Integrable Nonlinear Wave Research Articles

An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions

Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.

Lump and soliton on certain spatially-varying backgrounds for an integrable (3+1) dimensional fifth-order nonlinear oceanic wave model

The dynamics of nonlinear waves with controllable dispersion, nonlinearities, and background continues to be an exciting line of active research in recent years. In this work, we focus to investigate an integrable (3+1)-dimensional nonlinear model describing the evolution of water waves with higher-order temporal dispersion by characterizing the dynamics of lump and soliton waves on different spatially-varying backgrounds. Particularly, we construct some explicit lump wave and kink-soliton solutions for the considered model through its trilinear and bilinear equations with appropriate forms of the polynomial and exponential type initial seed solutions, respectively. Additionally, we obtain hyperbolic and periodic waves through bilinear Bäcklund transformation. We explore their propagation and transformation/modulation dynamics due to different spatial backgrounds through categorical analysis and clear graphical demonstrations for a complete understanding of the resulting solutions. Our analysis shows that the lump solution results into a simultaneous existence of well-localized spike and declining (coupled bright-dark wave) structures on a constant background, while the soliton solution exhibits kink and anti-kink wave patterns along different spatio-temporal domains with controllable properties. When the arbitrary spatial background is incorporated appropriately, we can witness the manifestation of soliton into manipulated periodic and interacting dynamical structures. On the other hand, the lump wave results into the coexistence, interacting wave and breather formation due to periodic and localized type arbitrary spatial backgrounds. The present results will be an important addition to the context of engineering nonlinear waves with non-vanishing controllable backgrounds in higher-dimensional models. Further, the present study can be extended to investigate several other nonlinear systems to understand the physical insights of the variable background in their dynamics.

Data-driven soliton mappings for integrable fractional nonlinear wave equations via deep learning with Fourier neural operator

In this paper, we firstly extend the Fourier neural operator (FNO) to discovery the mapping between two infinite-dimensional function spaces, where one is the fractional-order index space {ϵ|ϵ∈(0,1)} in the fractional integrable nonlinear wave equations while another denotes the soliton solution in the spatio-temporal function space. In other words, once the FNO network is trained, for any given ϵ∈(0,1), the corresponding soliton solution can be quickly obtained. To be specific, the soliton solutions are learned for the fractional nonlinear Schrödinger (fNLS), fractional Korteweg–de Vries (fKdV), fractional modified Korteweg–de Vries (fmKdV) and fractional sine-Gordon (fsineG) equations. The FNO architecture is utilized to learn the soliton mappings of the above four equations. The data-driven solitons are also compared with exact solutions to illustrate the powerful approximation capability of the FNO. Moreover, we study the influences of several critical factors (e.g., activation functions containing Relu(x), Sigmoid(x), Swish(x) and the new one xtanh(x), channels of fully connected layer) on the performance of the FNO algorithm. As a result, we find that the xtanh(x) and Swish(x) functions perform better than the Relu(x) and Sigmoid(x) functions in the FNO, and the FNO network with a more-channel fully-connected layer performs better as we expect. These results obtained in this paper will be useful to further understand the neural networks for fractional integrable nonlinear wave equations and the mappings between two infinite-dimensional function spaces.

Data-driven rogue waves and parameters discovery in nearly integrable [formula omitted]-symmetric Gross–Pitaevskii equations via PINNs deep learning

In this paper, we explore the forward and inverse problems for the generalized Gross–Pitaevskii (GP) equation with complex PT-symmetric potentials via the deep physics-informed neural networks (PINNs). The data-driven rogue waves (RWs) are mainly studied in the forward problem, where the comparisons between the data-driven RWs and numerical ones via the spectral method are used to present the PINNs solution accuracies. Besides, we also focus on the influences of several critical factors (e.g., the depths of neural networks, numbers of training points) on the performance of the PINNs algorithm. Finally, the inverse problem is also investigated such that the system parameters can be identified from the training data. The results obtained in this paper can be useful to further understand the neural networks on rogue wave structures in the nearly integrable PT-symmetric nonlinear wave systems.

Fully integrable one-dimensional nonlinear wave equation: Solution of a general initial value problem

A new integrable case of the one-dimensional nonlinear wave equation was found. A general solution for this case depending on two arbitrary functions was derived. The functional form of the speed of sound can be used to model weak nonlinear waves in non-dispersive media. For the initial value problem the nonlinear generalization of the d’Alembert’s formula was obtained.

Number of solitons produced from a large initial pulse in the generalized NLS dispersive hydrodynamics theory.

We show that the number of solitons produced from an arbitrary initial pulse of the simple wave type can be calculated analytically if its evolution is governed by a generalized nonlinear Schrödinger (NLS) equation provided this number is large enough. The final result generalizes the asymptotic formula derived for completely integrable nonlinear wave equations such as the standard NLS equation with the use of the inverse scattering transform method.

GROUP-INVARIANT SOLUTIONS, NON-GROUP-INVARIANT SOLUTIONS AND CONSERVATION LAWS OF QIAO EQUATION

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.

Integrability of a Generalized Short Pulse Equation Revisited

We further generalize the generalized short pulse equation studied recently in [Commun. Nonlinear Sci. Numer. Simulat. 39 (2016) 21-28; arXiv:1510.08822], and find in this way two new integrable nonlinear wave equations which are transformable to linear Klein-Gordon equations.

Nonlinear Fourier Methods for Ocean Waves

Multiperiodic Fourier series solutions of integrable nonlinear wave equations are applied to the study of ocean waves for scientific and engineering purposes. These series can be used to compute analytical formulae for the stochastic properties of nonlinear equations, in analogy to the standard approach for linear equations. Here I emphasize analytically computable results for the correlation functions, power spectra and coherence functions of a nonlinear random process associated with an integrable nonlinear wave equation. The multiperiodic Fourier series have the advantage that the coherent structures of soliton physics are encoded in the formulation, so that solitons, breathers, vortices, etc. are contained in the temporal evolution of the nonlinear power spectrum and phases. I illustrate the method for the Korteweg-deVries and nonlinear SchrÖdinger equations. Applications of the method to the analysis of data are discussed.

SINGULAR PERIODIC WAVES OF AN INTEGRABLE EQUATION FROM SHORT CAPILLARY-GRAVITY WAVES

The effects of parabola singular curves in the integrable nonlinear wave equation are studied by using the bifurcation theory of dynamical system. We find new singular periodic waves for a nonlinear wave equation from short capillary-gravity waves. The new periodic waves possess weaker singularity than the periodic peakon. It is shown that the second derivatives of the new singular periodic wave solutions do not exist in countable number of points but the first derivatives exist. We show that there exist close connection between the new singular periodic waves and parabola singular curve in phase plane of traveling wave system for the first time.

Generalized negative flows in hierarchies of integrable evolution equations

A one-parameter generalization of the hierarchy of negative flows is introduced for integrable hierarchies of evolution equations, which yields a wider (new) class of non-evolutionary integrable nonlinear wave equations. As main results, several integrability properties of these generalized negative flow equation are established, including their symmetry structure, conservation laws, and bi-Hamiltonian formulation. (The results also apply to the hierarchy of ordinary negative flows). The first generalized negative flow equation is worked out explicitly for each of the following integrable equations: Burgers, Korteweg-de Vries, modified Korteweg-de Vries, Sawada-Kotera, Kaup-Kupershmidt, Kupershmidt.

Bifurcations and Single Peak Solitary Wave Solutions of an Integrable Nonlinear Wave Equation

AbstractDynamical system theory is applied to the integrable nonlinear wave equation ut±(u3–u2)x+(u3)xxx=0. We obtain the single peak solitary wave solutions and compacton solutions of the equation. Regular compacton solution of the equation correspond to the case of wave speed c=0. In the case of c≠0, we find smooth soliton solutions. The influence of parameters of the traveling wave solutions is explored by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these soliton solutions of the nonlinear wave equation.

The Bethe ansatz and the Tzitzéica–Bullough–Dodd equation

The theory of classically integrable nonlinear wave equations and the Bethe ansatz systems describing massive quantum field theories defined on an infinite cylinder are related by an important mathematical correspondence that still lacks a satisfactory physical interpretation. In this paper, we shall extend this link to the case of the classical and quantum versions of the Tzitzéica-Bullough-Dodd model.

Instability of Optical Solitons in the Boundary Value Problem for a Medium of Finite Extension

We consider an integrable nonlinear wave system (anisotropic chiral field model) which exhibits a soliton solution when the Cauchy problem for an infinitely long medium is posed. Whenever the boundary value problem is formulated for the same system but for a medium of finite extension, we reveal that the soliton becomes unstable and the true attractor is a different structure which is called polarization attractor. In contrast to the localized nature of solitons, the polarization attractor occupies the entire length of the medium. By demonstrating the qualitative difference between nonlinear wave propagation in an infinite medium and in a medium of finite extension (with simultaneous change of the initial value problem to the boundary value problem), we would like to point out that solitons may loose their property of being stable attractors. Additionally, our findings show the interest of developing methods of integration for boundary value problems.

Bifurcation studies on travelling wave solutions for an integrable nonlinear wave equation

By using the method of planar dynamical systems to an integrable nonlinear wave equation, the existence of periodic travelling wave, solitary wave and kink wave solutions is proved in the different parametric conditions. The phase portraits of the travelling wave system are given. It can be shown that the existence of singular curves in the travelling wave system is the reason why the travelling wave solutions lose their smoothness. Moreover, the so-called W/M-shaped solitary wave solutions are obtained.

New exact solutions for a generalization of the Korteweg–de Vries equation (KdV6)

The generalized tanh–coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg–de Vries equation with a source and the second equation is a third-order linear differential equation.

From bell-shaped solitary wave to W/M-shaped solitary wave solutions in an integrable nonlinear wave equation

The bifurcation theory of dynamical systems is applied to an integrable nonlinear wave equation. As a result, it is pointed out that the solitary waves of this equation evolve from bell-shaped solitary waves to W/M-shaped solitary waves when wave speed passes certain critical wave speed. Under different parameter conditions, all exact explicit parametric representations of solitary wave solutions are obtained.

Two-component description of dynamical systems that can be approximated by solitons: The case of the ion acoustic wave equations of plasma physics

A new approach to the perturbative analysis of dynamical systems, which can be described approximately by soliton solutions of integrable non-linear wave equations, is employed in the case of small-amplitude solutions of the ion acoustic wave equations of plasma physics. Instead of pursuing the traditional derivation of a perturbed KdV equation, the ion velocity is written as a sum of two components: elastic and inelastic. In the single-soliton case, the elastic component is the full solution. In the multiple-soliton case, it is complemented by the inelastic component. The original system is transformed into two evolution equations: An asymptotically integrable Normal Form for ordinary KdV solitons, and an equation for the inelastic component. The zero-order term of the elastic component is a single-soliton or multiple-soliton solution of the Normal Form. The inelastic component asymptotes into a linear combination of single-soliton solutions of the Normal Form, with amplitudes determined by soliton interactions, plus a second-order decaying dispersive wave. Satisfaction of a conservation law by the inelastic component and of mass conservation by the disturbance to the ion density is determined solely by the initial data and/or boundary conditions imposed on the inelastic component. The electrostatic potential is a first-order quantity. It is affected by the inelastic component only in second order. The charge density displays a triple-layer structure. The analysis is carried out through the third order.

On the propagation of tsunami waves, with emphasis on the tsunami of 2004

In this review paper we discuss the range of validity of nonlinear dispersiveintegrable equations for the modelling of the propagation of tsunami waves. For the 2004tsunami the available measurements and the geophyiscal scales involvedrule out a connection between integrable nonlinear wave equations and tsunami dynamics.

A new integrable generalization of the Korteweg–de Vries equation

A new integrable sixth-order nonlinear wave equation is discovered by means of the Painlevé analysis, which is equivalent to the Korteweg–de Vries equation with a source. A Lax representation and an auto-Bäcklund transformation are found for the new equation, and its traveling wave solutions and generalized symmetries are studied.