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Numerical Integrators for Dispersion-Managed kdv equation

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Communications in Computational Physics 2022年第4期31卷 1180-1214页

作者： Ying He Xiaofei Zhao School of Mathematics and Statistics Wuhan UniversityWuhan 430072P.R.China Computational Sciences Hubei Key Laboratory Wuhan UniversityWuhan 430072P.R.China

In this paper,we consider the numerics of the dispersion-managed Kortewegde Vries(DM-kdv)equation for describingwave propagations in inhomogeneous media.The DM-kdv equation contains a variable dispersion map with discontinuity,which makes the solution non-smooth in time.We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-kdv equation,where a necessary constraint on the time step has been identified.Then,two exponential-type dispersionmap integrators up to second order accuracy are derived,which are efficiently incorporatedwith the Fourier pseudospectral discretization in space,and they can converge regardless the discontinuity and the step size.Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast&strong dispersion-management regime.

关键词： kdv equation dispersion management discontinuous coefficient convergence order finite difference time-splitting exponential integrator pseudospectral method

Number of Solitons Emerged in the Initial Profile of Shallow Water Using Convolutional Neural Networks

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Journal of Systems Science & Complexity 2024年第2期37卷 463-479页

作者： WANG Zhen CUI Shikun School of Mathematical Sciences Dalian University of TechnologyDalian 116024China School of Mathematical Sciences Beihang UniversityBeijing 100191China

The soliton resolution conjecture proposes that the initial value problem can evolve into a dispersion part and a soliton part.However,the problem of determining the number of solitons that form in a given initial profile remains unsolved,except for a few specific cases.In this paper,the authors use the deep learning method to predict the number of solitons in a given initial value of the Korteweg-de Vries(kdv)equation.By leveraging the analytical relationship between Asech^(2)(x)initial values and the number of solitons,the authors train a Convolutional Neural Network(CNN)that can accurately identify the soliton count from spatio-temporal data.The trained neural network is capable of predicting the number of solitons with other given initial values without any additional assistance.Through extensive calculations,the authors demonstrate the effectiveness and high performance of the proposed method.

关键词： Convolutional neural network deep learning method kdv equation soliton

A novel approach to study generalized coupled cubic Schrödinger-Korteweg-de Vries equations

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Journal of Ocean Engineering and Science 2024年第1期9卷 13-24页

作者： Lanre Akinyemi P.Veeresha M.T.Darvishi Hadi Rezazadeh Mehmet Senol Udoh Akpan Department of Mathematics Lafayette CollegeEastonPAUSA Department of Mathematics Christ Deemed to be UniversityBengaluru 560029India Department of Mathematics Razi UniversityKermanshah 67149Iran Faculty of Engineering Technology Amol University of Special Modern TechnologiesAmolIran Department of Mathematics Nev¸s ehir HacıBekta¸s Veli UniversityNev¸s ehirTurkey Drexel University Department of MathematicsPhiladelphiaPAUSA

The Kortewegde Vries(kdv)equation represents the propagation of long waves in dispersive media,whereas the cubic nonlinear Schrödinger(CNLS)equation depicts the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves.A model that couples these two equations seems in-triguing for simulating the interaction of long and short waves,which is important in many domains of applied sciences and engineering,and such a system has been investigated in recent decades.This work uses a modified Sardar sub-equation procedure to secure the soliton-type solutions of the generalized cubic nonlinear Schrödinger-Korteweg-de Vries system of equations.For various selections of arbitrary parameters in these solutions,the dynamic properties of some acquired solutions are represented graph-ically and analyzed.In particular,the dynamics of the bright solitons,dark solitons,mixed bright-dark solitons,W-shaped solitons,M-shaped solitons,periodic waves,and other soliton-type solutions.Our re-sults demonstrated that the proposed technique is highly efficient and effective for the aforementioned problems,as well as other nonlinear problems that may arise in the fields of mathematical physics and engineering.

关键词： CNLS equation Modiﬁed Sardar sub-equation method kdv equation Solitons Long and short waves

A FAMILY OF DIFFERENCE SCHEMES WITH FOURNEAR-CONSERVED QUANTITIES FOR THE kdv equation

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Journal of Computational Mathematics 1998年第2期16卷 129-140页

作者： Han, Z Shen, LJ No Jiaotong Univ Dept Comp Sci & Technol Beijing 100044 Peoples R China Inst Appl Phys & Computat Math Beijing 100088 Peoples R China

We construct and analyze a family of semi-discretized difference schemes with two parameters for the Korteweg-de Vries (kdv) equation. The scheme possesses the first four near-conserved quantities for periodic boundary conditions. The existence and the convergence of its global solution in Sobolev space L-infinity(0,T;H-3) are proved and the scheme is also stable about initial values. Furthermore, the scheme conserves exactly the first two conserved quantities in the special case.

关键词： convergence difference scheme kdv equation conserved quantity

A LATTICE BOLTZMANN METHOD FOR kdv equation

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Acta Mechanica Sinica 1998年第1期14卷 18-26页

作者：阎广武陈耀松胡守信 Laboratory for Nonlinear Mechanics of Continuous Media Institute of Mechanics Chinese Academy of Sciences Beijing 100080China Department of MathematicsJilin UniversityChangchun 130023China Department of Mechanics Peking UniversityBeijing 100871China Department of Mathematics Jilin UniversityChangchun 130023China

We prepose a 5-bit lattice Boltzmann model for kdv equation. Using Chapman-Enskog expansion and multiscale technique, we obtained high order moments of equilibrium distribution function, and the 3rd dispersion coeffic... 详细信息

关键词： lattice Boltzmann method kdv equation multiscale technique 5-bit lattice conservational law in time scalet 0

Element-free Galerkin method for a kind of kdv equation

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Chinese Physics B 2010年第6期19卷 7-12页

作者：王聚丰孙凤欣程荣军 Ningbo Institute of Technology Zhejiang University Ningbo University of Technology

The present paper deals with the numerical solution of the third-order nonlinear kdv equation using the element-free Calerkin （EFG） method which is based on the moving least-squares approximation. A variational method is used to obtain discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for kdv equations needs only scattered nodes instead of meshing the domain of the problem. It does not require any element connectivity and does not suffer much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for the kdv equation is investigated by two numerical examples in this paper.

关键词： element-free Galerkin method meshless method kdv equation

A new auxiliary equation method for finding travelling wave solutions to kdv equation

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Applied Mathematics and Mechanics(English Edition) 2010年第7期31卷 929-936页

作者：庞晶边春泉朝鲁 College of Science Inner Mongolia University of Technology Department of Mathematics Shanghai Maritime University

In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional kdv equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.

关键词： auxiliary equation method travelling wave solution kdv equation homogeneous balance method

Alternating segment explicit-implicit scheme for nonlinear third-order kdv equation

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Applied Mathematics and Mechanics(English Edition) 2007年第7期28卷 973-980页

作者：曲富丽王文洽 Accounting Department Women's Academy at Shandong School of Mathematics and System Science Shandong University

A group of asymmetric difference schemes to approach the Korteweg-de Vries （kdv） equation is given here. According to such schemes, the full explicit difference scheme and the full implicit one, an alternating segment explicit-implicit difference scheme for solving the kdv equation is constructed. The scheme is linear unconditionally stable by the analysis of linearization procedure, and is used directly on the parallel computer. The numerical experiments show that the method has high accuracy.

关键词： kdv equation intrinsic parallelism alternating segment explicit-implicit difference scheme unconditionally linear stable

GLOBAL ATTRACTOR AND ITS DIMENSION ESTIMATES FOR THE GENERALIZED DISSIPATIVE kdv equation ON R

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Acta Mathematicae Applicatae Sinica 1998年第3期14卷 252-259页

作者：郭柏灵吴永辉 Institute of Applied Physics and Computational Mathematics Beijing China LASG Institute of Atmospheric Physics the chinese Academy of Sciences Beijing China

In this paper we prove the existence of global attractor for the generalized dissipative kdvequation on R, and give an upper bound for its Hausdorff and fractal dimensions.

关键词： Global attractor Hausdorff and fractal dimensions kdv equation weighted norm