This dissertation deals with the internal waves generated by a submergedmoving body in stratified fluids by combining theoretical and experimental methods. Our purpose isto provide some scientific evidences for non-ac...This dissertation deals with the internal waves generated by a submergedmoving body in stratified fluids by combining theoretical and experimental methods. Our purpose isto provide some scientific evidences for non-acoustic detection of underwater moving bodies based onthe principles of dynamics of the internal waves. An approach to velocity potentials obtained bysuperposing Green''s functions of sources and sinks was proposed for Kelvin waves at the free surfaceor interface in a two-layer fluid. The effects of interacting surface- and internal-wave modesinduced by a dipole on the surface divergence field were investigated. A new theoretical modelformulating the interaction of a two-dimensional submerged moving body with the conjugate flow in athree-layer fluid was established. An exact solution satisfying the two-dimensional Benjamin-Onoequation was obtained and the vertically propagating properties of the weakly nonlinear long waveswere studied by means of the ray theory and WKB method. The above theoretical results arequalitatively consistent with those obtained in the experiments conducted by the author.展开更多
This work investigates three nonlinear equations that describe waves on the oceans which are the Kadomtsev Petviashvili-modified equal width(KP-MEW)equation,the coupled Drinfel’d-Sokolov-Wilson(DSW)equation,and the B...This work investigates three nonlinear equations that describe waves on the oceans which are the Kadomtsev Petviashvili-modified equal width(KP-MEW)equation,the coupled Drinfel’d-Sokolov-Wilson(DSW)equation,and the Benjamin-Ono(BO)equation using the modified(G′/G^(2))-expansion approach.The solutions of proposed equations by modified(G′/G^(2))-expansion approach can be trigonometric,hyperbolic,or rational solutions.As a result,some new exact solutions are obtained and plotted.展开更多
In this paper,we study the Cauchy problem for the Benjamin-Ono-Burgers equation ∂_(t)u−ϵ∂^(2)/_(x)u+H∂^(2)_(x)u+uu_(x)=0,where H denotes the Hilbert transform operator.We obtain that it is uniformly locally well-posed...In this paper,we study the Cauchy problem for the Benjamin-Ono-Burgers equation ∂_(t)u−ϵ∂^(2)/_(x)u+H∂^(2)_(x)u+uu_(x)=0,where H denotes the Hilbert transform operator.We obtain that it is uniformly locally well-posed for small data in the refined Sobolev space H~σ(R)(σ■0),which is a subspace of L2(ℝ).It is worth noting that the low-frequency part of H~σ(R)is scaling critical,and thus the small data is necessary.The high-frequency part of H~σ(R)is equal to the Sobolev space Hσ(ℝ)(σ■0)and reduces to L2(ℝ).Furthermore,we also obtain its inviscid limit behavior in H~σ(R)(σ■0).展开更多
The current paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation.By establishing the bilinear and trilinear estimates in some Bourgain spaces,we prove that the Cauchy problem for...The current paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation.By establishing the bilinear and trilinear estimates in some Bourgain spaces,we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data u0(x,w)∈L^(2)(Ω;H^(s)(R))which is F0-measurable with s≥1/2-α/4 andΦ∈L20,s.In particular,whenα=1,we prove that it is globally well-posed for the initial data u0(x,w)∈L2(Ω;H1(R))which is F0-measurable andΦ∈L20,1.The key ingredients that we use in this paper are trilinear estimates,the Ito formula and the Burkholder-Davis-Gundy(BDG)inequality as well as the stopping time technique.展开更多
We show for the Benjamin-Ono equation an existence uniqeness theorem in Sobolev spaces of arbitrary fractional order s greater-than-or-equal-to 2, provided the initial data is given in the same space.
In this paper,the Fourier collocation method for solving the generalized Benjamin-Ono equation with periodic boundary conditions is analyzed.Stability of the semi-discrete scheme is proved and error estimate in H^(1/2...In this paper,the Fourier collocation method for solving the generalized Benjamin-Ono equation with periodic boundary conditions is analyzed.Stability of the semi-discrete scheme is proved and error estimate in H^(1/2)-norm is obtained.展开更多
The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data i...The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data in H^s（R）（s 〉 -3/4）. For k = 2, local result for data in H^S（R）（s 〉1/4） is obtained. For k = 3, local result for data in H^S（R）（s 〉 -1/6） is obtained. Moreover, the solutions of generalized Korteweg-de Vries-Benjamin-Ono equation converge to the solutions of KdV equation if the term of Benjamin-Ono equation tends to zero.展开更多
In this article, the author sets up the abundant traveling wave solutions for time fractional Benjamin–Ono equation which was introduced to describe internal waves in stratified fluids by using Jacobi elliptic functi...In this article, the author sets up the abundant traveling wave solutions for time fractional Benjamin–Ono equation which was introduced to describe internal waves in stratified fluids by using Jacobi elliptic function expansion method. The traveling wave solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. It can be seen that the obtained results are found to be important for the statement of some physical demonstrations of problems in mathematical physics and ocean engineering. In ocean engineering Benjamin–Ono equations are generally used in computer simulation for the water waves in deep water and open seas.展开更多
By using perturbation methods, the evolution equation is derived for the second-order internal solitarywaves in stratified fluids of great depth, which is a kind of inhomogeneous linearized Belljamin-Ono equation.
The algebraic solitary wave and its associated eigenvalue problem in a deep stratified fluid with a free surface and a shallow upper layer were studied. And its vertical structure was examined. An exact solution for t...The algebraic solitary wave and its associated eigenvalue problem in a deep stratified fluid with a free surface and a shallow upper layer were studied. And its vertical structure was examined. An exact solution for the derived 2D Benjamin-Ono equation was obtained, and physical explanation was given with the corresponding dispersion relation. As a special case, the vertical structure of the weakly nonlinear internal wave for the Holmboe density distribution was numerically investigated, and the propagating mechanism of the internal wave was studied by using the ray theory.展开更多
The Benjamin-ono (BO) equation is an important nonlinear wave model which can describe the deep oceanic internal wave propagation. In this paper, the multi-algebraic solitary wave solutions for the internal wave BO eq...The Benjamin-ono (BO) equation is an important nonlinear wave model which can describe the deep oceanic internal wave propagation. In this paper, the multi-algebraic solitary wave solutions for the internal wave BO equation including the linear velocity term in matrix form are given by the bilinear form. Based on the analytic solutions of the BO equation obtained in this paper and considering the hydrological parameters, the propagation of one-solitary wave and different kinds of interaction for the two-solitary waves are discussed and illustrated.展开更多
文摘This dissertation deals with the internal waves generated by a submergedmoving body in stratified fluids by combining theoretical and experimental methods. Our purpose isto provide some scientific evidences for non-acoustic detection of underwater moving bodies based onthe principles of dynamics of the internal waves. An approach to velocity potentials obtained bysuperposing Green''s functions of sources and sinks was proposed for Kelvin waves at the free surfaceor interface in a two-layer fluid. The effects of interacting surface- and internal-wave modesinduced by a dipole on the surface divergence field were investigated. A new theoretical modelformulating the interaction of a two-dimensional submerged moving body with the conjugate flow in athree-layer fluid was established. An exact solution satisfying the two-dimensional Benjamin-Onoequation was obtained and the vertically propagating properties of the weakly nonlinear long waveswere studied by means of the ray theory and WKB method. The above theoretical results arequalitatively consistent with those obtained in the experiments conducted by the author.
基金The research is supported by the Scientific Research Foundation of Yunnan ProvincialDepartment and the Natural Science Foundation of Yunnan Province(2005A0026M)
文摘This work investigates three nonlinear equations that describe waves on the oceans which are the Kadomtsev Petviashvili-modified equal width(KP-MEW)equation,the coupled Drinfel’d-Sokolov-Wilson(DSW)equation,and the Benjamin-Ono(BO)equation using the modified(G′/G^(2))-expansion approach.The solutions of proposed equations by modified(G′/G^(2))-expansion approach can be trigonometric,hyperbolic,or rational solutions.As a result,some new exact solutions are obtained and plotted.
基金supported by National Natural Science Foundation of China(Grant No.12001236)supported by National Natural Science Foundation of China(Grant No.11731014)supported by National Natural Science Foundation of China(Grant No.11971166)。
文摘In this paper,we study the Cauchy problem for the Benjamin-Ono-Burgers equation ∂_(t)u−ϵ∂^(2)/_(x)u+H∂^(2)_(x)u+uu_(x)=0,where H denotes the Hilbert transform operator.We obtain that it is uniformly locally well-posed for small data in the refined Sobolev space H~σ(R)(σ■0),which is a subspace of L2(ℝ).It is worth noting that the low-frequency part of H~σ(R)is scaling critical,and thus the small data is necessary.The high-frequency part of H~σ(R)is equal to the Sobolev space Hσ(ℝ)(σ■0)and reduces to L2(ℝ).Furthermore,we also obtain its inviscid limit behavior in H~σ(R)(σ■0).
基金supported by Young Core Teachers Program of Henan Province(Grant No.5201019430009)supported by National Natural Science Foundation of China(Grant No.11771449)。
文摘The current paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation.By establishing the bilinear and trilinear estimates in some Bourgain spaces,we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data u0(x,w)∈L^(2)(Ω;H^(s)(R))which is F0-measurable with s≥1/2-α/4 andΦ∈L20,s.In particular,whenα=1,we prove that it is globally well-posed for the initial data u0(x,w)∈L2(Ω;H1(R))which is F0-measurable andΦ∈L20,1.The key ingredients that we use in this paper are trilinear estimates,the Ito formula and the Burkholder-Davis-Gundy(BDG)inequality as well as the stopping time technique.
文摘We show for the Benjamin-Ono equation an existence uniqeness theorem in Sobolev spaces of arbitrary fractional order s greater-than-or-equal-to 2, provided the initial data is given in the same space.
基金supported by NSF of China(60874039)Leading Academic Discipline Project of Shanghai Municipal Education Commission(J50101)
文摘In this paper,the Fourier collocation method for solving the generalized Benjamin-Ono equation with periodic boundary conditions is analyzed.Stability of the semi-discrete scheme is proved and error estimate in H^(1/2)-norm is obtained.
文摘The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data in H^s（R）（s 〉 -3/4）. For k = 2, local result for data in H^S（R）（s 〉1/4） is obtained. For k = 3, local result for data in H^S（R）（s 〉 -1/6） is obtained. Moreover, the solutions of generalized Korteweg-de Vries-Benjamin-Ono equation converge to the solutions of KdV equation if the term of Benjamin-Ono equation tends to zero.
文摘In this article, the author sets up the abundant traveling wave solutions for time fractional Benjamin–Ono equation which was introduced to describe internal waves in stratified fluids by using Jacobi elliptic function expansion method. The traveling wave solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. It can be seen that the obtained results are found to be important for the statement of some physical demonstrations of problems in mathematical physics and ocean engineering. In ocean engineering Benjamin–Ono equations are generally used in computer simulation for the water waves in deep water and open seas.
文摘By using perturbation methods, the evolution equation is derived for the second-order internal solitarywaves in stratified fluids of great depth, which is a kind of inhomogeneous linearized Belljamin-Ono equation.
文摘The algebraic solitary wave and its associated eigenvalue problem in a deep stratified fluid with a free surface and a shallow upper layer were studied. And its vertical structure was examined. An exact solution for the derived 2D Benjamin-Ono equation was obtained, and physical explanation was given with the corresponding dispersion relation. As a special case, the vertical structure of the weakly nonlinear internal wave for the Holmboe density distribution was numerically investigated, and the propagating mechanism of the internal wave was studied by using the ray theory.
文摘The Benjamin-ono (BO) equation is an important nonlinear wave model which can describe the deep oceanic internal wave propagation. In this paper, the multi-algebraic solitary wave solutions for the internal wave BO equation including the linear velocity term in matrix form are given by the bilinear form. Based on the analytic solutions of the BO equation obtained in this paper and considering the hydrological parameters, the propagation of one-solitary wave and different kinds of interaction for the two-solitary waves are discussed and illustrated.
