Landau Equations
Lane Frisch
A unified framework for equations of Ginzburg-Landau and Cahn-Hilliard type is developed using, as a basis, a balance law for microforces in conjunction with constitutive equations consistent with a mechanical version of the second law.
More information on publishing policies can be found on the Publishing Ethics Policies page. View our Publishing with Elsevier: step-by-step page to learn more about the publishing process. For any questions on your own submission or other questions related to publishing an article, contact our Researcher support team.
Although we endeavor to make our web sites work with a wide variety of browsers, we can only support browsers that provide sufficiently modern support for web standards. Thus, this site requires the use of reasonably up-to-date versions of Google Chrome, FireFox, Internet Explorer (IE 9 or greater), or Safari (5 or greater). If you are experiencing trouble with the web site, please try one of these alternative browsers. If you need further assistance, you may write to he...@aps.org.
Multivortex solutions of the Ginzburg-Landau equations (or, equivalently, of the Abelian Higgs model) are considered for a special choice of parameters. It is shown that for every n there is a 2n-parameter family of n-vortex solutions. It is conjectured that the parameters are just those needed to specify the positions of the vortices and that the vortices behave very much like noninteracting particles.
In this talk, I will discuss regularity criteria of the Prodi-Serrin-Ladyzhenskaya type and some optimal boundary regularities for the non cut-off Boltzmann and Landau equations on gerenal bounded domains with any of the usual physical boundary conditions: in-flow, bounce-back, specular-reflection and diffuse-reflection.
The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.
We consider the Frhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle numbers and a suitably small coupling, we show that the dynamics of the system is approximately described by the Landau-Pekar equations. These describe a Bose-Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
In ultrafast optics, optical pulses are generated to be of shorter pulse duration, which has enormous significance to industrial applications and scientific research. The ultrashort pulse evolution in fiber lasers can be described by the higher-order Ginzburg-Landau (GL) equation. However, analytic soliton solutions for this equation have not been obtained by use of existing methods. In this paper, a novel method is proposed to deal with this equation. The analytic soliton solution is obtained for the first time and is proved to be stable against amplitude perturbations. Through the split-step Fourier method, the bright soliton solution is studied numerically. The analytic results here may extend the integrable methods and could be used to study soliton dynamics for some equations in other disciplines. It may also provide the other way to obtain two-soliton solutions for higher-order GL equations.
Investigations on solitons have been made great progress since the first report on inverse scattering transformation (IST) method for soliton solutions1. Among them, one of active subjects is the study on optical solitons in nonlinear optics governed by nonlinear Schrdinger (NLS) equations2,3. Optical solitons can maintain their shapes and velocities during their propagation under the balance between group-velocity dispersion (GVD) and Kerr nonlinearity4. By virtue of the advantage of shape preserving, optical solitons have been applied in the optical switching, phase shifter, amplifier and information storage5,6,7,8.
On the other hand, soliton solutions have been obtained in such nonlinear partial differential equations as NLS equation, Sine-Gordon equation, Gross-Pitaevskii equation, Korteweg-de Vries equation, Burgers equation, Kadomtsev-Petviashvili equation and so on9,10,11,12,13. Recently, the integrable nonlocal NLS equation with parity-time (PT) symmetry has been introduced and solved by the IST method14. In addition to the IST method, there are some other integrable methods, such as Backlnd transformations, bilinear method, separation variable method and Darboux transformation, can be used to solve those equations15,16,17. Among all those methods, the bilinear method may be more direct and effective to solve integrable equations.
According to the above mentioned problems, we will propose a novel method to deal with the higher-order GL equation, such as Eq. (1). This method will be built on the asymmetry of the bilinear operator directly and will offer more freedoms and possibilities for variation than the bilinear method. A bright soliton solution for Eq. (1) will be first obtained, which is stable against amplitude perturbations. Through the split-step Fourier method, the bright soliton will be studied numerically.
Here, Y is a state function of variable t. The asymmetric operator can be considered as an asymmetric remainder when the modified bilinear operator eliminates the regular bilinear operator.
The linear asymmetric operators have a simple linear representation of differentiable functions. It indicates that the symmetry of the conventional bilinear method is not necessary for solvability, which attributes to the asymmetric operator represented by the conventional bilinear operators. The nonlinear asymmetric operator can be generalized to a bilinear form to transfer into an advanced linearity.
Here, is a double-channel bilinear asymmetric (DCBA) operator. G and F are state functions. The symmetry in the bilinear method is broken. The new bilinear forms are more free and generalized and contain the symmetric situation. The asymmetric degrees of two states can be exchanged. According to the Eq. (7), the third-order differential function can be written as
For the case of , it is so complex due to the nonlinear expression of Yttt. The low-order operators including 0, 1 and 2 order fit to the traditional bilinear method. However, the trilinear operator can not be written as a bilinear symmetric representation, but as the asymmetric case. The third-order dispersion term is usually presented in the dissipative situation.
So far, the bilinear asymmetric representation is more general than the symmetric representation. It can deal with the dissipative case as well as the conservative one. In the following, we will present a solvable theorem to find some interesting structures in the bilinear asymmetric equations.
If , the soliton is in the sech form. Otherwise, the soliton is asymmetric. Even more, if the equation contains another variable, it will be more free to obtain one-soliton solution. The structure of the equation has soliton solutions without the bilinear symmetric representation, which extends the integrable structures. For the special values of parameters, we can show the soliton profiles in Fig. 1.
We substitute all above relations into Eq. (27) and extract the coefficients of different exponent functions. The coefficient extractions should be equal to zero to satisfy Eq. (27). At first, we extract the constant coefficients and set it to zero. Then, we can solve the intrapulse Raman scattering coefficient,
Moreover, we extract the coefficients of and separate it into two individual equations according to the real and imaginary parts. We can obtain the relations between the group velocity dispersion and third-order dispersion as follows,
Here, the asymmetric parameter a is equal to 1. Finally, we can extract the coefficients of Eq. (28) to obtain the wave vector, parameter η and imaginary frequency. For , we can solve the wave vector according to real and imaginary parts,
Through the split-step Fourier method4, we can numerically stimulate the bright soliton evolution as shown in Fig. 3. The soliton drift is due to the interaction between the third-order dispersion (TOD) and intrapulse Raman scattering. While the amplitude is perturbed by 10%, the soliton is stable still.
The asymmetric representation method has been put forward to handle the analytic bright soliton solution of higher-order GL equation (1). The intrinsic structures of equations have been asymmetric, which are more general than the symmetric cases. A series concepts and methods of asymmetric representation theory have been represented. An asymmetric function has been proposed and asymmetric operators have been constructed. Some linear operators have been presented. Furthermore, the double-channel operator has been defined and used to make the representation of the single-channel operator. The conventional bilinear operators have been generalized to more cases and represented by the channel operators. A solvable theorem about the structure of the asymmetric operator equation has been proved and we have found an asymmetric structure. Through the novel asymmetric bilinear method, we have obtained a bright soliton solution for Eq. (1). Using the split-step Fourier method, the bright soliton has been numerically studied. The results in this paper extend the integrable methods and the asymmetric representation method can be used to solve other equations in different physical systems so as to study the soliton dynamics. In addition, the method here may provide a new idea to study two-soliton solutions for the GL equation in the future research, which is still an unsolvable problem.
64591212e2