Non-linear Wave Equation Equation Formula Latex Code and Summary

Non-linear Wave Equation

Tags: #physics #non-linear wave

Equation

$$\frac{d^2x}{dt^2}-\varepsilon\omega_0(1-\beta x^2)\frac{dx}{dt}+\omega_0^2x=0 \\ \frac{d}{dt}\left\{\frac{1}{2}\left(\frac{dx}{dt}\right)^2+\frac{1}{2} \omega_0^2x^2\right\}= \varepsilon\omega_0(1-\beta x^2)\left(\frac{dx}{dt}\right)^2 \\ \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}-\underbrace{au\frac{\partial u}{\partial x}}_{\rm non-lin}+ \underbrace{b^2\frac{\partial ^3u}{\partial x^3}}_{\rm dispersive}=0 \\ u(x-ct)=\frac{-d}{\cosh^2(e(x-ct))}$$

Latex Code

                                 \frac{d^2x}{dt^2}-\varepsilon\omega_0(1-\beta x^2)\frac{dx}{dt}+\omega_0^2x=0 \\
            \frac{d}{dt}\left\{\frac{1}{2}\left(\frac{dx}{dt}\right)^2+\frac{1}{2} \omega_0^2x^2\right\}= \varepsilon\omega_0(1-\beta x^2)\left(\frac{dx}{dt}\right)^2 \\
            \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}-\underbrace{au\frac{\partial u}{\partial x}}_{\rm non-lin}+ \underbrace{b^2\frac{\partial ^3u}{\partial x^3}}_{\rm dispersive}=0 \\
            u(x-ct)=\frac{-d}{\cosh^2(e(x-ct))}

Have Fun

Let's Vote for the Most Difficult Equation!

Introduction

$\frac{d^2x}{dt^2}-\varepsilon\omega_0(1-\beta x^2)\frac{dx}{dt}+\omega_0^2x=0 \\ \frac{d}{dt}\left\{\frac{1}{2}\left(\frac{dx}{dt}\right)^2+\frac{1}{2} \omega_0^2x^2\right\}= \varepsilon\omega_0(1-\beta x^2)\left(\frac{dx}{dt}\right)^2 \\ \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}-\underbrace{au\frac{\partial u}{\partial x}}_{\rm non-lin}+ \underbrace{b^2\frac{\partial ^3u}{\partial x^3}}_{\rm dispersive}=0 \\ u(x-ct)=\frac{-d}{\cosh^2(e(x-ct))}$

Latex Code

            \frac{d^2x}{dt^2}-\varepsilon\omega_0(1-\beta x^2)\frac{dx}{dt}+\omega_0^2x=0 \\
            \frac{d}{dt}\left\{\frac{1}{2}\left(\frac{dx}{dt}\right)^2+\frac{1}{2} \omega_0^2x^2\right\}= \varepsilon\omega_0(1-\beta x^2)\left(\frac{dx}{dt}\right)^2 \\
            \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}-\underbrace{au\frac{\partial u}{\partial x}}_{\rm non-lin}+ \underbrace{b^2\frac{\partial ^3u}{\partial x^3}}_{\rm dispersive}=0 \\
            u(x-ct)=\frac{-d}{\cosh^2(e(x-ct))}

Explanation

Discussion

Comment to Make Wishes Come True

Leave your wishes (e.g. Passing Exams) in the comments and earn as many upvotes as possible to make your wishes come true

Heather Adams

I'm geared up to pass this exam.

2023-09-25 00:00
Reply

Florence Stone reply to Heather Adams

Gooood Luck, Man!

2023-10-13 00:00:00.0
Reply
Vincent Alexander

I'm just going to put it out there: I want to pass this test.

2023-12-11 00:00
Reply

Raymond Sanchez reply to Vincent Alexander

Best Wishes.

2024-01-08 00:00:00.0
Reply
Chris Burton

Please universe, let me pass this exam.

2023-03-16 00:00
Reply

Denise Richardson reply to Chris Burton

Gooood Luck, Man!

2023-04-06 00:00:00.0
Reply