On a System of Nonlinear Optical Wave Equations in Two-Spatial Dimension

We show that for a system of nonlinear weakly dispersive wave equations in two-spatial dimension, which is a model in nonlinear optics, the local L 2 norm of its solutions decays to zero as time approaches infinity.


Introduction
The equation which describes a model of nonlinear modulated dispersive wave was first discovered to have a very important application in optical communication by Wai, Menyuk, Lee, & Chen (1986), Wai, Menyuk, Chen, & Lee (1987), and Wai, Chen, & Lee (1990).
In this paper, we will discuss the asymptotic behavior of the wave of a weakly dispersive medium, which is described by the two-dimensional generalization of the equation (1.1).The wave is still propagating in the x-direction but with a weakly perturbation in the y-direction.
-  +   + i|| 2 u = -  , (1.2.a) We will show that the smooth solutions decay in time in the local  2 norm.The motivation to study this system of equations is similar to the study of the well-known Kadomtsev-Petviashivili equation (Kadomtsev & Petviashvil 1970)   + 6u  +   = -    =   to the well-known Korteweg-de Vries equation (Korteweg & de Vries 1895) This paper is the first paper to study the proposed system of equations (1.2 a) & (1.2 b).

Two Conservation Laws
Conservation laws are very important in the physics.The famous Noether's Theorem (Noether 1918) states that every invariance of motion of a physical system has a corresponding conservation law.Furthermore, every conservation law is a constraint to the governing system of equations of the physical system.Here we present two conservation laws of (1.2 a) and (1.2b).

 
Now we integrate both sides with respect to t from 0 to T, T > 0, to get where  1 is a constant depending on the initial data and the bound for A. Note that  1 doesn't depend on T.
We now choose A(x,y) = arctan( where  2 depends on the initial data, A, and r.

Then
for some constant  4 which is independent of t.

 
2 Mr u dxdy  goes to zero as t goes to infinity.M(r)

Conclusion
The result of this paper is the first paper in the weakly dispersion in another direction of the system of equations that models the zero-group-dispersion wavelength of a single-mode optical fiber.There remains several problems for the future study of this system of equations such as the well-posedness, special solutions, symmetries, etc.Moreover, we have found two conservation laws.It would be very interesting to find more conservation laws, if any additional conservation law exists.