Exponential of a vector field Lie Theory is definitely not my field, but I hope my question will be however meaningful. 
I'm studying an article in which the vector field (in the plane) $X_1 = \partial_x$ and $X_2 = |x|^\alpha \partial_y$ is considered, with $\alpha >0 $. Then two curves are defined, which are supposed to connect the points $(\psi(y),y)$ and $(\psi(y-h),y-h)$, where $\psi$ is a $\mathcal{C}^1$ function defined on a certain open interval of $\mathbb{R}$. The first curve is defined by $$\gamma_1 (t) := \exp(t(X_1 - b X_2))(\psi(y),y)$$and the author says that this last expression is equal to $$\left(\psi(y) + t,y-b\int_0^t |\psi(y) + \tau|^\alpha \, d \tau \right)$$where $b = \min \{1,1/L\}$ and $L:=\sup_{|y|<\delta} | \psi' (y)|$. How did he get this last expression? I'm not familiar at all with exponentials of vector fields...
Thank you in advance!
 A: It means you look at the linear combination of the two vector fields, which is the vector field 
$$X = X_1 - bX_2 = \partial_x - b|x|^{\alpha} \, \partial_y =  \frac{\partial}{\partial x} - b|x|^{\alpha} \, \frac{\partial}{\partial y}.$$ Finding the vector field's exponent means you are finding the phase flow of the vector fields $$(x,y) \mapsto \Phi^t(x,y) = \Big(\Phi^t_1(x,y), \, \Phi^t_2(x,y)\Big) $$ i.e. it describes the curves tangent to the vector field, i.e. satisfies the condition
$$\frac{d}{dt} \Phi^t(x,y) = X{\big(\Phi^t(x,y)\big)} \, , \, \,\,\, \Phi^0(x,y) = (x,y)$$ The phase flow $\Phi^t$, also called the exponent of the vector field $\exp(tX) = \Phi^t$ is obtained by solving the system of differential equations
\begin{align}
\frac{dx}{dt} &= 1 \\
\frac{dy}{dt} &= -b |x|^{\alpha}
\end{align}
with an arbitrary initial condition $x(0) = x_0$ and $y(0) = y_0$. This initial value problem is easy to solve:
\begin{align}
x(t) &= x_0 + t\\
y(t) &= y_0 + \int_{0}^t |x_0 + \tau|^{\alpha} \, d\tau
\end{align}
With initial conditions $x_0 = \psi(y)$ and $y_0 = y$ you get
\begin{align}
x(t) &= \psi(y) + t\\
y(t) &= y + \int_{0}^t |\psi(y) + \tau|^{\alpha} \, d\tau
\end{align} exactly what you have. The choices of $b$ and $L$ have to do with something else, related to the specific problem you are reading about. 
