Quasi-periodic Wave Solutions Research Articles

Wronskian solutions, bilinear Bäcklund transformation, quasi-periodic waves and asymptotic behaviors for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

In this paper, we investigate the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which is widely used in fluid mechanics and theoretical physics. The N-soliton solution is obtained via the Hirota’s bilinear method. The Wronskian solution is derived by using the Wronskian technique for the bilinear form. Through the exchange formula, we deduce the bilinear Bäcklund transformation consisting of four equations and six parameters. In order to consider the quasi-periodic wave having complex structure, one-, two- and three-periodic waves are investigated systemically by combining the Hirota’s bilinear method with Riemann theta function. Furthermore, the corresponding graphs of periodic wave are presented by considering the geometric properties between the characteristic lines. The propagation characteristics of periodic waves are investigated by virtue of the characteristic lines. Finally, the asymptotic relationships between quasi-periodic wave solutions and soliton solutions are established theoretically under a condition of the small amplitude limit. The analytical method used in this paper can be applied in other integrable systems.

Quasi-periodic waves to the defocusing nonlinear Schrödinger equation

A direct approach for the quasi-periodic wave solutions to the defocusing nonlinear Schrödinger equation is proposed based on the theta functions and Hirota’s bilinear method. We transform the problem into a system of algebraic equations, which can be formulated into a least squares problem and then solved by using numerical iterative methods. A rigorous asymptotic analysis demonstrates that these solutions can be classified into two categories: quasi-periodic oscillatory waves and quasi-periodic dark solitons. Singular behaviors may arise in the former case. The numerical results obtained for both the (1+1) -dimensional and (2+1) -dimensional equations are consistent with the theoretical results. Additionaly, the system of algebraic equations can be further extended to address the Riemann–Schottky problem for hyperelliptic curves with 2 infinite points.

Stabilization for the Klein–Gordon–Zakharov system

This paper deals with global stability dynamics for the Klein–Gordon–Zakharov system in R 2 . We first establish that this system admits a family of linear mode unstable explicit quasi-periodic wave solutions. Next, we prove that the Kelvin–Voigt damping can help to stabilize those linear mode unstable explicit quasi-periodic wave solutions for the Klein–Gordon–Zakharov system in the Sobolev space H s + 1 ( R 2 ) × H s + 1 ( R 2 ) × H s + 1 ( R 2 ) for any s ⩾ 1. Moreover, the Kelvin–Voigt damped Klein–Gordon–Zakharov system admits a unique Sobolev regular solution exponentially convergent to some special solutions (including quasi-periodic wave solutions) of it. Our result can be extended to the n-dimension dissipative Klein–Gordon–Zakharov system for any n ⩾ 1.

Extended Korteweg-de Vries equation for long gravity waves in incompressible fluid without strong limitation to surface deviation

We have derived the extended Korteweg-de Vries equation describing the long gravity waves without limitation to surface deviation. The only restriction to the surface deviation is connected with the stability condition for appropriate solutions. The derivation of extended KdV equation is based on the Euler equations for inviscid irrotational and incompressible fluid. It is shown that the extended KdV equation reduces to standard KdV equation for small amplitude of the waves. We have also generalized the extended KdV equation for describing the decaying effect of the waves. Quasi-periodic and solitary wave solutions for extended KdV equation with decaying effect are found as well. We also demonstrate that the fundamental approach based on the inverse scattering method is applicable for solving the extended KdV equation in the case when decaying effect is negligibly small. Such case always occur for restricted propagation distances of the waves.

Quasi-periodic wave solutions of the nonlocal coupled nonlinear Schrödinger equation

In this paper, the quasi-periodic waves of the defocusing nonlocal coupled nonlinear Schrödinger equation can be calculated theoretically through a Darboux-dressing transformation by a separation of variable approach. The quasi-periodic wave solutions are expressed in separation-of-variables form. Moreover, special choices of the free parameters allow us to discuss them graphically. Our results may contribute to explaining and enriching the corresponding nonlinear wave phenomena emerging in nonlocal wave modes.

Lie symmetry analysis and invariant solutions of (3+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this paper, the (3+1)-dimensional constant coefficient of Date–Jimbo–Kashiwara–Miwa (CCDJKM) equation is studied. All of the vector fields, infinitesimal generators, Lie symmetry reductions and different similarity reduction solutions are constructed. Due to the arbitrary functions in the infinitesimal generators, the (3+1)-dimensional CCDJKM equation can further be reduced to many (2+1)-dimensional partial differential equations. The explicit solutions of the similarity reduction equations, which include the quasi-periodic wave solution, the interaction solution between the periodic wave and a kink soliton and the trigonometric function solutions, are constructed with proper arbitrary function selection, and these new exact solutions are given out analytically and graphically.

The New Simulation of Quasiperiodic Wave, Periodic Wave, and Soliton Solutions of the KdV-mKdV Equation via a Deep Learning Method.

How to solve the numerical solution of nonlinear partial differential equations efficiently and conveniently has always been a difficult and meaningful problem. In this paper, the data-driven quasiperiodic wave, periodic wave, and soliton solutions of the KdV-mKdV equation are simulated by the multilayer physics-informed neural networks (PINNs) and compared with the exact solution obtained by the generalized Jacobi elliptic function method. Firstly, the different types of solitary wave solutions are used as initial data to train the PINNs. At the same time, the different PINNs are applied to learn the same initial data by selecting the different numbers of initial points sampled, residual collocation points sampled, network layers, and neurons per hidden layer, respectively. The result shows that the PINNs well reconstruct the dynamical behaviors of the quasiperiodic wave, periodic wave, and soliton solutions for the KdV-mKdV equation, which gives a good way to simulate the solutions of nonlinear partial differential equations via one deep learning method.

The chaotic, supernonlinear, periodic, quasiperiodic wave solutions and solitons with cascaded system

In this research article, the dynamics of optical solitons in cascaded system with spatio-temporal dispersion is discussed. The Kerr medium is considered for this investigation. Two versatile integration tools have been used to extricate different traveling wave solutions. Rapid convergent approximation method has been employed to extract bright, singular and periodic solitons whereas polynomial function solutions are obtained by unified method. The constraints on the parameters for the validity of these solutions are also provided. Galilean transformation is prosecuted to construct the planer dynamical system of the said model. Runge–Kutta fourth-order technique is used to extract the non-linear periodic waves of the considered problem and consequences are depicted through different portraits. Further, by employing an extrinsic periodic force, chaotic and quasiperiodic pattern of discussed model is analyzed for different values of the parameters. While chaotic and quasiperiodic behaviors are observed for some values of parameters of considered system by keeping the force and frequency of the perturbed dynamical system fixed. Sensitive anatomy is carried out for diverse initial value problems. It is investigated that the chaotic and quasiperiodic pattern is present in the perturbed dynamical system.

A numerical study on the N-periodic wave solutions of two coupled bilinear equations

In this paper, based on the direct method proposed by Akira Nakamura, we present an efficient numerical scheme to calculate the N-periodic wave solutions to the Tzitzeica equation and the (2 + 1)-dimensional modified Bogoyavlenskii-Schiff (mBS) equation which can be transformed into a coupled bilinear system with some dependent variable transformation. By using this numerical scheme, we calculate their 2-periodic wave solutions and 3-periodic wave solutions as examples. We also show the asymptotic behaviors under a “small amplitude” limit of these quasi-periodic wave solutions numerically.

Quasi-periodic waves and irregular solitary waves of the AB system

Investigated in this paper is the AB system, which is used to describe the baroclinic instability processes in the geophysical flows. By introducing an auxiliary function , we construct the quasi-periodic wave solutions of the AB system, and find that behaviors of the quasi-periodic waves are similar to those of the periodic waves of the AB system except for certain phase shift in each evolution period. Especially, when , the quasi-periodic waves develop into the periodic waves. Besides, it is found that under certain conditions the quasi-periodic waves degenerate into the irregular solitary waves, which are characterized by shape variations or appearances, splits, combinations and disappearances of certain bulges in the evolution.

Bäcklund transformations and Riemann–Bäcklund method to a (3 + 1)-dimensional generalized breaking soliton equation

In this paper, the bilinear method is employed to investigate the N-soliton solutions of a (3 + 1)-dimensional generalized breaking soliton equation. Three sets of bilinear Backlund transformations are obtained by means of gauge transformation. The Riemann–Backlund method is further extended to the (3 + 1)-dimensional nonlinear integrable systems. The quasiperiodic wave solutions of the (3 + 1)-dimensional generalized breaking soliton equation are systematically analyzed. The asymptotic properties of the quasiperiodic solutions are discussed by using a limiting procedure. The one-periodic and two-periodic waves tend to the 1-soliton and 2-soliton under a small amplitude limit, respectively. The dynamical characteristics of the one- and two-periodic waves are summarized by selecting different parameters. Furthermore, we obtain some new types of the quasiperiodic wave solutions of the variable coefficient (3 + 1)-dimensional generalized breaking soliton equation. These solutions present the dynamical behaviors of C-type, anti-C-type and Z-type periodic waves moving on the background of the periodic waves of bell type.

The quasi-periodic solutions for the supersymmetric variable-coefficient KdV equation

The supersymmetric variable-coefficient KdV equation is presented and it admits Painlevé property by the standard singularity analysis. Based on Hirota bilinear method and Riemann theta function, one and two quasi-periodic wave solutions for the supersymmetric variable-coefficient KdV equation are studied. In addition, we give the asymptotic relations between quasi-periodic wave solutions and soliton solutions.

The solitary waves, quasi-periodic waves and integrability of a generalized fifth-order Korteweg-de Vries equation

ABSTRACTUnder investigation in this paper is a fifth-order Korteweg-de Vries (fKdV) equation, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. Based on the binary Bell polynomials, a lucid and systematic approach is proposed to systematically study its bilinear representation, bilinear Bäcklund transformations and Lax pairs with explicit formulas, respectively. These results can be reduced to the ones of several integrable equations such as Sawada-Kotera equation, Caudrey-Dodd-Gibbon equation, Lax equation, Kaup-Kuperschmidt equation and Ito equation, etc. Furthermore, the N-solitary wave solutions formula and quasi-periodic wave solutions are obtained by using bilinear form of the fKdV equation. Finally, the relation between the periodic wave solution and solitary wave solution is rigorously established.

Three-dimensional quasi-periodic shifted Green function throughout the spectrum, including Wood anomalies.

This work, part II in a series, presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near what are commonly called 'Wood anomaly frequencies'. At these frequencies, there is a grazing Rayleigh wave, and the quasi-periodic Green function ceases to exist. We present a modification of the Green function by adding two types of terms to its lattice sum. The first type are transversely shifted Green functions with coefficients that annihilate the growth in the original lattice sum and yield algebraic convergence. The second type are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on this new Green function for scattering by doubly periodic surfaces at and around Wood frequencies. We believe this is the first solver able to handle Wood frequencies for doubly periodic scattering problems in three dimensions. We demonstrate the method by applying it to acoustic scattering.

Soliton and quasi-periodic wave solutions for b-type Kadomtsev–Petviashvili equation

In this paper, truncated Laurent expansion is used to obtain the bilinear equation of a nonlinear evolution equation. As an application of Hirota’s method, multisoliton solutions are constructed from the bilinear equation. Extending the application of Hirota’s method and employing multidimensional Riemann theta function, one and two-periodic wave solutions are also obtained in a straightforward manner. The asymptotic behavior of one and two-periodic wave solutions under small amplitude limits is presented, and their relations with soliton solutions are also demonstrated.

Exact Traveling Wave Solutions of the Krichever–Novikov Equation: A Dynamical System Approach

In this paper, we show that to find the traveling wave solutions for the Krichever–Novikov equation, we only need to consider a spatial form F-VI of the fourth-order differential equations in the polynomial class having the Painlevé property given by [Cosgrove, 2000]. By using the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions in some two-dimensional invariant manifolds, various exact solutions such as solitary wave solution, periodic wave solutions, quasi-periodic wave solutions and uncountably infinitely many unbounded wave solutions are obtained.

Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation**Project supported by the National Natural Science Foundation of China (Grant Nos. 51579040, 51379033, and 51522902), the National Basic Research Program of China (Grant No. 2013CB036101), and Liaoning Natural Science Foundation, China (Grant No. 201602172).

We construct the Hirota bilinear form of the nonlocal Boussinesq (nlBq) equation with four arbitrary constants for the first time. It is special because one arbitrary constant appears with a bilinear operator together in a product form. A straightforward method is presented to construct quasiperiodic wave solutions of the nlBq equation in terms of Riemann theta functions. Due to the specific dispersion relation of the nlBq equation, relations among the characteristic parameters are nonlinear, then the linear method does not work for them. We adopt the perturbation method to solve the nonlinear relations among parameters in the form of series. In fact, the coefficients of the governing equations are also in series form. The quasiperiodic wave solutions and soliton solutions are given. The relations between the periodic wave solutions and the soliton solutions have also been established and the asymptotic behaviors of the quasiperiodic waves are analyzed by a limiting procedure.

The Riemann–Bäcklund method to a quasiperiodic wave solvable generalized variable coefficient ( $$\varvec{2+1}$$ 2 + 1 )-dimensional KdV equation

In this paper, the Riemann–Backlund method is extended to a generalized variable coefficient ( $$2+1$$ )-dimensional Korteweg–de Vries equation. The soliton and quasiperiodic wave solutions are investigated systematically. The relations between the quasiperiodic wave solutions and the soliton solutions are rigorously established by a limiting procedure. It is proved that the periodic wave solutions tend to the soliton solutions under a small amplitude limit. Furthermore, the propagation characteristics of the soliton solutions and periodic wave solutions are discussed through the graphical analysis.

Constructing Quasi-Periodic Wave Solutions of Differential-Difference Equation by Hirota Bilinear Method

Abstract In the present paper, based on the Riemann theta function, the Hirota bilinear method is extended to directly construct a kind of quasi-periodic wave solution of a new integrable differential-difference equation. The asymptotic property of the quasi-periodic wave solution is analyzed in detail. It will be shown that quasi-periodic wave solution converge to the soliton solutions under certain conditions and small amplitude limit.

The Hamiltonian structure and quasi-periodic solutions for the generalized associated Camassa–Holm equation

In this paper, we study a integrable generalization of the associated Camassa–Holm equation. This equation is shown to be completely integrable and possesses bi-Hamiltonian structure. The bilinear equation and its solition solution are presented through the Bell polynomial technique. Meanwhile, the quasi-periodic wave solution is obtained by Riemann theta-function, and we show the asymptotic properties of the quasi-periodic wave solutions.