Talk:Three-wave equation

Intro is too technical

For the context, I've added the tag "too technical for most readers to understand" primarily because I feel the introduction could benefit from some higher-level context with less jargon on what three-wave-equations represent (for people without much knowledge on this area of mathematics). This could be, e.g., by explaining terms like "resonant interaction" and giving some more context on why they have broad applicability in science. From the intro, it's also not clear where the "Three/triad" comes from in the name. 7804j (talk) 20:57, 29 May 2025 (UTC)

Hmm. Well, this is physics/engineering, not mathematics; I expanded the intro to include shallow-water waves. If you found this article because of the recent twitter posts about China doing submarine hunting from microphones placed in littoral waters (i.e. preparing for the invasion of Taiwan), then I don't think this belongs in the lede to this article. It's too political. Too narrow a focus. So instead, I just said "acoustic coupling in littoral zone" and one of the existing refs covers this already. There is no coupling in deep water, BTW; this is listed further down in the article.

The only simple explanation for "resonant interaction" I can think of is to say "when things vibrate, it is called a resonance, and when vibrating things interact, it is called a resonant interaction." But this is baby-talk, I don't see that it would be appropriate. The reader is assumed to know the basics of differential equations.

The section "informal introduction" explains how to count to three.

The simplest informal explanation I can think of is that zero means "nothing happens"/solution is a constant, one derivative == trivially integrable aka the "anti-derivative", two derivatives == ordinary wave, therefore three is the smallest number above two that is non-trivial (aka non-linear). This is already covered in any class on differential eqns, as part of the definition of what it means for something to be a linear equation.

The jargony way of counting is zero==vacuum, one==blue sky catastrophe, two==propagator and three==nonlinear system. So I'm not sure how else to explain it. 67.198.37.16 (talk) 22:36, 31 May 2025 (UTC)

I didn't discover this article based on the Twitter posts, I was just doing some general cleanup and tagging articles that I felt could be improved across topics.

You are right that maybe this particular topic is too complex to be fundamentally simplified without making it feel like "baby talk", so maybe all it needs is a bit of additional context on usage/importance.

I have drafted below what I think could be an improved intro with more context. I have used LLM (Claude and ChatGPT) for this, so it needs to be reviewed by someone familiar with the topic. Could you take a look and let me know what you think? If you feel the text is good, feel free to publish it, otherwise you can ignore it and I'll just drop the issue tag I added.

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In nonlinear systems, the three-wave equations, also known as the three-wave resonant interaction equations or triad resonances, are a system of three completely integrable nonlinear partial differential equations that describe the resonant interaction of small-amplitude waves in various nonlinear media. These equations model fundamental wave phenomena across a range of physical systems, including water waves, capillary waves, plasma oscillations, and wave interactions in nonlinear optics.

The three-wave equations represent a fundamental deterministic model underlying wave turbulence theory and serve as a paradigmatic example of resonant interactions in dispersive media. They arise when three waves with wave vectors k₁, k₂, and k₃ satisfy both the resonance condition (commonly expressed as k₁ = k₂ + k₃) and the frequency matching condition ω₁ = ω₂ + ω₃, where ωᵢ denotes the angular frequency of each wave component. In some formulations, the resonance condition is equivalently written as k₁ + k₂ + k₃ = 0 and ω₁ + ω₂ + ω₃ = 0, depending on the convention adopted. These resonant triad interactions enable efficient energy transfer between the three wave modes.

Because they provide a direct and tractable example of resonant wave interactions, have broad applicability across the physical sciences, and possess the remarkable property of complete integrability, the three-wave equations have been extensively studied since the 1970s. Their integrability allows for exact analytical solutions via methods such as the inverse scattering transform, making them a cornerstone in the mathematical theory of integrable systems and in the study of soliton-like phenomena. The equations have also played a crucial role in the development of Hamiltonian formulations of wave dynamics and in advancing the understanding of energy cascades in weakly nonlinear wave systems. 7804j (talk) 11:41, 1 June 2025 (UTC)

Actually, that looks great! I'm merging it in now. 67.198.37.16 (talk) 17:15, 3 June 2025 (UTC)