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Talk:Linear and nonlinear waves - Scholarpedia

Talk:Linear and nonlinear waves

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    Reviewer A

    This review is written by Prof. Andrei Polyanin together with Dr. Alexei Zhurov.

    We hope our remarks will help to improve the article.

    Section "Linear waves"

    Equation (6) should be placed ahead of the others as it is the simplest. We believe the principle "from simple to complex" is best suited for this sort of article.

    Section "Nonlinear waves"

    The Korteweg-de Vries equation

    This subsection should not go in first. The Korteweg-de Vries equation is quite complex and is not a typical representative of a wave equation. It should be moved towards the end of the "Nonlinear waves" section.

    Closed form PDE solution methods

    1. A few very important and effective methods have been missed. These are:

    • The Hodograph transformation. It allows the exact linearization of some nonlinear wave equations and hyperbolic systems of equations in gas dynamics (e.g., see the "Handbook of Nonlinear PDEs" by A.D. Polyanin and V. F. Zaitsev, pp. 686-687).
    • Group analysis methods (the classical and nonclassical methods for symmetry reductions).
    • Method of generalized separation of variables.
    • Differential constraints method (the last two methods are outlined, for example, "Handbook of Nonlinear PDEs" by Polyanin and Zaitsev).

    2. The "Exp-function method", "Factorization", and "Tanh methods" are not classified as individual methods. In fact, they are techniques of searching for traveling-wave solutions of special form; they only allow finding particular solutions to some ordinary differential equation of special form. The items "Exp-function method", "Factorization", and "Tanh method" should better be treated as special cases of the "Traveling wave solutions".

    3. "Nonlinear transformations" cannot be called a method. The main problem is to specify transformations that may result in exact solutions; there are many different nonlinear transformations and they are not linked by a common idea. It makes sense to keep the item "Nonlinear transformations" only if its special cases are: "Point transformations" (then the "Hodograph transformation" should be included here as a special case), "Contact transformations", and "Backlund transformations" (in which case, this item should be removed from the top of the list).

    Nonlinear wave equation solutions

    1. The abbreviations "Apps" and "Soln" should be avoided—use "Applications" and "Solution(s)" instead.

    2. Following the principle "from simple to complex", we recommend that equations appear in order of increasing their complexity (according to the order of the highest derivative). Then the authors should start with the Hopf equation \[ u_t+uu_x=0\quad (\hbox{or}\quad u_t+f(u)u_x=0), \] which is the simplest nonlinear first-order wave equation. It is often used as a model equation of gas dynamics and admits general solution in closed parametric form (e.g., see the "Handbook of First-Order PDEs" by A.D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Taylor and Francis, 2002, pp. 275-277).

    3. Boussinesq equation: the left-hand side \(u(x,t)=\) is missing.

    4. The last equation, "Wave equation with exponential non-linearity", which is not typical, should be replaced with "Nonlinear wave equation of general form": \[ u_t=[f(u)u_x]_x \] It can be linearized in the general case; see the "Handbook of Nonlinear PDEs" by Polyanin and Zaitsev, pp. 252-255, which also contains exact solutions of this equation. "Wave equation with exponential non-linearity" can be mentioned as a special case of the general equation with \(f(u)=ae^{\lambda u}\).

    Section "Numerical solution methods"

    Initial conditions and boundary conditions

    The diffusion equation (46) should be replaced by the wave equation \(u_{tt}=au_{xx}\), as the article is about wave processes rather than diffusion.

    Response from Authors to Reviewer A comments

    Thank you for your constructive comments to which we have made appropriate changes.

    Re[2]: Reviewer A

    We have made some corrections in the article (see Closed form PDE solution methods) and the list of references — please have a look. Also please make sure the references style is consistent (e.g., author's initials appear on the left and on the right, etc.).

    Re[3]: Reviewer A


    Also, an acknowledgement added to reviewers Prof. Andrei Polyanin and Dr. Alexei Zhurov for their positive and constructive comments.

    User 3 (User:) (Assistant Editor User:) : History of Fourier series

    Fourier was the first to suggest that essentially any function could be represented by a Fourier series. However, his conjecture was not widely accepted at the time. Specifically, Lagrange argued that the Fourier series of a square wave would not converge. It turns out Lagrange had anticipated the Gibbs phenomenon, but falsely concluded that the series would diverge where the derivative was discontinuous. Euler, Laplace, Monge, Biot, Poisson, and Lacroix were other outspoken skeptics. The controversy raged for decades.

    Dirichlet in 1828, was the first to prove that all functions satisfying Dirichlet's conditions (i.e. "non-pathological piecewise continuous") could be represented by a convergent Fourier series.

    Re[5]: Assistant Editor User:

    Thank you for this clarification. I have amended the main article but kept the changes to a minimum.

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