Stability of Travelling Wave Solutions of the Burgers Equation

Graduation researchPublic notes

May 2026 — present (personal project / graduation research)

Overview

A maths site that brings together my graduation research in the Department of Mathematics, School of Education, Waseda University, as a paper translation, lecture notes and a graduation thesis. The subject is a classic of nonlinear partial differential equation theory:

K. Nishihara, A Note on the Stability of Travelling Wave Solutions of Burgers' Equation, Japan J. Appl. Math. 2 (1985), 27–35.

Its main theme is to show rigorously that travelling wave solutions (shock layers) of the viscous Burgers equation are stable in time, including how fast perturbations decay. Alongside a full translation of the original paper, the site contains a graduation thesis that carries the argument through to its conclusion without skipping any steps, and translations of related classical papers such as Hopf (1950), Il'in–Oleinik (1960) and Gelfand (1959), all arranged so that they can be read side by side.

The mathematics

Subject — the viscous Burgers equation

We study an equation that models shock waves in fluids, combining a nonlinear advection term with a viscous diffusion term.

v_t + \left(\tfrac{v^2}{2}\right)_x = \varepsilon\, v_{xx} \qquad (\varepsilon > 0)

For initial data that approach different values $v_- > v_+$ at the two ends, we ask what shape the solution eventually settles into.

Explicit form of the travelling wave solution (shock layer)

Considering a solution $V(x - st)$ that keeps its shape while moving at a constant speed $s$ , the partial differential equation reduces to an ordinary differential equation, which can be integrated to give an exact $\tanh$ -shaped profile.

s = \frac{v_+ + v_-}{2}

V(\xi) = \frac{v_+ + v_-}{2} - \frac{v_- - v_+}{2}\,\tanh\!\left(\frac{v_- - v_+}{4\varepsilon}\,(\xi - x_0)\right)

Stability — linearisation via the Hopf–Cole transformation

The equation satisfied by the perturbation $u = v - V$ around the travelling wave is nonlinear, but applying the Hopf–Cole transformation $u = -2\,(\log\varphi)_\xi$ turns it into a linear heat equation. Once it is linear, the long-time behaviour of the solution can be tracked directly through an explicit formula (the heat kernel) — this is the heart of the argument.

Conclusion — an exponential decay estimate

Under the condition that the perturbation has zero total mass ( $\int u_0\, d\xi = 0$ ), the solution converges to the travelling wave at an exponential rate, and moreover the decay rate is obtained as an explicit value.

v(x,t) - V(x - st) = O\!\left(\exp\!\left(-\frac{(v_- - v_+)^2}{16\varepsilon}\,t\right)\right) \qquad (t \to \infty)

What makes this research interesting

The earlier work of Il'in–Oleinik (1960) used the maximum principle to show that solutions “converge exponentially to the travelling wave”. However, its decay rate is only a qualitative, existential statement — “there exists some positive constant” — and no concrete value is given.

By contrast, Nishihara (1985) restricts attention to the viscous Burgers equation and, in exchange, goes through the explicit formula of the Hopf–Cole transformation to write the decay rate down as the concrete value $\dfrac{(v_- - v_+)^2}{16\varepsilon}$ . Its significance lies in sharpening the qualitative information “it decays exponentially” into the quantitative result “it decays exponentially at this rate”.

On this site, the derivation of the explicit travelling wave (1.8) (integrating the ODE, partial fractions, and the $\tanh$ identity), the derivation of the perturbation equation and the determination of $x_0$ , the linearisation by the Hopf transformation, and the asymptotic estimate using the heat kernel are all laid out step by step, filling in the gaps that the original paper leaves out.

Site contents

• Paper (Japanese translation) — a complete translated commentary on the original Nishihara (1985) paper (all 5 sections)
• Graduation thesis — the full version, derived through to the conclusion with nothing omitted (a PDF typeset in LaTeX is also provided)
• Lecture notes — overview and problem setting / derivation of the perturbation equation / linearisation by the Hopf transformation, explained session by session
• Translations of references — Hopf (1950), Il'in–Oleinik (1960), Gelfand (1959) and other classics cited by the paper, with translations and the original PDFs

Tech stack and implementation notes

TypeScript

Vite

KaTeXmarkdown-itWeb Crypto API (AES-GCM)

• Equations typeset with KaTeX and written in markdown-it, so paper-grade mathematics renders without breaking
• The thesis, lectures and supplementary notes are public. Translations and private notes are encrypted with AES-GCM before the build and decrypted with the Web Crypto API when opened, preserving existing passwords and saved logins
• Development loads plain-text Markdown directly; production loads encrypted content only for restricted materials

Access

The thesis, lecture notes, detailed proofs and visualisations are available without logging in. Paper translations and private notes remain restricted, with the same password as before.

Open the maths notes