A VARIATIONAL METHOD TO STUDY THE ZAKHAROV EQUATION

A variational method given by Ritz has been applied to the Zakharov equation to construct an analytical solution. The solution of Zakharov equation gives a good description of both linear and nonlinear evolutions of instabilities generated in waves due to modulation. The spatially periodic trial function is chosen in the form of combination of Jacobian elliptic functions with the dependence of its parameters subject to optimization. This Zakharov equation is reduced to nonlinear Schr¨ odinger equation in the static limit.

are the coupled partial differential equations. Here, E(x,t) is the slowly varying envelope of the high-frequency field, and n is the density of the media or ions in media. These Zakharov equations can be approximated by nonlinear Schrödinger equation [5,4].
The Zakharov equation (2.1) can be formulated as the variational problem corresponding to Lagrangian Lagrangian ℘(x, t) is given by where u t = n + |E| 2 . (2.4) The asterisks and c · c denote the complex conjugate, and the limit of integration is the periodicity length λ, which will later be assumed as constant. Now, first we will reproduce the Zakharov equations by using Euler-Lagrange equations: On substituting the values from (2.3) in (2.5), we obtain the Zakharov equation (2.1) which shows that the selection of Lagrangian density (2.2) is compatible. We employ the Ritz variational principle to the action integral S(t) = ∫ L(t)dt with respect to time-dependent parameters of the trial function which admits: (1) the shape of an unmodulated wave with a sinusoidal disturbance, (2) provide spatial periodicity of the Lagrangian with period λ. So, these features are provided by [10,11] The time-dependent functions, independent from one another, are A, β, c, φ, α, x 0 , and k.
Here, dn(z;β) and cn(z;β) are the Jacobian elliptic functions for z = α(x − x 0 ) and α = 4K(β)/λ, and K(β) is the complete elliptic integral of first kind. For small parameters β and c, one obtain from (2.6) Equation (2.7) describes envelops of a finite amplitude wave with (wave number k) slightly modulated by a plane wave with wave number α. We have seen that Zakharov Arun Kumar 3705 equation is reduced to NLS equation in the static limit n = −|E| 2 [7,9]. So the total Lagrangian (2.3) can be written as (Euler-Lagrange equation reproduces the NLS equation from ℘ NLS ) and is the additional part. So the action integral for Zakharov equation becomes Now we assume trial function for ℘ 1 as Now we substitute our trial function in (2.2) after changing α(x − x 0 ) = z and k(t) = V/2. We get (2.13) where , 3 3 , (2.14) 3706 A variational method to study the Zakharov equation Here, assumptions α = 4K/λ and (1) , are used to calculate above integrals. C 1 , C 2 , C 3 , and C 4 are the quantities in fraction and are the combination of complete elliptic integrals K, E, and their argument β. So the action integral S becomes (2.15) The variable x 0 is absent in the action integral except for the combination φ − Vx 0 /2, and hence all trajectories x 0 (t) are equivalent provided that φ(t) is approximately shifted. Now we proceed to study the variational equations for the parameters A, β, c, B, and φ with the help of Euler-Lagrange equations. Then, our system of equations, which follows from action integral, consists of the following conservation laws and Euler-Lagrange equations [1].

A direct numerical method
Zakharov equations can directly be solved by finite difference scheme [3], where E(x,t) is the slowly varying envelope of the high-frequency field and is given by . The initial conditions in static limit are A = 1, α = 1.2, β = c = 0.1, and V = 0, α(x − x 0 ) = z, x 0 = 0. A specific Crank-Nicholson scheme of finite difference with step length h = 0.4 in x, and l = 1 in t (time), is applied for first part, and a simple finite difference scheme is used for second part of the equation with same step lengths. The solution is given below. Figure 3.2 shows the example that corresponds to H − NV 2 /4 > U (β = 0). These results, with β = c = 0.1, can be compared with the outcome from the numerical integration 3708 A variational method to study the Zakharov equation

Conclusion
We applied Ritz variational principle based on the Zakharov-Lagrangian to solve the Zakharov equation, which may be a model for both linear and nonlinear evolution of some instabilities in a wave system or flow. Spatial variance of trial function was assumed a priori, while time dependence of its parameters was subject to optimization. The crucial point, finding an appropriate trial function, was solved by introducing the variability to parameters of a stationary solution of the Zakharov equation. We chose the solution in the form of a combination of Jacobian elliptic functions. The results of the theoretical model compare well the numerical solution to Zakharov equation.