Numerical investigation of guiding of Gaussian laser pulse in plasma channel with radial variation in presence of Kerr effect by Runge–Kutta and extrapolation methods

Abstract

In this paper, guiding of Gaussian laser pulse in plasma channel in the presence of Kerr effect is studied numerically. We assume that the plasma channel has radial variation. We obtain matched condition to guide intense laser pulse through plasma channel in presence of Kerr effect. Using the SDE method, four coupled equations for the pulse amplitude, phase, spot size and wave front curvature will be derived. We apply two numerical methods for solving these equations, Runge–Kutta and extrapolation methods. We give the functional form of the nonlinear initial value problem and obtain some properties about the convergence of the methods. Since these are pure mathematical considerations, they are mentioned in Appendix section. The solutions obtained by the methods are compared together and we see that the solutions are the same. Hence, the proposed methods are applicable and have the sufficient accuracy for the current system of equations.

Appendix

In Sect. 3, we have shown that the mathematical model of the problem is as follows

$$ \begin{gathered} \dot{X} = f(X), \hfill \\ X(0) = X_{0} , \hfill \\ \end{gathered} $$

(22)

where

$$ X(\tilde{z}) = \left[ {\begin{array}{*{20}c} {\tilde{r}_{s} (\tilde{z})} \\ {\alpha (\tilde{z})} \\ {\tilde{B}(\tilde{z})} \\ {\Psi (\tilde{z})} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {x_{1} (\tilde{z})} \\ {x_{2} (\tilde{z})} \\ {x_{3} (\tilde{z})} \\ {x_{4} (\tilde{z})} \\ \end{array} } \right],\begin{array}{*{20}c} {f(X) = \left[ {\begin{array}{*{20}c} {\frac{{x_{2} }}{{x_{1} }}} \\ {\frac{{1 + x_{2}^{2} }}{{x_{1}^{2} }} - \theta x_{1}^{2} - \beta x_{3}^{2} } \\ {\frac{{ - x_{2} x_{3} }}{{x_{1}^{2} }}} \\ {1 - \frac{1}{{x_{1}^{2} }} - \gamma + \frac{3}{2}\beta x_{3}^{2} } \\ \end{array} } \right]} \\ \end{array} , $$

(23)

where $\tilde{r}_{s} (\tilde{z}),\alpha (\tilde{z}),\tilde{B}(\tilde{z}),\Psi (\tilde{z})$ are the unknown physical functions and $\theta = \frac{{4\pi \,e^{2} N_{n} }}{{2m\,c\omega_{0} }} \cdot \frac{{r_{{s_{0} }}^{2} \,z_{0} }}{{R_{ch}^{2} }} \cdot \Delta \tilde{n}_{e} ,\beta = \frac{{\omega_{0}^{2} n_{0}^{2} n_{2} }}{{2\pi \,c^{2} }}\,P_{in} ,\gamma = \frac{{4\pi e^{2} N_{n} z_{0} }}{{2mc\omega_{0} }}\tilde{n}_{{e_{0} }}$ are known constants. In our sample problem $X(0) = X_{0} = (1,0,1,0)^{T}$. This is an autonomous system of differential equations. Hence, the extrapolation and the Runge–Kutta methods are simple and applicable. For example, we give the Runge–Kutta method with details in Sect. 4 and we see this simplicity by an algorithm. For the autonomous system (22), suppose $X = \left( {x_{1} ,...,x_{m} } \right)^{T}$ and $D$ is the domain of $f$, then $f$, is said to satisfy a Lipschitz condition on $D$ in the variables $x_{1} ,...,x_{m}$, if a constant $L > 0$ exists with

$$ \left| {f(x_{1} ,...,x_{m} ) - f(z_{1} ,...,z_{m} )} \right|\begin{array}{*{20}c} {\begin{array}{*{20}c} { \le L\sum\limits_{j = 1}^{m} {\left| {x_{j} - z_{j} } \right|} ,} & {(x_{1} ,...,x_{m} ),(z_{1} ,...,z_{m} ) \in D} \\ \end{array} } \\ \end{array} . $$

(24)

By using the mean value theorem, it can be shown that if $f$ and its first partial derivatives are continuous on $D$ and if

$$ \begin{array}{*{20}c} {\begin{array}{*{20}c} {\left| {\frac{{\partial f(x_{1} ,...,x_{m} )}}{{\partial x_{i} }}} \right| \le L,} & {i = 1,2,...,m,} \\ \end{array} (x_{1} ,...,x_{m} ) \in D} \\ \end{array} , $$

(25)

then $f$ satisfies a Lipschitz condition on $D$ with Lipschitz constant $L$ [19]. When $f$ is continuous and satisfies a Lipschitz condition, then the system (1.1) is a well-posed problem and has a unique solution [15].

In accordance with physical considerations, the domain of the function $f$ in (23) is $D = \left\{ {(x_{1} ,x_{2} ,x_{3} ,x_{4} )|0 < \mathop {\min }\limits_{{0 \le \tilde{z}}} x_{1} (\tilde{z}),\mathop {Max}\limits_{{0 \le \tilde{z}}} x_{i} (\tilde{z}) < \infty ,i = 1,2,3,4} \right\}$. Now we show that $f$ satisfies a Lipschitz condition on $D$. We define $m_{1} : = \mathop {\min }\limits_{{0 \le \tilde{z}}} x_{1} (\tilde{z}),M_{i} : = \mathop {Max}\limits_{{0 \le \tilde{z}}} x_{i} (\tilde{z}) < \infty ,i = 1,2,3,4$, then $\left| {\frac{{\partial f_{1} }}{{\partial x_{1} }}} \right| \le \left| {\frac{{x_{1} }}{{x_{1}^{2} }}} \right| \le \frac{{M_{2} }}{{m_{1}^{2} }} = :l_{11}$, $\left| {\frac{{\partial f_{1} }}{{\partial x_{2} }}} \right| \le \left| {\frac{1}{{x_{1}^{2} }}} \right| \le \frac{1}{{m_{1}^{2} }} = :l_{12}$,$\left| {\frac{{\partial f_{1} }}{{\partial x_{3} }}} \right| = \left| {\frac{{\partial f_{1} }}{{\partial x_{4} }}} \right| = 0 = :l_{13} = l_{14}$,$\left| {\frac{{\partial f_{2} }}{{\partial x_{1} }}} \right| \le \frac{{2\left( {1 + M_{2}^{2} } \right)}}{{m_{1}^{3} }} + 2\theta M_{1} = :l_{21}$,$\left| {\frac{{\partial f_{2} }}{{\partial x_{2} }}} \right| \le \frac{{2M_{2} }}{{m_{1}^{2} }} = :l_{22}$,$\left| {\frac{{\partial f_{2} }}{{\partial x_{3} }}} \right| \le 2\beta M_{3} = :l_{23}$,$\left| {\frac{{\partial f_{2} }}{{\partial x_{4} }}} \right| = 0 = :l_{24}$,$\left| {\frac{{\partial f_{3} }}{{\partial x_{1} }}} \right| \le \frac{{2M_{2} M_{3} }}{{m_{1}^{3} }} = :l_{31}$,$\left| {\frac{{\partial f_{3} }}{{\partial x_{2} }}} \right| \le \frac{{M_{3} }}{{m_{1}^{2} }} = :l_{32}$,$\left| {\frac{{\partial f_{3} }}{{\partial x_{3} }}} \right| \le \frac{{M_{2} }}{{m_{1}^{2} }} = :l_{33}$,$\left| {\frac{{\partial f_{3} }}{{\partial x_{4} }}} \right| = 0 = :l_{34}$,$\left| {\frac{{\partial f_{4} }}{{\partial x_{1} }}} \right| \le \frac{2}{{m_{1}^{3} }} = :l_{41}$,$\left| {\frac{{\partial f_{4} }}{{\partial x_{2} }}} \right| = \left| {\frac{{\partial f_{4} }}{{\partial x_{4} }}} \right| = 0 = :l_{42} = l_{44}$,$\left| {\frac{{\partial f_{4} }}{{\partial x_{3} }}} \right| \le 3\beta M_{3} = :l_{43}$. Hence, for $j = 1,2,3,4$, $\left| {\frac{\partial f}{{\partial x_{j} }}} \right| \le \left| {\left( {\frac{{\partial f_{1} }}{{\partial x_{j} }},\frac{{\partial f_{2} }}{{\partial x_{j} }},\frac{{\partial f_{3} }}{{\partial x_{j} }},\frac{{\partial f_{4} }}{{\partial x_{j} }}} \right)^{T} } \right| = \left( {\sum\limits_{i = 1}^{4} {\left| {\frac{{\partial f_{i} }}{{\partial x_{j} }}} \right|^{2} } } \right)^{1/2} \le \left( {\sum\limits_{i = 1}^{4} {l_{ij}^{2} } } \right)^{1/2} = :L_{j}$.

And by defining $L = \mathop {Max}\limits_{1 \le j \le 4} L_{j}$, $f$ satisfies a Lipschitz condition on $D$ with Lipschitz constant $L$. Obviously f is continuous and the proposed methods are convergent and have a good accuracy [15] [18]. Indeed, in accordance with [20] page 136 Eq. (21), the extrapolation method has a super convergence accuracy.