Homework Problems
1) Consider the PDE
xut+uux= 0.
Upon a simple transformation that makes it look like an equation we have
studied, solve it with initial conditions u(x, t) = x.
2) Consider the KdV equation:
ut+ 6uux+uxxx = 0.
•Let us look for traveling wave solutions u(x, t) = f(x−ct). Write the
resulting ODE for f=f(ξ) where ξ=x−ct.
•Integrate this equation once, assuming that both fand its derivatives
vanish at x→ ±∞.
•Write the resulting ODE as f′′ =... and solve it via quadrature, to find
the prototypical KdV solution that decays (and has decaying derivatives)
at ±∞.
3) Still for the KdV equation:
ut+ 6uux+uxxx = 0.
By passing the derivative inside the integral as a partial derivative, and assuming
that uand its derivatives vanish sufficiently fast at ±∞, show that the quantities
•M=R∞
−∞ udx,
•P=R∞
−∞ u2dx
•E=R∞
−∞ u3−(ux)2
2dx,
are conserved quantities, i.e., that their time derivatives vanish. Interestingly,
as we’ll discuss further below, the KdV has infinitely many such conserved
quantities, of these are only the first (and more physical) few. By the way,
notice that it sounds plausible that the mass is defined as above (area under
the field profile). However, illustrate that it also makes sense to define the
