Basic concept Q ,non-linear PDE , kdv

In summary, the conversation revolves around solving the Korteweg-de Vries equation (Ut+6UUx+Uxxx=0) and the need for an initial condition, U(x,t=0), in order to solve the equation. The question arises whether the solution is only valid for a small interval of t or for all values of t. The expert clarifies that while having an initial condition would be sufficient, it is not necessary for solving the equation.
  • #1
binbagsss
1,265
11
Ut+6UUx+Uxxx=0 [kdv eq]

Why to solve this do you need U(x,t=0)?
Why is it a initial value problem?

This should probably be really obvious. I think I've forgotten some basic background stuff, just starting my course in solitons...

Thanks for your help.​
 
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  • #2
Sorry I'm not sure how , if you can, edit your threads on the new layout..

Some thoughts:
- If we have U(x,t=0) and solve for U(x,t) , is our solution only valid for t in a small interval around t=0? Or can we solve for all t from this?
 
  • #3
bump.
 
  • #4
Your question doesn't make sense. You don't "have to have" a value for U(x, 0). While that, together with values for U(a, t) and U(a, b), would be sufficient since the partial differential equation is of first order in t and second order in x, it is NOT necessary.
 

Related to Basic concept Q ,non-linear PDE , kdv

1. What is the basic concept of a non-linear PDE?

A non-linear PDE (partial differential equation) is an equation that contains both the unknown function and its derivatives in a non-linear form. This means that the equation cannot be solved using simple algebraic techniques, and instead requires more advanced mathematical methods.

2. What is the KdV equation and how is it related to non-linear PDEs?

The KdV (Korteweg-de Vries) equation is a non-linear PDE that describes the behavior of certain types of waves, such as shallow water waves. It is one of the most well-known and studied non-linear PDEs, and has been used to model a variety of physical phenomena.

3. How do non-linear PDEs differ from linear PDEs?

In a linear PDE, the unknown function and its derivatives are only present in a linear form, meaning that the equation can be solved using simple algebraic techniques. Non-linear PDEs are more complex, as they involve nonlinear terms that cannot be simplified in the same way.

4. What are some real-world applications of non-linear PDEs?

Non-linear PDEs have a wide range of applications in physics, engineering, and other fields. They are commonly used to model fluid dynamics, heat transfer, and electromagnetic fields. They have also been applied to biological systems, such as modeling the spread of diseases.

5. Can non-linear PDEs be solved analytically?

In most cases, non-linear PDEs cannot be solved analytically (i.e. using exact mathematical formulas). Instead, numerical methods are used to approximate solutions. However, there are some special cases where non-linear PDEs can be solved analytically, such as the KdV equation.

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