Solution of a differential equation?

In summary, the conversation discusses the concept of "a solution" to a differential equation, specifically the Korteweg-de Vries (KdV) equation. The speakers mention that for an initial condition of a Sech^2 pulse, the solution is a traveling wave known as a solitary wave. They also mention that using other initial conditions, such as a Sech^3 pulse, would still result in a solution of the KdV equation, but it may not have an analytical or exact solution. The speakers also note that soliton solutions are not the only type of solution to the KdV equation, but they are surprising because they have a Sech^2-profile.
  • #1
hanson
319
0
Hi all. I am getting confused about the notion of "a solution" to a differential equation.
Let's consider the KdV equation, ut+uux+uxxx=0.
So, if the initial condition is a Sech^2 pulse, then the solution would be a traveling wave solution and this is the well known solitary wave.

So, what if I arbitrarily use another initial condition? say, a Sech^3 pulse or anything? This initial profile shall be also governed by the KdV equation and the evolution of this strange initial profile shall be still a solution of the KdV equation, right? just that we cannot find the analytical or exact solution?

Say if I use an excellent numerical scheme to see the evolution of this Sech^3, theoretically, the evolution generated using this arbitrary initial condition shall be called a solution of the KdV, right?
 
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  • #2
Certainly.

Solitons are not the only solution to the KdV equation.

What IS surprising is that soliton solutions DO exist..:smile:

The soliton solution is gained by hypothesizing the existence of a non-dispersive solution of KdV; calculations then reveal that:

Insofar as such solutions exist, they need to have a Sech^2-profile.
 
  • #3


Yes, you are correct. A solution to a differential equation is any function that satisfies the equation and its initial conditions. In the case of the KdV equation, the initial condition of a Sech^2 pulse produces a traveling wave solution. However, any other initial condition, such as a Sech^3 pulse, will also produce a solution to the KdV equation, but it may not be a traveling wave. As you mentioned, it may not be possible to find an analytical or exact solution for this new initial condition, but it can still be considered a solution to the equation. Using a numerical scheme to approximate the evolution of this new initial condition is a valid way to find a solution to the KdV equation. So, in summary, a solution to a differential equation can take on many different forms, depending on the initial conditions, but as long as it satisfies the equation, it can be considered a solution.
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It describes the relationship between a function and its rate of change.

2. What is a solution of a differential equation?

A solution of a differential equation is a function that satisfies the equation. It is a function that, when substituted into the equation, makes it a true statement.

3. How do you solve a differential equation?

There is no one set method for solving a differential equation. It depends on the type of equation and its properties. Generally, solutions can be found through analytical methods, such as separation of variables or using integrating factors, or through numerical methods, such as Euler's method or Runge-Kutta methods.

4. What is the difference between an ordinary and a partial differential equation?

An ordinary differential equation involves a single independent variable, while a partial differential equation involves multiple independent variables. This means that the solution of a partial differential equation is a function of several variables, while the solution of an ordinary differential equation is a function of a single variable.

5. How are differential equations used in real life?

Differential equations are used to model and understand a wide range of physical, biological, and social phenomena. They are used in fields such as physics, engineering, economics, and biology to predict and analyze the behavior of systems that change over time, such as population growth, chemical reactions, and electrical circuits.

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