What can be an interesting problem of differential equations involving the definition of the Gudermannian function? [closed]

Question

In this post I denote the Gudermannian function as $$\operatorname{gd}(x)=\int_0^x\frac{dt}{\cosh t}$$ and its inverse as $\operatorname{gd}^{-1}(x)$, please see if you need it the definitions, alternative definitions and identities, history and meaning in mathematics from the Wikipedia with title Gudermannian function, or the article Gudermannian from the encyclopedia MathWorld.

Few days ago I wondered about what can be interesting problems of differential equation involving these special functions , I say ordinary differential equations and also partial differential equations (where potentially one can provide initial values or boundary conditions, to get a well-posed problems).

My firt attempt was a pendulum-like equation, because I know that the pendulum equation has a good mathematical/physical content. Thus one can try to solve $$y''(x)+y'(x)+\operatorname{gd}(x)=0.\tag{1}$$ And it can be integrated easily, see the code solve y''+y'+gd(x)=0 in Wolfram Alpha online calculator.

And using the method of variation of parameters one can to compute, easily (I used Wolfram Alpha to compute two integrals), the general solution of $$y''(x)+y(x)+\operatorname{gd}(x)=0.\tag{2}$$

Question. I wondered if it is possible to set some interesting problem involving differential equations and the Gudermannian function or its inverse. If possible with a good mathematical content (I say that maybe is feasible to propose an interesting problem invoking the special meaning of some of these functions). If it is feasible, feel free to add your answer adding your proposal of problem and/or remarks about it (if it is required in your problem you can to work with special value conditions and special domains). Many thanks.

I don't add any tag related to geometry, but I know that partial differential equations are closely-related to geometry of manifolds.

Answer 1

The OP asks for a "a problem or two" involving the Gudermannian function. Here are three applications from physics:

• An Alternative Lattice Field Theory Formulation Inspired by Lattice Supersymmetry (identifies the lattice derivative operator as the inverse Gudermannian function of the lattice momentum)

• The Rise of Solitons in Sine-Gordon Field Theory: From Jacobi Amplitude to Gudermannian Function (expresses the soliton solution of the classical sine-Gordon field theory in terms of the Gudermannian function)

• A diagrammatic expansion of the Casimir energy in multiple reflections: theory and applications (relates the Casimir energy from vacuum fluctuations to the Gudermannian function)