In this work, with the help of the symbolic computation system Maple and the Riccati mapping approach and a linear variable separation approach, a new family of traveling wave solutions of the (2 + 1)-dimensional diss...In this work, with the help of the symbolic computation system Maple and the Riccati mapping approach and a linear variable separation approach, a new family of traveling wave solutions of the (2 + 1)-dimensional dissipative Zabolotskaya-Khokhlov equation (DZK) is derived. Based on the derived solitary wave solution, some novel kind wave excitations are investigated.展开更多
Cylindrical waveguides without end surfaces can serve as two-dimensional resonant cavities. In such cavities the electromagnetic oscillations corresponding to an eigenfrequency can always be taken as TM or TE modes ev...Cylindrical waveguides without end surfaces can serve as two-dimensional resonant cavities. In such cavities the electromagnetic oscillations corresponding to an eigenfrequency can always be taken as TM or TE modes even when the walls have a finite conductivity and the medium is absorptive. This paper obtains analytic solutions to the field equations when the cylinder has a circular cross section. Some nonperturbative conclusions are drawn from the eigenvalue equation. Approximate analytic results for the resonant frequencies are obtained when the absorption of the medium is small and the walls are good conductors. Stability of the eigen modes is discussed. Similar results for the coaxial line are presented.展开更多
The lump solution is one of the exact solutions of the nonlinear evolution equation.In this paper,we study the lump solution and lump-type solutions of(2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure(AKNS)equ...The lump solution is one of the exact solutions of the nonlinear evolution equation.In this paper,we study the lump solution and lump-type solutions of(2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure(AKNS)equation by the Hirota bilinear method and test function method.With the help of Maple,we draw three-dimensional plots of the lump solution and lump-type solutions,and by observing the plots,we analyze the dynamic behavior of the(2+1)-dimensional dissipative AKNS equation.We find that the interaction solutions come in a variety of interesting forms.展开更多
This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipatio...This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves.The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail.The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied,and the critical values that can describe the magnitude of the dissipation effect are,respectively,found for the two cases of b_3<0 and b_3>0 in the equation.The results show that,when the dissipation effect is significant(i.e.,r is greater than the critical value in a certain situation),the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution;when the dissipation effect is small(i.e.,r is smaller than the critical value in a certain situation),the traveling wave solution to the equation appears as the oscillation attenuation solution.By using the hypothesis undetermined method,all possible solitary wave solutions to the equation when there is no dissipation effect(i.e.,r=0)and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained;in particular,when the dissipation effect is small,an approximate solution of the oscillation attenuation solution can be achieved.This paper is further based on the idea of the homogenization principles.By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper,and by investigating the asymptotic behavior of the solution at infinity,the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained,which is an infinitesimal amount that decays exponentially.The influence of the dissipation coefficient on the amplitude,frequency,period,and energy of the bounded traveling wave solution of the equation is also discussed.展开更多
The effects of different wind input and wave dissipation formulations on the steady Ekman current solution are described. Two formulations are considered: one from the wave modeling(WAM) program proposed by Hasselmann...The effects of different wind input and wave dissipation formulations on the steady Ekman current solution are described. Two formulations are considered: one from the wave modeling(WAM) program proposed by Hasselmann and Komen and the other provided by Tsagareli and Babanin. The solution adopted for our study was presented by Song for the wave-modifi ed Ekman current model that included the Stokes drift, wind input, and wave dissipation with eddy viscosity increasing linearly with depth. Using the Combi spectrum with tail effects, the solutions are calculated using two formulations for wind input and wave dissipation, and compared. Differences in the results are not negligible. Furthermore, the solution presented by Song and Xu for the eddy viscosity formulated using the K-Profi le Parameterization scheme under wind input and wave dissipation given by Tsagareli and Babanin is compared with that obtained for a depth-dependent eddy viscosity. The solutions are further compared with the available well-known observational data. The result indicates that the Tsagareli and Babanin scheme is more suitable for use in the model when capillary waves are included, and the solution calculated using the K-Profi le Parameterization scheme agrees best with observations.展开更多
By using the complete discrimination system for the polynomial method, the classification of single traveling wave solutions to the generalized Kadomtsev-Petviashvili equation without dissipation terms in p=2?is obtai...By using the complete discrimination system for the polynomial method, the classification of single traveling wave solutions to the generalized Kadomtsev-Petviashvili equation without dissipation terms in p=2?is obtained.展开更多
By the complete discrimination system for polynomial method, we obtained the classification of single traveling wave solutions to the generalized strong nonlinear Boussinesq equation without dissipation terms in p=1.
This paper is concerned with the existence and the nonlinear asymptotic stabil- ity of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {θt=vζx＋（ζθ）x＋aθxx,ζt=-θ...This paper is concerned with the existence and the nonlinear asymptotic stabil- ity of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {θt=vζx＋（ζθ）x＋aθxx,ζt=-θx＋βζxx;with initial data and end states （ζθ）（x,0）=（ζ0,θ0）（x）→（ζ±,θ±）as x→∞.We obtain the existence of traveling wave solutions by phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without restrictions on the coeffi- cients a and v by the method of energy estimates.展开更多
This communication is devoted to analyze elastic deformation on electrically conducted viscoelastic fluid in the presence of viscous dissipation effects. Non-linear analysis is computed through exact solutions for vel...This communication is devoted to analyze elastic deformation on electrically conducted viscoelastic fluid in the presence of viscous dissipation effects. Non-linear analysis is computed through exact solutions for velocity,temperature and concentration profiles. Special emphasis is provided for elastic deformation in the presence of magnetohydrodynamics effects. Concentration profile is discussed significantly in the presence constructive and destructive chemical reaction. Results are displayed through graphs and discussed for physical parameters that are used in present analysis. Notable findings include that temperature and thermal boundary layer thickness is an increasing function of Prandtl number and a decreasing function of elastic deformation. In addition, heat transfer rate is enhanced by increasing the conjugate parameter(γ) which measures the strength of surface heating.展开更多
This paper dwells upon the asymptotic behavior, as t→∞, of the integral solution u(t) to the nonhaear evolution equation u'(t) ∈A(t)u(f) + g(t),t≥s,u(s) =xo∈D(A(a)),where {A(t)}t≥s is a family m-ipative oper...This paper dwells upon the asymptotic behavior, as t→∞, of the integral solution u(t) to the nonhaear evolution equation u'(t) ∈A(t)u(f) + g(t),t≥s,u(s) =xo∈D(A(a)),where {A(t)}t≥s is a family m-ipative operator in a Hlilbert space H,and g∈L(loc)(0,∞;H). We prove that converges weakly, as t→∞, uniforluly in h≥0, which applies that u(t) is weak convergence if and only if u(t) is weakly asymptotically regular i.e., u(t + h) -u(f) →0 for h≥0.展开更多
In this paper the authors consider Cauchy problem of first order quasilinear hyperbolic and prove that existence of its periodic solutions and estimates of life span of solutions. This results reveals the relationship...In this paper the authors consider Cauchy problem of first order quasilinear hyperbolic and prove that existence of its periodic solutions and estimates of life span of solutions. This results reveals the relationship of dissipation and smoothness of periodic solutions.展开更多
This paper studies the Cauchy problem of the dissipative quasi-geostrophic equation in pseudomeasure space PM^n＋1-2α（R^n） or Lorentz space L n/2α-1,∞（R^n）, which admit the singular solutions. The global well-p...This paper studies the Cauchy problem of the dissipative quasi-geostrophic equation in pseudomeasure space PM^n＋1-2α（R^n） or Lorentz space L n/2α-1,∞（R^n）, which admit the singular solutions. The global well-posedness is established provided initial data θ0（x） are small enough in these spaces. Moreover, the asymptotic stability of solutions in pseudomeasure space is proved. In particular, if the initial data are homo-geneous functions of degree 1 - 2α, the self-similar solutions are also obtained.展开更多
A numerical study is performed to investigate the flow and heat transfer at the surface of a permeable wedge immersed in a copper （Cu）-water-based nanofluid in the presence of magnetic field and viscous dissipation ...A numerical study is performed to investigate the flow and heat transfer at the surface of a permeable wedge immersed in a copper （Cu）-water-based nanofluid in the presence of magnetic field and viscous dissipation using a nanofluid model proposed by Tiwari and Das （Tiwari I K and Das M K 2007 Int. J. HeatMass Transfer 50 2002）. A similarity solution for the transformed governing equation is obtained, and those equations are solved by employing a numerical shooting technique with a fourth-order Runge-Kutta integration scheme. A comparison with previously published work is carried out and shows that they are in good agreement with each other. The effects of velocity ratio parameter ~, solid volume fraction tp, magnetic field M, viscous dissipation Ec, and suction parameter Fw on the fluid flow and heat transfer characteristics are discussed. The unique and dual solutions for self-similar equations of the flow and heat transfer are analyzed numerically. Moreover, the range of the velocity ratio parameter for which the solution exists increases in the presence of magnetic field and suction parameter.展开更多
This article considers the global regularity to the initial-boundary value problem for the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion. To overcome the difficulty caused by the vanishin...This article considers the global regularity to the initial-boundary value problem for the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion. To overcome the difficulty caused by the vanishing viscosities, we first establish the elliptic system for ux and by, which are estimated by 7 × us and × by, respectively. Then, we establish the global estimates for × u and ×b.展开更多
In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physi- cally different types of materials, one component i...In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physi- cally different types of materials, one component is a Kirchhoff type wave equation with nonlinear time dependent localized dissipation which is effective only on a neighborhood of certain part of the boundary, while the other is a Kirchhoff type wave equation with nonlinear memory.展开更多
In this paper we show the decay of solutions to the initial-boundary value problem for some nonlinear hyperbolic equation with a nonlinear dissipative term, by using a difference inequality.
For a kind of partially dissipative quasilinear hyperbolic systems without Shizuta-Kawashima condition,in which all the characteristics,except a weakly linearly degenerate one,are involved in the dissipation,the globa...For a kind of partially dissipative quasilinear hyperbolic systems without Shizuta-Kawashima condition,in which all the characteristics,except a weakly linearly degenerate one,are involved in the dissipation,the global existence of H 2 classical solution to the Cauchy problem with small initial data is obtained.展开更多
文摘In this work, with the help of the symbolic computation system Maple and the Riccati mapping approach and a linear variable separation approach, a new family of traveling wave solutions of the (2 + 1)-dimensional dissipative Zabolotskaya-Khokhlov equation (DZK) is derived. Based on the derived solitary wave solution, some novel kind wave excitations are investigated.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10675174)
文摘Cylindrical waveguides without end surfaces can serve as two-dimensional resonant cavities. In such cavities the electromagnetic oscillations corresponding to an eigenfrequency can always be taken as TM or TE modes even when the walls have a finite conductivity and the medium is absorptive. This paper obtains analytic solutions to the field equations when the cylinder has a circular cross section. Some nonperturbative conclusions are drawn from the eigenvalue equation. Approximate analytic results for the resonant frequencies are obtained when the absorption of the medium is small and the walls are good conductors. Stability of the eigen modes is discussed. Similar results for the coaxial line are presented.
文摘The lump solution is one of the exact solutions of the nonlinear evolution equation.In this paper,we study the lump solution and lump-type solutions of(2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure(AKNS)equation by the Hirota bilinear method and test function method.With the help of Maple,we draw three-dimensional plots of the lump solution and lump-type solutions,and by observing the plots,we analyze the dynamic behavior of the(2+1)-dimensional dissipative AKNS equation.We find that the interaction solutions come in a variety of interesting forms.
基金Project supported by the National Natural Science Foundation of China(No.11471215)。
文摘This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves.The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail.The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied,and the critical values that can describe the magnitude of the dissipation effect are,respectively,found for the two cases of b_3<0 and b_3>0 in the equation.The results show that,when the dissipation effect is significant(i.e.,r is greater than the critical value in a certain situation),the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution;when the dissipation effect is small(i.e.,r is smaller than the critical value in a certain situation),the traveling wave solution to the equation appears as the oscillation attenuation solution.By using the hypothesis undetermined method,all possible solitary wave solutions to the equation when there is no dissipation effect(i.e.,r=0)and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained;in particular,when the dissipation effect is small,an approximate solution of the oscillation attenuation solution can be achieved.This paper is further based on the idea of the homogenization principles.By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper,and by investigating the asymptotic behavior of the solution at infinity,the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained,which is an infinitesimal amount that decays exponentially.The influence of the dissipation coefficient on the amplitude,frequency,period,and energy of the bounded traveling wave solution of the equation is also discussed.
基金Supported by the National Natural Science Foundation of China(No.41176016)the National Basic Research Program of China(973 Program)(Nos.2012CB417402,2011CB403501)the Fund for Creative Research Groups by National Natural Science Foundation of China(No.41121064)
文摘The effects of different wind input and wave dissipation formulations on the steady Ekman current solution are described. Two formulations are considered: one from the wave modeling(WAM) program proposed by Hasselmann and Komen and the other provided by Tsagareli and Babanin. The solution adopted for our study was presented by Song for the wave-modifi ed Ekman current model that included the Stokes drift, wind input, and wave dissipation with eddy viscosity increasing linearly with depth. Using the Combi spectrum with tail effects, the solutions are calculated using two formulations for wind input and wave dissipation, and compared. Differences in the results are not negligible. Furthermore, the solution presented by Song and Xu for the eddy viscosity formulated using the K-Profi le Parameterization scheme under wind input and wave dissipation given by Tsagareli and Babanin is compared with that obtained for a depth-dependent eddy viscosity. The solutions are further compared with the available well-known observational data. The result indicates that the Tsagareli and Babanin scheme is more suitable for use in the model when capillary waves are included, and the solution calculated using the K-Profi le Parameterization scheme agrees best with observations.
文摘By using the complete discrimination system for the polynomial method, the classification of single traveling wave solutions to the generalized Kadomtsev-Petviashvili equation without dissipation terms in p=2?is obtained.
文摘By the complete discrimination system for polynomial method, we obtained the classification of single traveling wave solutions to the generalized strong nonlinear Boussinesq equation without dissipation terms in p=1.
基金supported by the Natural Science Foundation of China(11001095)the Ph.D.specialized grant of the Ministry of Education of China(20100144110001)+2 种基金the Special Fund for Basic Scientific Research of Central Colleges(CCNU12C01001)supported by the Fundamental Research Funds for the Central Universities(2015IA009)the Natural Science Foundation of China(61573012)
文摘This paper is concerned with the existence and the nonlinear asymptotic stabil- ity of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {θt=vζx＋（ζθ）x＋aθxx,ζt=-θx＋βζxx;with initial data and end states （ζθ）（x,0）=（ζ0,θ0）（x）→（ζ±,θ±）as x→∞.We obtain the existence of traveling wave solutions by phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without restrictions on the coeffi- cients a and v by the method of energy estimates.
文摘This communication is devoted to analyze elastic deformation on electrically conducted viscoelastic fluid in the presence of viscous dissipation effects. Non-linear analysis is computed through exact solutions for velocity,temperature and concentration profiles. Special emphasis is provided for elastic deformation in the presence of magnetohydrodynamics effects. Concentration profile is discussed significantly in the presence constructive and destructive chemical reaction. Results are displayed through graphs and discussed for physical parameters that are used in present analysis. Notable findings include that temperature and thermal boundary layer thickness is an increasing function of Prandtl number and a decreasing function of elastic deformation. In addition, heat transfer rate is enhanced by increasing the conjugate parameter(γ) which measures the strength of surface heating.
文摘This paper dwells upon the asymptotic behavior, as t→∞, of the integral solution u(t) to the nonhaear evolution equation u'(t) ∈A(t)u(f) + g(t),t≥s,u(s) =xo∈D(A(a)),where {A(t)}t≥s is a family m-ipative operator in a Hlilbert space H,and g∈L(loc)(0,∞;H). We prove that converges weakly, as t→∞, uniforluly in h≥0, which applies that u(t) is weak convergence if and only if u(t) is weakly asymptotically regular i.e., u(t + h) -u(f) →0 for h≥0.
文摘In this paper the authors consider Cauchy problem of first order quasilinear hyperbolic and prove that existence of its periodic solutions and estimates of life span of solutions. This results reveals the relationship of dissipation and smoothness of periodic solutions.
基金B. Yuan was partially supported by the China postdoctoral Science Foundation （No. 20060390530）, Natural Science Foundation of Henan Province （No. 0611055500）, Science Foundation of the Education Department of Henan Province （200510460008） and Doctor Foundation of Henan Polytechnic University.
文摘This paper studies the Cauchy problem of the dissipative quasi-geostrophic equation in pseudomeasure space PM^n＋1-2α（R^n） or Lorentz space L n/2α-1,∞（R^n）, which admit the singular solutions. The global well-posedness is established provided initial data θ0（x） are small enough in these spaces. Moreover, the asymptotic stability of solutions in pseudomeasure space is proved. In particular, if the initial data are homo-geneous functions of degree 1 - 2α, the self-similar solutions are also obtained.
文摘A numerical study is performed to investigate the flow and heat transfer at the surface of a permeable wedge immersed in a copper （Cu）-water-based nanofluid in the presence of magnetic field and viscous dissipation using a nanofluid model proposed by Tiwari and Das （Tiwari I K and Das M K 2007 Int. J. HeatMass Transfer 50 2002）. A similarity solution for the transformed governing equation is obtained, and those equations are solved by employing a numerical shooting technique with a fourth-order Runge-Kutta integration scheme. A comparison with previously published work is carried out and shows that they are in good agreement with each other. The effects of velocity ratio parameter ~, solid volume fraction tp, magnetic field M, viscous dissipation Ec, and suction parameter Fw on the fluid flow and heat transfer characteristics are discussed. The unique and dual solutions for self-similar equations of the flow and heat transfer are analyzed numerically. Moreover, the range of the velocity ratio parameter for which the solution exists increases in the presence of magnetic field and suction parameter.
基金supported by the Scientific Research Funds of Huaqiao University(14BS309)the National Natural Science Foundation of China(11526091)
文摘This article considers the global regularity to the initial-boundary value problem for the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion. To overcome the difficulty caused by the vanishing viscosities, we first establish the elliptic system for ux and by, which are estimated by 7 × us and × by, respectively. Then, we establish the global estimates for × u and ×b.
文摘In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physi- cally different types of materials, one component is a Kirchhoff type wave equation with nonlinear time dependent localized dissipation which is effective only on a neighborhood of certain part of the boundary, while the other is a Kirchhoff type wave equation with nonlinear memory.
文摘In this paper we show the decay of solutions to the initial-boundary value problem for some nonlinear hyperbolic equation with a nonlinear dissipative term, by using a difference inequality.
基金supported by the Fudan University Creative Student Cultivation Program in Key Disciplinary Areas (No. EHH1411208)
文摘For a kind of partially dissipative quasilinear hyperbolic systems without Shizuta-Kawashima condition,in which all the characteristics,except a weakly linearly degenerate one,are involved in the dissipation,the global existence of H 2 classical solution to the Cauchy problem with small initial data is obtained.
