The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （NLS） equation i...The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （NLS） equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation （in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data）, or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.展开更多
In this article, a nonconforming quadrilateral element （named modified quasi- Wilson element） is applied to solve the nonlinear schrSdinger equation （NLSE）. On the basis of a special character of this element, tha...In this article, a nonconforming quadrilateral element （named modified quasi- Wilson element） is applied to solve the nonlinear schrSdinger equation （NLSE）. On the basis of a special character of this element, that is, its consistency error is of order O（ha） for broken Ha-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover, the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.展开更多
The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odin...The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.展开更多
基金supported by the National Science Foundation under grant DMS-0807653
文摘The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （NLS） equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation （in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data）, or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.
基金supported by the National Natural Science Foundation of China(11271340,11101381)
文摘In this article, a nonconforming quadrilateral element （named modified quasi- Wilson element） is applied to solve the nonlinear schrSdinger equation （NLSE）. On the basis of a special character of this element, that is, its consistency error is of order O（ha） for broken Ha-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover, the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.
文摘The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.
