Euler's equation for one-dimensional flow (Landau Lifshitz)

One page 5 in Landau & Lifshitz Fluid Mechanics (2nd edition), the authors pose the following problem:

Quote:

Write down the equations for one-dimensional motion of an ideal fluid in terms of the variables [itex]a[/itex], [itex]t[/itex], where [itex]a[/itex] (called a Lagrangian variable) is the [itex]x[/itex] coordinate of a fluid particle at some instant [itex]t=t0[/itex].

The authors then go on to give their solutions and assumptions. Here are the important parts:

Quote:

The coordinate [itex]x[/itex] of a fluid particle at an instant [itex]t[/itex] is regarded as a function of [itex]t[/itex] and its coordinate a at the initial instant: [itex]x=x(a,t)[/itex].

For the condition of mass conversation the authors arrive at (where [itex]ρ_0=ρ(a)[/itex] is the given initial density distribution):

[tex]

ρ\mathrm{d}x=ρ_0 \mathrm{d}a

[/tex]



or alternatively:

[tex]

ρ\left(\frac{∂x}{∂a}\right)_t=ρ_0

[/tex]



Now the authors go on to write out Euler's equation, where I start to miss something. With the velocity of the fluid particle [itex]v=\left(\frac{∂x}{∂t}\right)_a[/itex] and [itex]\left(\frac{∂v}{∂t}\right)_a[/itex] the rate of change of the velocity of the particle during its motion, they write for Euler's equation:

Quote:

[tex] \left(\frac{∂v}{∂t}\right)_a=−1ρ_0 \left(\frac{∂p}{∂a}\right)_t [/tex]

How are the authors arriving at that equation?



In particular, when looking at the full Euler's equation:

[tex]

\frac{∂v}{∂t}+(\mathbf{v}⋅\textbf{grad})\mathbf{v}=−1 ρ\, \textbf{grad}\, p

[/tex]



what happens with the second term on the LHS, [itex](\mathbf{v}⋅\textbf{grad})\mathbf{v}[/itex]? Why does it not appear in the authors' solution?