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Zero-Dispersion Limits

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Encyclopedia of Nonlinear Science

ZERO-DISPERSION LIMITS

The Korteweg-de Vries (KdV) equation with small dispersion

(1)

is a model for the formation and propagation of dispersive shock waves in one dimension. Let u(x, t; ε) denote the solution of Cauchy problem (1), where the initial data u₀(x) is smooth and decreases at infinity sufficiently fast. It is known that for ε>0, no matter how small, the solution of (1) remains smooth for all t>0. For ε=0, (1) becomes the Cauchy problem for the Hopf equation

(2)

The solution of the Hopf equation can be obtained by the method of characteristics. If the initial data u₀(x) is somewhere increasing, the solution u(x, t) of Equation (2) always has a point (x_c, t_c) of gradient catastrophe where an infinite derivative develops.

After the time of gradient catastrophe t_c, the solution u(x, t, ε) of (1) develops in the neighborhood of x_c an expanding region filled with rapid modulated oscillations of wavelength of order 1/ε. These oscillations are called dispersive shock waves.

Lax and Levermore (1998), performing the zero-dispersion asymptotics for the inverse-scattering problem for the KdV equation, showed that as ε tends to zero, u(x, t; ε) tends uniformly to the smooth solution u(x, t) of (2) as long as t<t_c. For t>t_c, the solution u(x,t; ε) converges weakly in the oscillation region to a limit that is not a solution of conservation law (2).

The first example describing dispersive shock waves was proposed by Gurevich and Pitaevski (1973). Their description was rigorously proved by Venakides (1990) who derived the general form of the rapid oscillations. The oscillation zone is approximately described for a short time t>t_c by a modulated periodic wave solution of the KdV equation:

(3)

In the above formula, the term is the weak limit of u(x, t, ε) as ε→0, while the remaining term describes the rapid oscillations. The function Θ is 2π-periodic with zero average, and it can be expressed in terms of elliptic functions. The quantity α defined below and the phase depend on some functions u_i(x, t), i=1, 2, 3. The functions u₁(x, t)>u₂(x, t)>μ₃(x, t) solve the Whitham (1974) modulation equations

∂_tu_i(x, t)−λ_i(u₁, u₂, u₃)∂_xu_i(x, t)=0,

i=1, 2, 3,

(4)

where

(5)

(6)

and K(s) and E(s) are the complete elliptic integrals of the first and second kind with modulus s=(u₂−u₁)/(u₃−u₁).

The solution u₁(x, t)>u₂(x, t)>u₃(x, t) of the Whitham equations can be plotted in the (x, u) plane as branches of a multivalued function. The solutions of the Hopf equation and the Whitham equations are connected to one another as illustrated in Figure 1(a). The function u₂(x, t) can vary from u₃(x, t) to u₁(x, t). On the (x, t≥0) plane, the oscillation region is bounded on one side by the curve x⁻(t) where u₂(x, t)=u₁(x, t), and on the other side by the curve x⁺(t) where u₂(x, t)=μ₃(x, t) (see Figure 1(a)). For x⁻(t)<x<x⁺(t), the solution of (1) for small ε is approximately given by (3) while outside the interval [x⁻(t), x⁺(t)] is given by the solution u(x, t) of the Hopf equation (2). At edge x=x⁻(t) of the oscillation region, the amplitude of the oscillations vanishes and (3) goes to When x =x⁺(t), solution (3) goes to the one-soliton solution of the KdV equation. In general, the oscillation zone grows with time. For generic analytic initial data with a cubic inflection point, the growth in the (x, t) plane of the oscillation zone near the point of gradient catastrophe (x_c, t_c) is described, up to shifts and rescaling, by the semi-cubic law

x^±(t)=x_c±a^±(t−t_c)^3/2,

Figure 1. (a) The dashed line represents the formal solution of the Hopf equation; the continuous line represents the solution of the Whitham equations. The solution (u₁(x, t), u₂(x, t), u₃(x, t)) of the Whitham equations and the position of the boundaries x⁻(t) and x⁺(t) are to be determined from the conditions u(x⁻(t), t)=u₃(x⁻(t), t), u(x⁺(t), t)=u₁(x⁺(t), t), where u(x, t) is the solution of the Hopf equation, (b) The oscillations in the region x⁻(t₁)<x<x⁺(t₁).

where a^± are two positive numbers. A completely different behavior appears in the zero-diffusion case. The simplest equation that combines nonlinearity and diffusion is the Burgers equation

u_t−uu_x=εu_xx,

(7)

where ε>0. For smooth initial data u₀(x), the Burgers equation can be integrated through the Cole-Hopf transformation to

(8)

where

The behavior of the exact solution (8) as ε→0 can be obtained by observing that the dominant contributions to the integrals in (8) come from the neighborhood of the stationary points of G where

(9)

If (9) has only one stationary point, by the application of the steepest descent method, the asymptotic solution u(x, t; ε) as ε→0⁺ converges strongly to

u(x, t)=u₀(ξ), x=ξ−u₀(ξ)t.

(10)

The above is exactly the solution of the Cauchy problem for the Hopf equation (2). The stationary point ξ(x, t) of (9) becomes the characteristic variable in (10). For bump-like initial data the solution of the Hopf equation (2) has a point of gradient catastrophe (x_c, t_c). After the time t=t_c of gradient catastrophe, (10) gives a multivalued solution: the characteristics of the Hopf equation begin to intersect. For a typical initial pulse, there are usually three characteristics that intersect at each point of the multivalued region; that is, (9) has three solutions ξ₁(x, t), ξ₂(x, t), ξ₃(x, t) ξ₁>ξ₂>ξ₃. The dominant behavior of the solution of the Burgers equation will be given by the following contributions:

(11)

Let us suppose that for x_c<x<x_s and t>t_c, the function G(ξ₁(x, t)) is less than G(ξ₂(x, t)) and G(ξ₃(x, t)). Then the above expression for u in the limit ε→0 reads

(12)

while assuming that for x>x_s G(ξ₂(x, t))< G(ξ₁(x, t)), G(ξ₃(x, t)), we have

(13)

In each case, (10) applies to both ξ₁ and ξ₂. Therefore, the solution of the Burgers equation converges as ε→0⁺ to the solution of the Hopf equation (2) almost everywhere except at the points (x, t) where G(ξ_i(x, t))=G(ξ_j(x, t)), i≠j, i, j=1, 2, 3. For example, in the case treated above, the change over from ξ₁ to ξ₂ occurs when x=x_s where G(ξ₁)=G(ξ₂) Near x=x_s the solution of the Burgers equation as ε→0⁺ has a transition from (12) to (13) which is called a shock wave. In other words, the solution of the Burgers equation in the zero viscosity limit is given by two different branches of the solution of the Hopf equation joined by a jump at the point xs. The condition G(ξ₁)=G(ξ₂) reads

Because of (10), the above relation is equivalent to

(14)

which describes the shock wave. Since the shock occurs at x=x_s(t), t>t_c, we also have

x_s(t)=ξ₁−u₀(ξ₁)t, x_s(t)=ξ₂−u₀(ξ₂)t.

The above three equations determine the functions x_s(t), ξ₁(t), and ξ₂(t). ^The values of u(x, t) on the two sides of the shock are u⁻(x, t)=u₀(ξ₁(x, t)) and u⁺(x, t)=u₀(ξ₂(x, t)). The shock speed can be derived by taking the time derivative of the above two equations and reads

Comparison of the above relation with (14) shows that the modulus of the shock speed is equal to the average value of the characteristics velocity u₀(η) over the interval [ξ₁, ξ₂].

While the zero-dispersion limits have been studied only for integrable equations such as the KdV or the nonlinear Schrödinger equation, the zero-viscosity limits have been studied for the parabolic equation of the form

The scalar case in several spatial dimensions was investigated by Kruzhkov (1970). The two-component case in one spatial dimension has been studied by DiPerna (1983), while the n-component case in one spatial dimension has been investigated by Bressan (2002).

TAMARA GRAVA

Zero-Dispersion Limits

ZERO-DISPERSION LIMITS

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