Solving a Higher-Order PDE: Traveling Wave & Phase Portrait

In summary, we are considering the PDE ut + 6u3ux + uxxx = 0, a higher-order variant of the KdV. We are asked to derive the 3rd-order ODE for the traveling wave solution u = f(x-ct), which is -cf'(x-ct) + 6f(x-ct)3f'(x-ct) + f'''(x-ct) = 0. Next, we reduce the order of this ODE to obtain the expression for the polynomial g(f), where g(f) = (f')2/2. The phase portrait for real solutions can be sketched by considering the maximum number of real roots of g(f), which is
  • #1
squenshl
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Homework Statement


Consider the PDE ut + 6u3ux + uxxx = 0
which may be thought of as a higher-order variant of the KdV.
a) Assume a traveling wave u = f(x-ct) and derive the 3rd-order ODE for that solution.
b) Reduce the order of this ODE and obtain the expression for the polynomial g(f), where g(f) = (f')2/2
c) Sketch the (f,f') phase portrait for real solutions, assuming that g(f) has the maximum number of real roots.

Homework Equations





The Attempt at a Solution


a) I let u = f(x-ct) and got -cf'(x-ct) + 6f(x-ct)3f'(x-ct) + f'''(x-ct) = 0, is this the 3rd-order ODE required.
b) I got (f')2/2 = g(f) where g(f) = -f(x-ct)5/4 + cf(x-ct)2/6 + A1f(x-ct) + A2
c) I know that this has 5 solutions (5th-order polynomial), do I write it in the form (f')2/2 = g(f) = 1/4*(f-F1)(f-F2)(f-F3)(f-F4)(F5-f) and if so how does this look on phase portrait.
 
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  • #2
I just need to do c). Any ideas?
 

1. What is a higher-order PDE?

A higher-order PDE (partial differential equation) is a type of mathematical equation that involves multiple variables and their partial derivatives. It describes a relationship between these variables and how they change over time or space.

2. What is a traveling wave?

A traveling wave is a specific type of solution to a higher-order PDE where a disturbance or pattern moves through a medium without changing its shape. This type of solution is often used to model physical phenomena such as sound waves or electromagnetic waves.

3. How do you solve a higher-order PDE?

Solving a higher-order PDE involves finding a function that satisfies the equation and any given boundary conditions. This can be done using various mathematical techniques such as separation of variables, Fourier series, or numerical methods.

4. What is a phase portrait?

A phase portrait is a visual representation of the behavior of a system over time. In the context of higher-order PDEs, it is a graph that shows the relationship between different variables and how they change over time or space.

5. Why is solving a higher-order PDE important?

Higher-order PDEs are used to model and understand complex physical phenomena in fields such as physics, engineering, and economics. Solving these equations allows us to make predictions and gain insights into the behavior of these systems, leading to advancements in various industries and fields of study.

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