Rewriting the Fermi-Pasta-Ulam problem as a Korteveg-de Vries equation

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Solving for u, we get: u(x',t') = f(x - u(x',t')t), where f is any differentiable function. This is different from equation (4) given in the instructions, which appears to be a typo.
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furryscience
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1. Homework Statement
Hello, I've been given an assignment of solving the Fermi-Pasta-Ulam problem
(involving a chain of oscillators, details found here: http://en.wikipedia.org/wiki/Fermi%E...93Ulam_problem ). I have a problem with a few details, as described below.

2. Homework Equations
I have to rewrite
q_tt=c^2*q_xx + e*q_x*q_xx + b*q_xxxx (1) (c, e and b are constants)
into
u_t' + u*u_x' + d^2*u_x'x'x' = 0 (2) (d is a constant)
using the substitution
x' = x - c_1*t, t'=c_2*t, q_x=c_3*u(x',t')
and a smart choice of constants c_i.

I also have to find the implicit solution of (2) without the
dissipative term, that is
u_t' + u*u_x' = 0 (3)

3. The Attempt at a Solution
To rewrite the derivatives in (1) with respect to x is pretty straightforward:
q_x=c_3*u, q_xx=c_3*u_x', q_xxxx=c_3*u_x'x'x'
if I'm not mistaken. What I don't get is how to rewrite q_tt, since both
x' and t' involve the variable t.

As for solving (3), I really have no clue. The instructions claim that the implicit
solution is of the form
u(x',t') = f(x - u(x',t')t) (4)
where f is any differentiable function, but I'm pretty sure this is a typo, since (4)
involves a function u referring to itself.

The instructor isn't responding to my email queries about this, so any help is most
appreciated.
 
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  • #2
4. The Attempt at a Solution To rewrite equation (1) into equation (2), we need to use the substitution given: x' = x - c_1*t, t'=c_2*t, q_x=c_3*u(x',t'). Substituting in equation (1) gives us: q_tt=c^2*q_xx + e*q_x*q_xx + b*q_xxxx = c^2*c_3*u_x' + e*c_3*u*u_x' + b*c_3*u_x'x'x' = c_3*(c^2*u_x' + e*u*u_x' + b*u_x'x'x'). Now, since t'=c_2*t, q_tt can be rewritten as q_t't' = c_2*q_tt. Thus, q_t't' = c_2*c_3*(c^2*u_x' + e*u*u_x' + b*u_x'x'x'). Substituting this into equation (2) gives us: u_t' + u*u_x' + d^2*u_x'x'x' = 0 = c_2*c_3*(c^2*u_x' + e*u*u_x' + b*u_x'x'x') + u*u_x' + d^2*u_x'x'x' = 0. We can then solve for c_2 and c_3 to get: c_2 = 1/(c^2 + e*u + b*d^2), c_3 = 1/(d^2). To solve equation (3) implicitly, we can rearrange it to get: u_t' = -u*u_x'. We can then integrate both sides with respect to t' to get: ∫u_t'dt' = ∫
 

1. What is the Fermi-Pasta-Ulam problem and why is it important?

The Fermi-Pasta-Ulam (FPU) problem is a mathematical and physical paradox that arose in the mid-20th century. It involves the behavior of a one-dimensional lattice of particles, connected by springs, when subjected to a certain type of external force. The problem is important because it helped to shed light on the limitations of traditional methods of statistical mechanics and led to the development of new methods for studying nonlinear systems.

2. How does the Korteveg-de Vries equation relate to the FPU problem?

The Korteveg-de Vries (KdV) equation is a nonlinear partial differential equation that describes the evolution of a certain type of wave known as a soliton. It was originally derived to describe waves in shallow water, but it has since been found to have applications in various other areas, including the FPU problem. In particular, the KdV equation can be used to model the behavior of the FPU system in certain regimes, providing insights into the problem that were previously unavailable.

3. What are the advantages of rewriting the FPU problem as a KdV equation?

One of the main advantages of rewriting the FPU problem as a KdV equation is that it allows for a simpler and more elegant mathematical description of the system. The KdV equation is a well-understood and extensively studied equation, so using it to model the FPU problem can provide deeper insights into the behavior of the system. Additionally, the KdV equation can be solved using a variety of analytical and numerical methods, making it easier to study the FPU problem in different scenarios.

4. What are the limitations of using the KdV equation to model the FPU problem?

While the KdV equation has been found to be useful in modeling certain aspects of the FPU problem, it is not a perfect representation of the system. The KdV equation is a one-dimensional model, whereas the FPU problem is inherently multidimensional. This means that the KdV equation may not accurately capture all of the dynamics and behaviors of the FPU system, especially in more complex scenarios.

5. How has the understanding of the FPU problem evolved with the use of the KdV equation?

The use of the KdV equation to model the FPU problem has greatly expanded our understanding of the system. It has allowed for the identification of new phenomena and behaviors that were previously unknown, and has also provided a more comprehensive understanding of the transitions between different states in the system. Furthermore, the insights gained from using the KdV equation have led to advancements in other areas of physics, such as the study of turbulence and chaos.

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