the 1792 Unzen-Mayuyama mega slide and generated tsunami. This event remains the largest volcanic disaster in the history of Japan. Landslide dynamics simulated by TS coincide with historical reports and the known geometry of landslide deposits. The model output of tsunami runup heights well match the observed values recorded by the locations of historical ‘Tsunami Stones’. This well documented and data-rich case further validates the ‘Tsunami Squares’ model and the excellent reproduction of the 1792 event provides new insights for better understanding of landslide tsunami hazards.

On the utility of C/S as an indicator of nonlinearity in finite-amplitude dispersive waves

A frequency-domain nonlinearity indicator C/S, recently introduced by Ohm et al. [Proc. Mtgs. Acoust. 29, 045003 (2016)], is a companion indicator to the well-known Morfey-Howell Q/S [AIAA J. 19, 986–992 (1981)]. The two indicators can be regarded as “two sides of the same coin,” because C/S and Q/S are, respectively, defined by the real and imaginary parts of the cross-spectrum of the pressure and squared-pressure waveforms. Despite the common origin, however, C/S and Q/S describe the pertinent nonlinear wave process in a different light. Furthermore, for nonlinear waves in media with strong dispersion only C/S may be applicable, because the use of Q/S could lead to erroneous interpretations. In this paper, we demonstrate the utility of C/S in the context of the Korteweg-de Vries (KdV) equation, which is one of the most celebrated model equations for nonlinear waves in dispersive media. The success of C/S is juxtaposed with the limitation of Q/S in describing nonlinear dispersive wave processes, especially in the case of solitons.

Sufficient conditions for orbital stability of periodic traveling waves

The present paper deals with sufficient conditions for orbital stability of periodic waves of a general class of evolution equations supporting nonlinear dispersive waves. Our method can be seen as an extension to spatially periodic waves of the theory of solitary waves recently developed in [25]. Firstly, our main result do not depend on the parametrization of the periodic wave itself. Secondly, motived by the well known orbital stability criterion for solitary waves, we show that the same criterion holds for periodic waves. In addition, we show that the positiveness of the principal entries of the Hessian matrix related to the “energy surface function” are also sufficient to obtain the stability. Consequently, we establish the orbital stability of periodic waves for several nonlinear dispersive models. We believe our method can be applied in a wide class of evolution equations; in particular it can be extended to regularized dispersive wave equations.

Accelerating dynamical peakons and their behaviour

A wide class of nonlinear dispersive wave equations are shown to possess a novel type of peakon solution in which the amplitude and speed of the peakon are time-dependent. These novel dynamical peakons exhibit a wide variety of different behaviours for their amplitude, speed, and acceleration, including an oscillatory amplitude and constant speed which describes a peakon breather. Examples are presented of families of nonlinear dispersive wave equations that illustrate various interesting behaviours, such as asymptotic travelling-wave peakons, dissipating/anti-dissipating peakons, direction-reversing peakons, runaway and blow up peakons, among others.

On time relaxed schemes and formulations for dispersive wave equations

The numerical simulation of nonlinear dispersive waves is a central research topic of many investigations in the nonlinear wave community. Simple and robust solvers are needed for numerical studies of water waves as well. The main difficulties arise in the numerical approximation of high order derivatives and in severe stability restrictions on the time step, when explicit schemes are used. In this study we propose new relaxed system formulations which approximate the initial dispersive wave equation. However, the resulting relaxed system involves first order derivatives only and it is written in the form of an evolution problem. Thus, many standard methods can be applied to solve the relaxed problem numerically. In this article we illustrate the application of the new relaxed scheme on the classical Korteweg-de Vries equation as a prototype of stiff dispersive PDEs.

Local-in-space blow-up criteria for two-component nonlinear dispersive wave system

We investigate the blow-up phenomena for the two-component generalizations of Camassa-Holm equation on the real line. We establish some a local-in-space blow-up criterion for system of coupled equations under certain natural initial profiles. Presented result extends and specifies the earlier blow-up criteria for such type systems.

A MODEL IN FREQUENCY DOMAIN FOR TRANSFORMATION OF FULLY DISPERSIVE NONLINEAR WAVES

In this study, mathematical derivation and numerical verification of a wave transformation model in frequency domain is discussed. This wave model is fully dispersive and nonlinear; and is derived based on the WKB assumptions. Transforming the problem into the frequency domain and using multiple scale analysis in space and perturbation theory, the model is expanded up to second order in wave steepness. This fully dispersive nonlinear wave model is a set of evolution equations which explicitly contains quadratic near-resonant interactions. The comparison between the presented model, the existing fully dispersive model and a nearshore model with different set of laboratory and field data shows that the presented model provides significant improvements particularly at higher frequencies.

Homoclinic breather waves, rogue waves and solitary waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation

PurposeThe purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which can be used to describe the propagation of weakly nonlinear dispersive long waves on the surface of a fluid.Design/methodology/approachThe authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional gKP equation.FindingsThe results imply that the gKP equation admits rogue waves, homoclinic breather waves and soliton waves. Moreover, the authors also find that rogue waves can come from the extreme behavior of the breather solitary wave. The authors analyze the propagation and interaction properties of these solutions to better understand the dynamic behavior of these solutions.Originality/valueThese results may help us to further study the local structure and the interaction of waves in KP-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

Dynamics of surface and internal long waves generated by atmospheric pressure disturbances

Different from non-dispersive wave models such as the shallow water equations, MCC (Miyata, Choi, and Camassa) type strongly nonlinear dispersive wave model can describe not only the dynamics of surface waves, but also those of internal waves. The MCC model for a two-layer fluid system is applied to study dynamics of both surface waves and internal waves. The resonance mechanisms with atmospheric pressure disturbances are numerically studied not only for surface waves, so-called Proudman resonance, but also for internal waves. Baroclinic internal waves are generated when the speed of the atmospheric pressure disturbance is comparable to the critical linear internal wave speed. We compare the amplitudes of generated internal waves to the resonance theory for surface waves. Although most baroclinic internal wave phenomena can be explained by this linear theory, the relative magnitudes of amplitudes for forced and free waves do not agree with the theory. In addition, sequentially generated surface waves with resonance are observed when the radius of the atmospheric pressure disturbance is small. Finally, the dynamics of both surface and internal waves over non-uniform bottom topography are discussed in terms of Froude numbers to understand how they propagate and interact with each other when the Proudman resonance condition is satisfied while they travel.

New closed form soliton and other solutions of the Kundu–Eckhaus equation via the extended sinh-Gordon equation expansion method

In this paper, the extended sinh-Gordon equation expansion method (ShGEEM) is applied to look for new complex hyperbolic and complex trigonometric function solutions, especially dark, bright, combined dark–bright, singular, combined singular soliton and other soliton solutions are obtained of the Kundu–Eckhaus (KE) equation which demonstrates the physical explanation in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. By choosing randomly the special values of free parameters, the solitary wave solutions are shown different patterns. The reported results might play an important role in the quantum field theory and nonlinear optics for explaining the physical meaning of the studied model. The obtained results also demonstrate that the extended ShGEEM is more competent than the other applied method applied by authors in the literature. Therefore, it is shown that the applied method provides a more powerful mathematical tool for constructing exact traveling wave solutions for many other nonlinear evolution equations arises in mathematical physics and nonlinear optics.

A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel

The mathematical model of breaking of non-linear dispersive water waves with memory effect is very important in mathematical physics. In the present article, we examine a novel fractional extension of the non-linear Fornberg-Whitham equation occurring in wave breaking. We consider the most recent theory of differentiation involving the non-singular kernel based on the extended Mittag-Leffler-type function to modify the Fornberg-Whitham equation. We examine the existence of the solution of the non-linear Fornberg-Whitham equation of fractional order. Further, we show the uniqueness of the solution. We obtain the numerical solution of the new arbitrary order model of the non-linear Fornberg-Whitham equation with the aid of the Laplace decomposition technique. The numerical outcomes are displayed in the form of graphs and tables. The results indicate that the Laplace decomposition algorithm is a very user-friendly and reliable scheme for handling such type of non-linear problems of fractional order.

Conserved norms and related conservation laws for multi-peakon equations

All nonlinear dispersive wave equations in the general class are known to possess multi-peakon weak solutions. A classification is presented for families of multi-peakon equations in this class that possess conserved momentum; conserved H1 norm of u; conserved H2 norm of u; conserved L2 norm of m; related conservation laws. The results yield, among others, two interesting wide families of equations: for which the H1 norm of u is conserved; for which the L2 norm of m is conserved. The overlap of these two families yields a singular equation which is nevertheless found to possess both smooth solitary wave solutions and peakon travelling wave solutions.

A parametric study of tsunami fission generation in a bay geometry based on numerical simulation

ABSTRACTTsunami fission generates short waves at the front face of tsunami and these short waves may expand the inundation area, increase the hydrodynamic forces exerted on structures, and amplify the damage to coastal communities. It is therefore important to understand to what extent the tsunami fission generates such short wave components under variable conditions for development of better tsunami disaster mitigation strategies and plannings. This study carried out numerical experiments to describe under which conditions the tsunami fission occurs in a bay. The experiments indicated that the amplification of the nearshore tsunami height induced by the narrowing bay geometry and the bay-scale resonance enhance the tsunami fission. Based on the computations, a parametric diagram for the occurrence of tsunami fission was proposed in terms of the tsunami properties, the effect of bay geometry and the bay resonance. The developed diagram was applied to Sendai, Ryori, Yoshihama, and Toni Bays, which were affected by the 2011 Tohoku Tsunami. The proposed diagram consistently explained the occurrence of tsunami fission in Sendai Bay and in Ryori Bay as observed in the video footage. The diagram also suggested that the tsunami fission would have occurred in Yoshihama Bay. A high-resolution numerical computation was performed to simulate the tsunami deformation in the Yoshihama Bay and Toni Bay based on a nonlinear dispersive wave model. The result supported the validity of the diagram in describing the generation of tsunami fission in Yoshihama Bay.

Dispersive Shallow Water Wave Modelling. Part I: Model Derivation on a Globally Flat Space

In this paper, we review the history and current state-of-the-art in the modelling of long nonlinear dispersive waves. For the sake of conciseness of this review, we omit the unidirectional models and focus especially on some classical and improved Boussinesq-type and Serre-Green-Naghdi equations. Finally, we propose also a unified modelling framework which incorporates several well-known and some less known dispersive wave models. The present manuscript is the first part of a series of two papers. The second part will be devoted to the numerical discretization of a practically important model on moving adaptive grids.

Optical soliton and rogue wave solutions of the ultra-short femto-second pulses in an optical fiber via two different methods and its applications

In this research, we study exact and solitary traveling solutions of the Kundu–Eckhaus equation with variable coefficients and its relation with famous Schrödinger equation and the complex Burgers equation. Nonlinear complex fractional Kundu–Eckhaus equation has vital role in description of the propagation of the ultra-short femtosecond pulses in an optical fiber. This model has a vital role in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. We apply a new auxiliary equation method and novel G′G-expansion method on this model to get new exact and solitary traveling wave solutions. The aim of these solutions is showing the slowly varying amplitude of the pulse envelope and discover new physical properties of ultra-short femto second pulses and that happened by using the new from of soliton solutions which obtained by mentioned methods.

Variational Boussinesq model for strongly nonlinear dispersive waves

For wave tank, coastal and oceanic applications, a fully nonlinear Variational Boussinesq model with optimized dispersion is derived and a simple Finite Element implementation is described. Improving a previous weakly nonlinear version, high waves over flat and varying bottom are shown to be simulated with high accuracy for a number of different cases consisting of solitary wave propagation, a focused wave above flat bottom, irregular waves over deep water, waves over a submerged bar with steep slopes (up to 1:2) and waves over shoals.

New conservative difference schemes with fourth‐order accuracy for some model equation for nonlinear dispersive waves

In this article, some high‐order accurate difference schemes of dispersive shallow water waves with Rosenau‐KdV‐RLW‐equation are presented. The corresponding conservative quantities are discussed. Existence of the numerical solution has been shown. A priori estimates, convergence, uniqueness, and stability of the difference schemes are proved. The convergence order is in the uniform norm without any restrictions on the mesh sizes. At last numerical results are given to support the theoretical analysis.

On blow-up criteria for a class of nonlinear dispersive wave equations with dissipation

We study the Cauchy problem for the nonlinear dispersive wave equation with dissipative term on the real line, $$u_{t}-u_{xxt}+\left[ f\left( u\right) \right] _{x}-\left[ f\left( u\right) \right] _{xxx}+\left[ g\left( u\right) + \frac{f^{\prime \prime }\left( u\right) }{2}u_{x}^{2}\right] _{x}+\lambda \left( u-u_{xx}\right) =0$$ that includes the corresponding dissipative version of the Camassa–Holm equation as well as the hyperelastic-rod wave equation with dissipative term ( $$f(u)=ku^{2}/2$$ and $$g(u)=\left( 3-k\right) u^{2}/2$$ ) as special cases. In the present paper we demonstrate the simple and new conditions on the initial data that lead to blow-up of the solution. In particular, we establish local in space criterion (i.e., a criterion involving only the properties of the data $$u_{0}$$ in a neighborhood of a single point) which guarantees that solutions blow up in finite time.