Are Solitons the Only Type of Waves in Wave Equations?

[Klaus_Hoffmann]

Solitons=Only waves??

given any wave equation (linear or not) could we always find a solution.

$$\Psi (x,t)= f(x-ct)$$

is this the so-called only wave or soliton ?? , and what would be the shape of f(r) r=x-ct given a certain wave equation ?.. does the Schröedinguer equation admit such solutions or only if this SE is non-linear ??

 

[olgranpappy]

given any wave equation (linear or not) could we always find a solution.

$$\Psi (x,t)= f(x-ct)$$

no.

67890

 

[Claude Bile]

I would say that the term "wave-equation" implies the existence of wave-like solutions (i.e. functions of the form f(x-c.t)). Olgranpappy, I'm a little confused by your post - I would think that you would at least attempt to qualify such a statement! (either that or I am missing something clever).

Functions of the form f(x-c.t) are not solitons, though they must satisfy this condition, as they are propagating waves. A solition is a wave that does not change its shape as it propagates, hence not only must it be of the form f(x-c.t), its profile in the y and z directions cannot vary with x.

The Schrödinger equation being a linear equation does not have any soliton solutions for finite waves (i.e. not including the artificial examples of infinite plane waves and so forth). The non-linear Schordinger equation does have soliton solutions. As I understand it, nonlinearity is crucial for soliton formation and propagation.

Claude.

 

[olgranpappy]

I would say that the term "wave-equation" implies the existence of wave-like solutions (i.e. functions of the form f(x-c.t)). Olgranpappy, I'm a little confused by your post - I would think that you would at least attempt to qualify such a statement! (either that or I am missing something clever).

What?...

The form $$\psi(x,t)=f(x-ct)$$ necessarily implys a relation between the space and time derivatives. Namely,

$$\frac{\partial \psi}{\partial x}=\frac{-1}{c}\frac{\partial \psi}{\partial t}$$

Why would such a thing be true for a solution of a general differential equation? For example, I could explicitly forbid such solutions in the differential equation.

 

[Claude Bile]

What?...

The form $$\psi(x,t)=f(x-ct)$$ necessarily implys a relation between the space and time derivatives. Namely,

$$\frac{\partial \psi}{\partial x}=\frac{-1}{c}\frac{\partial \psi}{\partial t}$$

Why would such a thing be true for a solution of a general differential equation? For example, I could explicitly forbid such solutions in the differential equation.

Ah, I see where you are coming from now - though I think the OP was talking specifically about wave-equations, not DEs in general.

Claude.

 

[olgranpappy]

oh, he did say "wave equation." But then what exactly is the difference between a "wave equation" and some other type of differential equation? Are "wave equations" hyperbolic? I don't get it.

 

[Claude Bile]

Yes, this is a little confusing for me too - I thought that the term wave-equation is defined on the basis of having solutions of the form f(x-vt) because such functions describe a propagating disturbance, though this is more of a physicists classification than one a mathematician would use.

Claude.

 

[olgranpappy]

Yes, this is a little confusing for me too - I thought that the term wave-equation is defined on the basis of having solutions of the form f(x-vt)...

In that case, the answer to the original question is: "Yes, by definition."