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9 Wave Equations
160

at some finite blow-up time. Defocusing equations are typically better behaved,
do not exhibit blow-up and in many cases have global-in-time solutions [18].
Another important nonlinear problem is represented by the so called KdV-
11
equation (named after D.J. Korteweg and G. de Vries, [9]), modeling the one-
dimensional wave motion in shallow water, i.e. shallow water waves in a chan-
12
nel .Itisaquadraticallynonlinearpartialdifferentialequationofthirddifferen-
tial order for the real-valued function u = u(x, t), representing the wave profile:

u
2
2
u t + + ε u xxx = 0, x ∈ R , t> 0 . (9.24)
2
x
Here the equation is presented already in dimensionless, scaled form, and ε
represents an aspect ratio parameter.
Of particular interest is the zero-dispersion limit of the KdV equation, ob-
tained by taking ε to 0 in the solution of (9.24), subject to appropriate initial
data. Note that setting ε to 0 in (9.24) leads to the inviscid Burgers equation 13

U
2
U t + = 0, x ∈ R , t> 0 , (9.25)
2
x
which is the prototype for a one-dimensional hyperbolic conservation law. The
Burgers equation has straight line characteristics in time-position space, along
which the value of the initial state U(t = 0) is transported. Thus, this equation
exhibits solutions, which become discontinuous in finite time (at points of inter-
section of characteristics), unless the initial datum U(t = 0) is a nondecreasing
function of the spatial variable x. Similarly to the WKB limit of the Schrödinger
equation, the term which was neglected when passing from the KdV to the
Burgers equation, is a third order derivative of u and the formal limit procedure
breaks down at the onset of shock-type singularities of the Burgers solution.
In fact, for sufficiently smooth initial data, the KdV equation has smooth so-
lutionsand,for ε small, breakdown of regularity is migitated by the onset of
1
fast oscillations with frequency of order when the derivatives of u get large. In
ε
other words, the nonlinear convection term u 2 tends to create discontinuities
2 x
2
in the solution and the third derivative term ε u xxx tends to smooth the solution.
The latter wins the competition but at the prize of developing high frequency
oscillations if ε is small. This limit process was made mathematically rigorous in
aseriesofthreedeeppapers[10],[11],[12]bytheAbelPrizerecipientPeterLax 14
15
and his then Ph.D. student David Levermore . Their mathematical methodol-
ogy was based on the fact that the KdV equation can be written as an infinite
11 http://staff.science.uva.nl/∼janwieg/korteweg/
12 http://mathworld.wolfram.com/Korteweg-deVriesEquation.html
13 http://en.wikipedia.org/wiki/Burgers’_equation
14
http://mathworld.wolfram.com/news/2005-03-18/abelprize/
15
http://www.math.umd.edu/∼lvrmr/