# System of coupled first order PDE

• blue2script
In summary, the conversation discusses a problem related to QFT in a research project, specifically a Dirac equation with a space-time dependent mass in 2 spatial dimensions. This equation is represented mathematically as a PDE involving matrices and a given function. The conversation also mentions trying to solve the equation using Fourier-method but ultimately finding a simpler method to decouple the linear PDE.
blue2script
Hello everybody,

I have a problem here related to QFT in a research project. I end up with some Dirac equation with space-time dependent mass in 2 spatial dimensions.

More mathematically, the PDE to solve is

$$\left( {i\left( {\sigma ^i \otimes I_2 } \right)\partial _i + g_y \varphi ^a \left( {I_2 \otimes \sigma ^a } \right)} \right)\psi = 0$$

where $$\varphi = \varphi\left(x,y\right)$$ is a (given) function of x,y.

More explicit, the system looks like:
$$\left[ {i\left( {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array} } \right)\partial _1 + i\left( {\begin{array}{*{20}c} 0 & { - i} & 0 & 0 \\ i & 0 & 0 & 0 \\ 0 & 0 & 0 & { - i} \\ 0 & 0 & i & 0 \\ \end{array} } \right)\partial _2 + g_y \left\{ {\varphi ^1 \left( {\begin{array}{*{20}c} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} } \right) + \varphi ^2 \left( {\begin{array}{*{20}c} 0 & 0 & { - i} & 0 \\ 0 & 0 & 0 & { - i} \\ i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ \end{array} } \right) + \varphi ^3 \left( {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & { - 1} & 0 \\ 0 & 0 & 0 & { - 1} \\ \end{array} } \right)} \right\}} \right]\left( {\begin{array}{*{20}c} {\psi _1 } \\ {\psi _2 } \\ {\psi _3 } \\ {\psi _4 } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right)$$

with $$\psi_i = \psi_i\left(x,y\right)$$. I tried some Fourier-method, but that wouldn't work out. I am completely stuck here and have no idea how to proceed. Could anyone give me a hint?

Blue2script

Ok, sorry guys, I got it. Its pretty simple to decouple this linear PDE. I am left now with four second order PDEs.

Thanks!

Blue2script

## 1. What is a system of coupled first order PDE?

A system of coupled first order PDE (partial differential equations) is a set of equations that involve multiple dependent variables and their partial derivatives with respect to one or more independent variables. These equations are typically used to model complex physical systems and their behavior.

## 2. How is a system of coupled first order PDE different from a single PDE?

A single PDE involves only one dependent variable and its derivatives, whereas a system of coupled first order PDE involves multiple dependent variables and their derivatives. This means that the solution to a system of coupled PDEs will involve finding values for all of the dependent variables, rather than just one.

## 3. What are some common examples of systems of coupled first order PDE?

Some common examples of systems of coupled first order PDE include the Navier-Stokes equations for fluid dynamics, the Maxwell's equations for electromagnetism, and the Schrödinger equation for quantum mechanics. These systems are used to model a wide range of physical phenomena in fields such as engineering, physics, and biology.

## 4. How are systems of coupled first order PDEs solved?

The most common method for solving systems of coupled first order PDEs is through numerical methods, such as finite difference or finite element methods. These methods involve discretizing the equations into a set of algebraic equations, which can then be solved using computer algorithms. Analytical solutions are also possible for simpler systems, but they are rare for complex systems.

## 5. What are the applications of systems of coupled first order PDE?

Systems of coupled first order PDEs have a wide range of applications in various fields, including fluid dynamics, electromagnetism, heat transfer, and quantum mechanics. They are used to model and predict the behavior of complex systems, and are essential for understanding and developing new technologies in areas such as aerospace, energy, and materials science.

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