In this paper, the authors consider a system of degenerate Davey-Stewartson equations. They prove the global existence of weak solutions in some weighted function spaces and the decay of weak solutions in some anisotr...In this paper, the authors consider a system of degenerate Davey-Stewartson equations. They prove the global existence of weak solutions in some weighted function spaces and the decay of weak solutions in some anisotropic spaces for appropriate initial data.展开更多
The nonlinear D-S equations on R^d, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space H^s whenever the exponent s is lower than that predi...The nonlinear D-S equations on R^d, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space H^s whenever the exponent s is lower than that predicted by scaling or Galilean invariance, or when the regularity is too low to support distributional solutions. Authors analyze a class of solutions for which the zero-dispersion limit provides good approximations.展开更多
The homoclinic solutions problem of the Davey-Stewartson ( DS) Equations were studied. By using the Hirota's bilinear method, the homoclinic orbits of the Davey-Stewartson Equations were obtained through the depen...The homoclinic solutions problem of the Davey-Stewartson ( DS) Equations were studied. By using the Hirota's bilinear method, the homoclinic orbits of the Davey-Stewartson Equations were obtained through the dependent variable transformation. The homoclinic orbits of the Davey-Stewartson Equations were discussed.展开更多
We discuss the mechanism of formation of singularities of solutions to the Novikov-Veselov,modified Novikov-Veselov,and Davey-Stewartson II(DSII)equations obtained by the Moutard type transformations.These equations a...We discuss the mechanism of formation of singularities of solutions to the Novikov-Veselov,modified Novikov-Veselov,and Davey-Stewartson II(DSII)equations obtained by the Moutard type transformations.These equations admit the L,A,B-triple presentation,the generalization of the L,Apairs for 2+1-soliton equations.We relate the blow-up of solutions to the non-conservation of the zero level of discrete spectrum of the L-operator.We also present a class of exact solutions,of the DSII system,which depend on two functional parameters,and show that all possible singularities of solutions to DSII equation obtained by the Moutard transformation are indeterminancies,i.e.,points when approaching which in different spatial directions the solution has different limits.展开更多
We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-Ⅰ singularities of solutions with■ can never happen at time T for all adiabatic number γ 1. Here ...We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-Ⅰ singularities of solutions with■ can never happen at time T for all adiabatic number γ 1. Here κ > 0 does not depend on the initial data.This is achieved by proving the regularity of solutions under■ This new scaling invariance also motivates us to construct an explicit type-Ⅱ blowup solution for γ > 1.展开更多
In this study,the sine-Gordon equation method is modified to deal with variable-coefficient systems containing imaginary parts,such as nonlinear Schrödinger systems.These are of considerable importance in many fi...In this study,the sine-Gordon equation method is modified to deal with variable-coefficient systems containing imaginary parts,such as nonlinear Schrödinger systems.These are of considerable importance in many fields of research,including ocean engineering and optics.As an example,we apply the modified method to variable-coefficient coupled nonlinear Schrödinger equations and Davey-Stewartson system with variable coefficients,treating them as one-dimensional and two-dimensional systems,respectively.As a result of this application,novel solitary wave solutions are obtained for both cases.Moreover,some figures are provided to illustrate how the solitary wave propagation is determined by the different values of the variable group velocity dispersion terms,which can be used to model various phenomena.展开更多
Abstract Considering the generalized Davey-Stewartson equation $i\mathop u\limits^. - \Delta u + \lambda \left| u \right|^p u + \mu E\left( {\left| u \right|^q } \right)\left| u \right|^{q - 2} u = 0$ where $\lambda &...Abstract Considering the generalized Davey-Stewartson equation $i\mathop u\limits^. - \Delta u + \lambda \left| u \right|^p u + \mu E\left( {\left| u \right|^q } \right)\left| u \right|^{q - 2} u = 0$ where $\lambda > 0,\mu \ge 0,E = F^{ - 1} \left( {\xi _1^2 /\left| \xi \right|^2 } \right)F$ we obtain the existence of scattering operator in ^(A↑^n) := { u ] H1(A↑^n) : |x|u ] L2(A↑^n)}.展开更多
We consider a Bianchi type Ⅰ physical metric g, an auxiliary metric q and a density matter ρ in Eddington-inspired-Born-Infeld theory. We first derive a system of second order nonlinear ordinary differential equatio...We consider a Bianchi type Ⅰ physical metric g, an auxiliary metric q and a density matter ρ in Eddington-inspired-Born-Infeld theory. We first derive a system of second order nonlinear ordinary differential equations. Then, by a suitable change of variables, we arrive at a system of first order nonlinear ordinary differential equations. Using both the solution-tube concept for the first order nonlinear ordinary differential equations and the nonlinear analysis tools such as the Arzelá-Ascoli theorem, we prove an existence result for the nonlinear system obtained. The resolution of this last system allows us to obtain new exact solutions for the model considered.Finally, by studying the asymptotic behaviour of the exact solutions obtained, we conclude that this solution is the counterpart of the Friedman-Lemaitre-Robertson-Walker spacetime in Eddington-inspired-Born-Infeld theory.展开更多
In the present paper we study the long time behavior of solutions to the Davey-Stewartson system in the Banach spaces. We make use of the properties of the semigroup generated by the linear principal operator in Lp() ...In the present paper we study the long time behavior of solutions to the Davey-Stewartson system in the Banach spaces. We make use of the properties of the semigroup generated by the linear principal operator in Lp() and prove that the Davey-Stewartson system possesses a compact global attractor Ap in Lp(). Furthermore, one show that the attractor is in fact independent of p and prove the attractor has finite Hausdorff and fractal dimensions.展开更多
文摘In this paper, the authors consider a system of degenerate Davey-Stewartson equations. They prove the global existence of weak solutions in some weighted function spaces and the decay of weak solutions in some anisotropic spaces for appropriate initial data.
基金supported by the Science Foundation of Jiangsu University (07JDG038)
文摘The nonlinear D-S equations on R^d, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space H^s whenever the exponent s is lower than that predicted by scaling or Galilean invariance, or when the regularity is too low to support distributional solutions. Authors analyze a class of solutions for which the zero-dispersion limit provides good approximations.
文摘The homoclinic solutions problem of the Davey-Stewartson ( DS) Equations were studied. By using the Hirota's bilinear method, the homoclinic orbits of the Davey-Stewartson Equations were obtained through the dependent variable transformation. The homoclinic orbits of the Davey-Stewartson Equations were discussed.
文摘We discuss the mechanism of formation of singularities of solutions to the Novikov-Veselov,modified Novikov-Veselov,and Davey-Stewartson II(DSII)equations obtained by the Moutard type transformations.These equations admit the L,A,B-triple presentation,the generalization of the L,Apairs for 2+1-soliton equations.We relate the blow-up of solutions to the non-conservation of the zero level of discrete spectrum of the L-operator.We also present a class of exact solutions,of the DSII system,which depend on two functional parameters,and show that all possible singularities of solutions to DSII equation obtained by the Moutard transformation are indeterminancies,i.e.,points when approaching which in different spatial directions the solution has different limits.
基金supported by National Natural Science Foundation of China (Grant No. 11725102)National Support Program for Young Top-Notch Talents+3 种基金SGST 09DZ2272900 from Shanghai Key Laboratory for Contemporary Applied Mathematicssupported by Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants (Grant Nos. CUHK-14305315, CUHK-14300917 and CUHK-14302917)NSFC/RGC Joint Research Scheme Grant (Grant No. N-CUHK 443-14)a Focus Area Grant from the Chinese University of Hong Kong
文摘We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-Ⅰ singularities of solutions with■ can never happen at time T for all adiabatic number γ 1. Here κ > 0 does not depend on the initial data.This is achieved by proving the regularity of solutions under■ This new scaling invariance also motivates us to construct an explicit type-Ⅱ blowup solution for γ > 1.
基金The authors would like to thank the Deanship of Scientific Research,Majmaah University,Saudi Arabia,for funding this work under project No.R-1441-26.
文摘In this study,the sine-Gordon equation method is modified to deal with variable-coefficient systems containing imaginary parts,such as nonlinear Schrödinger systems.These are of considerable importance in many fields of research,including ocean engineering and optics.As an example,we apply the modified method to variable-coefficient coupled nonlinear Schrödinger equations and Davey-Stewartson system with variable coefficients,treating them as one-dimensional and two-dimensional systems,respectively.As a result of this application,novel solitary wave solutions are obtained for both cases.Moreover,some figures are provided to illustrate how the solitary wave propagation is determined by the different values of the variable group velocity dispersion terms,which can be used to model various phenomena.
文摘Abstract Considering the generalized Davey-Stewartson equation $i\mathop u\limits^. - \Delta u + \lambda \left| u \right|^p u + \mu E\left( {\left| u \right|^q } \right)\left| u \right|^{q - 2} u = 0$ where $\lambda > 0,\mu \ge 0,E = F^{ - 1} \left( {\xi _1^2 /\left| \xi \right|^2 } \right)F$ we obtain the existence of scattering operator in ^(A↑^n) := { u ] H1(A↑^n) : |x|u ] L2(A↑^n)}.
文摘We consider a Bianchi type Ⅰ physical metric g, an auxiliary metric q and a density matter ρ in Eddington-inspired-Born-Infeld theory. We first derive a system of second order nonlinear ordinary differential equations. Then, by a suitable change of variables, we arrive at a system of first order nonlinear ordinary differential equations. Using both the solution-tube concept for the first order nonlinear ordinary differential equations and the nonlinear analysis tools such as the Arzelá-Ascoli theorem, we prove an existence result for the nonlinear system obtained. The resolution of this last system allows us to obtain new exact solutions for the model considered.Finally, by studying the asymptotic behaviour of the exact solutions obtained, we conclude that this solution is the counterpart of the Friedman-Lemaitre-Robertson-Walker spacetime in Eddington-inspired-Born-Infeld theory.
基金This project is supported Supported by National Natural Science Foundation of China.
文摘In the present paper we study the long time behavior of solutions to the Davey-Stewartson system in the Banach spaces. We make use of the properties of the semigroup generated by the linear principal operator in Lp() and prove that the Davey-Stewartson system possesses a compact global attractor Ap in Lp(). Furthermore, one show that the attractor is in fact independent of p and prove the attractor has finite Hausdorff and fractal dimensions.
