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Hello,
I am reading the book, The Quantum Theory of Fields II by Weinberg.
In page 426 of this book (about soliton, domain wall stuffs), we have Eq(23.1.5) as the solution that minimizes Eq(23.1.3).
The paragraph below Eq(23.1.5), the author said "The advantage of the derivation based on the formula (23.1.3) is that it shows immediately that the solution (23.1.5) is stable against small perturbations that maintain the flatness of the boundary. ... By adding a term [itex]\frac{1}{2} (\frac{\textrm{d}\phi}{\textrm{d}y})^2 +\frac{1}{2} (\frac{\textrm{d}\phi}{\textrm{d}z})^2[/itex] in the integrand of Eq (23.1.2), we can see that this solution is also stable against any perturbation ..."
Here, I don't understand why they have to be stable against small perturbation in both cases. I guess I don't have any good idea about the stability of differential equation or action. Could you guys explain how we can show they are stable explicitly?
Thank you for your help.
I am reading the book, The Quantum Theory of Fields II by Weinberg.
In page 426 of this book (about soliton, domain wall stuffs), we have Eq(23.1.5) as the solution that minimizes Eq(23.1.3).
The paragraph below Eq(23.1.5), the author said "The advantage of the derivation based on the formula (23.1.3) is that it shows immediately that the solution (23.1.5) is stable against small perturbations that maintain the flatness of the boundary. ... By adding a term [itex]\frac{1}{2} (\frac{\textrm{d}\phi}{\textrm{d}y})^2 +\frac{1}{2} (\frac{\textrm{d}\phi}{\textrm{d}z})^2[/itex] in the integrand of Eq (23.1.2), we can see that this solution is also stable against any perturbation ..."
Here, I don't understand why they have to be stable against small perturbation in both cases. I guess I don't have any good idea about the stability of differential equation or action. Could you guys explain how we can show they are stable explicitly?
Thank you for your help.