An analytical solution for long waves propagating over a submerged atoll is established. The atolls involved in this study are annular coral reefs with large lagoons in the middle, and the expression of the cross sect...An analytical solution for long waves propagating over a submerged atoll is established. The atolls involved in this study are annular coral reefs with large lagoons in the middle, and the expression of the cross section is a trinomial function of the radial distance, i.e., h=ar(2s)-br~s+h_0, where s is the positive rational number. This analytical solution extends the theory by Wang et al.(2018) as s is no longer limited to s=2/m, where m is the positive integer. In addition, by adjusting the terrain parameters properly, the analytic solution can be degenerated to describe the wave propagation over topography with a hump or pit. According to the relationship between wave rays and wave energy, the distribution characteristics and formation mechanism of energy over the topography are expounded. When the lagoon is non-existent, all wave rays converge at the x-axis, which results in an abrupt amplification of the wave amplitude around the convergence point. When a lagoon is mounted on the top of the atoll, the rays are scattered due to the refraction of the lagoon, and only some rays converge at the symmetrical axis and the ridges on both sides,which results in the amplification of wave amplitudes in these areas.展开更多
A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a (1+1)-dimensional dispersive long wave equation with general coefficients to an elementary integral fo...A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a (1+1)-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.展开更多
The mixed covolume method for the regularized long wave equation is devel- oped and studied. By introducing a transfer operator γh, which maps the trial function space into the test function space, and combining the ...The mixed covolume method for the regularized long wave equation is devel- oped and studied. By introducing a transfer operator γh, which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.展开更多
The global fast dynamics for the generalized symmetric regularized long wave equation with damping term is considered. The squeezing property of the nonlinear semi_group associated with this equation and the existence...The global fast dynamics for the generalized symmetric regularized long wave equation with damping term is considered. The squeezing property of the nonlinear semi_group associated with this equation and the existence of exponential attractor are proved. The upper bounds of its fractal dimension are also estimated.展开更多
This investigation examines long wave reflection and transmission induced by a sloping step. Bellman and Kalaba's (1959) invariant imbedding is introduced to find wave reflection. An alternative method matching bo...This investigation examines long wave reflection and transmission induced by a sloping step. Bellman and Kalaba's (1959) invariant imbedding is introduced to find wave reflection. An alternative method matching both the surface elevation and its surface slope of each region at the junction is applied to the determination of wave reflection and transmission. The proposed methods are compared with the accurate numerical results of Porter and Porter (2000) and those of Mei (1983) for a vertical step. The wave reflection obtained for a mildly sloping step differs significantly from the result of Mei. The wave reflection is found to fluctuate owing to wave trapping for the mild sloping step. The height and the face slope of the step are important for determining wave reflection and transmission coefficients.展开更多
A stability of a nonlinear ultra-long wave and its solution are discussed in this paper by employing Burger model which is subject to heat resource. It is of interest noted that the wave solution can be described by a...A stability of a nonlinear ultra-long wave and its solution are discussed in this paper by employing Burger model which is subject to heat resource. It is of interest noted that the wave solution can be described by an equation of KDV or MKDV and that conditions for the existence of the solution are related to characteristic divergences. In addition, a wave velocity expression for nonlinear ultra-long waves and some diagnostic correlations among wave parameters have been obtained.展开更多
A numerical model of low frequency waves is presented. The model is based on that of Roelvink (1993), but the numerical techniques used in the solution are based on the so-called Weighted- Average Flux (WAF) method wi...A numerical model of low frequency waves is presented. The model is based on that of Roelvink (1993), but the numerical techniques used in the solution are based on the so-called Weighted- Average Flux (WAF) method with Time-Operator- Splitting JOS) used for the treatment of the source terms. This method allows a small number of computational points to be used, and is particularly efficient in modeling wave setup. The short wave (or primary wave) energy equation is solved with a traditional Lax-Wendroff technique. A nonlinear wave theory is introduced. The model described in this paper is found to be satisfactory in modeling low frequency waves associated with incident bichromatic waves.展开更多
This paper presents a meshless method for the nonlinear generalized regularized long wave (GRLW) equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtain...This paper presents a meshless method for the nonlinear generalized regularized long wave (GRLW) equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method. A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm. Compared with numerical methods based on mesh, the meshless method for the GRLW equation only requires the.scattered nodes instead of meshing the domain of the problem. Some examples, such as the propagation of single soliton and the interaction of two solitary waves, are given to show the effectiveness of the meshless method.展开更多
In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for (2+1)-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurca...In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for (2+1)-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.展开更多
With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the (24-1)-dimensional dispersive long wave equation. When selecting appropriate mult...With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the (24-1)-dimensional dispersive long wave equation. When selecting appropriate multi-valued functions in the variable separation solution, we investigate the interactions among special multi-dromions, dromion-like multi-peakons, and dromion-like multi-semifoldons, which all demonstrate non-completely elastic properties.展开更多
A global localization system of in-pipe robot is introduced in this paper.Global position system(GPS)is applied to monitor the motion of robot along the whole pipeline which is equally divided intomany segments by tra...A global localization system of in-pipe robot is introduced in this paper.Global position system(GPS)is applied to monitor the motion of robot along the whole pipeline which is equally divided intomany segments by tracking stations.The definite segment in which robot existing can be detected and thisis long-range localization.Ultra-long wave(ULW)is adopted to solve the problem of metallic shieldingand realize effective communication between inside and outside of pipeline.ULW emitter is carried byrobot.When the plant is broken or defects on pipe-wall are inspected,the robot will stop moving.Anten-na array is presented and disposed upon the definite segment to search the accurate location of robot,andthis is short-range localization.In this paper,five-antenna array is adopted and an effective linear signalfusion algorithm is presented.The localization precision reaches R < 25cm.By tests in Shengli oil field,the whole system is verified with robust solutions.展开更多
Long-period waves pose a threat to coastal communities as they propagate from deep ocean to shallow coastal waters. At the coastline, such waves have a greater height and longer period in comparison with local storm w...Long-period waves pose a threat to coastal communities as they propagate from deep ocean to shallow coastal waters. At the coastline, such waves have a greater height and longer period in comparison with local storm waves, and can cause severe inundation and damage. In this study,we considered linear long waves in a two-dimensional(vertical-horizontal) domain propagating towards a shoreline over a shallowing shelf.New solutions to the linear shallow water equations were found, through the separation of variables, for two forms of transition shelf morphology: deep water and shallow coastal water horizontal shelves connected by linear and parabolic transition, respectively. Expressions for the transmission and reflection coefficients are presented for each case. The analytical solutions were used to test the results from a novel computational scheme, which was then applied to extending the existing results relating to the reflected and transmitted components of an incident wave. The solutions and computational package provide new tools for coastal managers to formulate improved defence and riskmitigation strategies.展开更多
In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations....In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.展开更多
By using the homogeneous balance principle(HBP),we derive a B■cklund trans- formation(BT)to the generalized dispersive long wave equation with variable coefficients. Based on the BT,we give many kinds of the exact so...By using the homogeneous balance principle(HBP),we derive a B■cklund trans- formation(BT)to the generalized dispersive long wave equation with variable coefficients. Based on the BT,we give many kinds of the exact solutions of the equatioh,such as,single solitary solutions,multi-soliton solutions and generalized exact solutions.展开更多
In this article, we study the Lax pairs of (2+1)-dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion fo...In this article, we study the Lax pairs of (2+1)-dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formulas of the first part of the Lax pairs. Then by further discussion and doing some revisional work, we make the recursion formulas fit for the second part of Lax pairs. At last, some solutions to the MGDLW equation are worked out by using the recursion formula.展开更多
A new method using group-induced second-order long waves (GSLW) to describe wave groups is presented in this paper on the basis of the GSLW theory by Longuet- Higgins and Steward (1964) . In the method , the parabolic...A new method using group-induced second-order long waves (GSLW) to describe wave groups is presented in this paper on the basis of the GSLW theory by Longuet- Higgins and Steward (1964) . In the method , the parabolic relationship between GSLW and the wave envelope is first deduced , and then the distribution function of GSLW amplitude is derived . Thus, the formulae in terms of the moments of GSLW and short wave spectra for the average time duration and the mean length of runs of wave heights exceeding a certain level can be derived . A new groupiness factor equivalent to half the mean wave number in wave groups is defined by taking into account the widths of spectra of GSLW and short waves . Compared with theoretical results of others , ours are closer to measured wave data .展开更多
基金financially supported by Hainan Provincial Natural Science Foundation of China (Grant No. 422MS090)Fujian Provincial Natural Scienceof China (Grant No. 2022J05282)2020 Xiamen Youth Innovation Fund Project of China (Grant No.3502Z20206069)。
文摘An analytical solution for long waves propagating over a submerged atoll is established. The atolls involved in this study are annular coral reefs with large lagoons in the middle, and the expression of the cross section is a trinomial function of the radial distance, i.e., h=ar(2s)-br~s+h_0, where s is the positive rational number. This analytical solution extends the theory by Wang et al.(2018) as s is no longer limited to s=2/m, where m is the positive integer. In addition, by adjusting the terrain parameters properly, the analytic solution can be degenerated to describe the wave propagation over topography with a hump or pit. According to the relationship between wave rays and wave energy, the distribution characteristics and formation mechanism of energy over the topography are expounded. When the lagoon is non-existent, all wave rays converge at the x-axis, which results in an abrupt amplification of the wave amplitude around the convergence point. When a lagoon is mounted on the top of the atoll, the rays are scattered due to the refraction of the lagoon, and only some rays converge at the symmetrical axis and the ridges on both sides,which results in the amplification of wave amplitudes in these areas.
文摘A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a (1+1)-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.
基金supported by the National Natural Science Fundation of China (No. 11061021)the Science Research of Inner Mongolia Advanced Education (Nos. NJ10006, NJ10016, and NJZZ12011)the National Science Foundation of Inner Mongolia (Nos. 2011BS0102 and 2012MS0106)
文摘The mixed covolume method for the regularized long wave equation is devel- oped and studied. By introducing a transfer operator γh, which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.
基金ProjectsupportedbytheNationalNaturalScienceFoundationofChina (No .1 0 2 71 0 3 4)
文摘The global fast dynamics for the generalized symmetric regularized long wave equation with damping term is considered. The squeezing property of the nonlinear semi_group associated with this equation and the existence of exponential attractor are proved. The upper bounds of its fractal dimension are also estimated.
文摘This investigation examines long wave reflection and transmission induced by a sloping step. Bellman and Kalaba's (1959) invariant imbedding is introduced to find wave reflection. An alternative method matching both the surface elevation and its surface slope of each region at the junction is applied to the determination of wave reflection and transmission. The proposed methods are compared with the accurate numerical results of Porter and Porter (2000) and those of Mei (1983) for a vertical step. The wave reflection obtained for a mildly sloping step differs significantly from the result of Mei. The wave reflection is found to fluctuate owing to wave trapping for the mild sloping step. The height and the face slope of the step are important for determining wave reflection and transmission coefficients.
文摘A stability of a nonlinear ultra-long wave and its solution are discussed in this paper by employing Burger model which is subject to heat resource. It is of interest noted that the wave solution can be described by an equation of KDV or MKDV and that conditions for the existence of the solution are related to characteristic divergences. In addition, a wave velocity expression for nonlinear ultra-long waves and some diagnostic correlations among wave parameters have been obtained.
基金This project was supported by the British Councilthe National Natural Science Foundation of China (Grant No.59809001 and 1 9732004)
文摘A numerical model of low frequency waves is presented. The model is based on that of Roelvink (1993), but the numerical techniques used in the solution are based on the so-called Weighted- Average Flux (WAF) method with Time-Operator- Splitting JOS) used for the treatment of the source terms. This method allows a small number of computational points to be used, and is particularly efficient in modeling wave setup. The short wave (or primary wave) energy equation is solved with a traditional Lax-Wendroff technique. A nonlinear wave theory is introduced. The model described in this paper is found to be satisfactory in modeling low frequency waves associated with incident bichromatic waves.
基金supported by the National Natural Science Foundation of China (Grant No. 10871124)the Innovation Program of the Shanghai Municipal Education Commission,China (Grant No. 09ZZ99)
文摘This paper presents a meshless method for the nonlinear generalized regularized long wave (GRLW) equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method. A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm. Compared with numerical methods based on mesh, the meshless method for the GRLW equation only requires the.scattered nodes instead of meshing the domain of the problem. Some examples, such as the propagation of single soliton and the interaction of two solitary waves, are given to show the effectiveness of the meshless method.
基金Supported by the National Natural Science Foundation of China (10871206)Program for Excellent Talents in Guangxi Higher Education Institutions
文摘In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for (2+1)-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.
基金Supported by the National Natural Science Foundation of China (Grant No. 11005092)the Program for Innovative Research Team of Young Teachers,China (Grant No. 2009RC01)+1 种基金the Undergraduate Innovative Base of Zhejiang A & F University,Chinathe Zhejiang Province Undergraduate Scientific and Technological Innovation Project,China (Grant No. 2012R412018)
文摘With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the (24-1)-dimensional dispersive long wave equation. When selecting appropriate multi-valued functions in the variable separation solution, we investigate the interactions among special multi-dromions, dromion-like multi-peakons, and dromion-like multi-semifoldons, which all demonstrate non-completely elastic properties.
基金Supported by the High Technology Research and Development Programme of China (No. 2006AA04Z205)
文摘A global localization system of in-pipe robot is introduced in this paper.Global position system(GPS)is applied to monitor the motion of robot along the whole pipeline which is equally divided intomany segments by tracking stations.The definite segment in which robot existing can be detected and thisis long-range localization.Ultra-long wave(ULW)is adopted to solve the problem of metallic shieldingand realize effective communication between inside and outside of pipeline.ULW emitter is carried byrobot.When the plant is broken or defects on pipe-wall are inspected,the robot will stop moving.Anten-na array is presented and disposed upon the definite segment to search the accurate location of robot,andthis is short-range localization.In this paper,five-antenna array is adopted and an effective linear signalfusion algorithm is presented.The localization precision reaches R < 25cm.By tests in Shengli oil field,the whole system is verified with robust solutions.
基金The project supported by China Postdoctoral Science Foundation, Natural Science Foundation of Zhejiang Province of China under Grant No. Y604056, and Doctor Foundation of Ningbo City under Grant No. 2005A610030
基金The project supported in part by National Natural Science Foundation of China under Grant No. 10272071 and the Science Research Foundation of Huzhou University under Grant No. KX21025
基金supported by a Researcher Links Grant from the British Council,the Royal Academy of Engineering(Grant No.IAAP1/100086)the EFRaCC Project funded through the British Council's Global Innovation Initiative Program
文摘Long-period waves pose a threat to coastal communities as they propagate from deep ocean to shallow coastal waters. At the coastline, such waves have a greater height and longer period in comparison with local storm waves, and can cause severe inundation and damage. In this study,we considered linear long waves in a two-dimensional(vertical-horizontal) domain propagating towards a shoreline over a shallowing shelf.New solutions to the linear shallow water equations were found, through the separation of variables, for two forms of transition shelf morphology: deep water and shallow coastal water horizontal shelves connected by linear and parabolic transition, respectively. Expressions for the transmission and reflection coefficients are presented for each case. The analytical solutions were used to test the results from a novel computational scheme, which was then applied to extending the existing results relating to the reflected and transmitted components of an incident wave. The solutions and computational package provide new tools for coastal managers to formulate improved defence and riskmitigation strategies.
文摘In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.
基金Supported by the Natural Science Foundation of Education Committee of Henan Province(2003110003)
文摘By using the homogeneous balance principle(HBP),we derive a B■cklund trans- formation(BT)to the generalized dispersive long wave equation with variable coefficients. Based on the BT,we give many kinds of the exact solutions of the equatioh,such as,single solitary solutions,multi-soliton solutions and generalized exact solutions.
基金The project supported by National Natural Science Foundation of China under Grant No.10101025
文摘In this article, we study the Lax pairs of (2+1)-dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formulas of the first part of the Lax pairs. Then by further discussion and doing some revisional work, we make the recursion formulas fit for the second part of Lax pairs. At last, some solutions to the MGDLW equation are worked out by using the recursion formula.
基金The project supported by the China Postdoctoral Science Foundation under Grant No. 2004036086, K.C. Wong Education Foundation, Hong Kong, and partially supported by the State Key Basic Research Program of China under Grant No. 2004CB318000 . The authors are grateful to professor Gao Xiao-Shan for his enthusiastic guidance and help.
基金This project was funded by the National Natural Science Foundation of China
文摘A new method using group-induced second-order long waves (GSLW) to describe wave groups is presented in this paper on the basis of the GSLW theory by Longuet- Higgins and Steward (1964) . In the method , the parabolic relationship between GSLW and the wave envelope is first deduced , and then the distribution function of GSLW amplitude is derived . Thus, the formulae in terms of the moments of GSLW and short wave spectra for the average time duration and the mean length of runs of wave heights exceeding a certain level can be derived . A new groupiness factor equivalent to half the mean wave number in wave groups is defined by taking into account the widths of spectra of GSLW and short waves . Compared with theoretical results of others , ours are closer to measured wave data .