Nonlinear-wave equation conventions

KdV formulas agree only after their field, space, time and spectral conventions agree. This reference translates the real KdV equation used in the soliton sequence. It covers the decaying line problem, constant backgrounds and changes of units; it does not assign a common convention to every nonlinear-wave equation.

The positive-pulse KdV equation

The model uses

$$u_t+6u u_x+u_{xxx}=0, \qquad x\in\mathbb R,$$

with a real, smooth field decaying sufficiently rapidly at both ends for the operation being performed. A travelling pulse is

$$u(x,t)=2\kappa^2\operatorname{sech}^2\!\left[\kappa(x-4\kappa^2t-x_0)\right], \qquad \kappa\gt0.$$

Its peak height is $2\kappa^2$, speed is $4\kappa^2$ and inverse width is $\kappa$. These three quantities are linked. The full width at half maximum is

$$\mathrm{FWHM}=\frac{2\operatorname{arcosh}\sqrt2}{\kappa}.$$

The factor of two here distinguishes a full width from the distance between the center and one half-height point. For $\kappa=1$, the height is $2$ and the speed is $4$; doubling $\kappa$ multiplies both by four and halves the width.

The travelling-wave lesson derives this formula from the differential equation and decay conditions. In references using $q=-u$, the pulse is a negative well and the equation is $q_t-6q q_x+q_{xxx}=0$. The sign conversion agrees with Grunert and Teschl 2009, eq. (1.1), author PDF p. 1.

Changing the field sign or the direction of time

Substitution, including every derivative, is safer than matching the name “KdV.” Starting from the equation above gives:

New field	Equation satisfied	Pulse in the new convention
$q(x,t)=-u(x,t)$	$q_t-6q q_x+q_{xxx}=0$	Negative well moving right
$v(x,t)=u(x,-t)$	$v_t-6v v_x-v_{xxx}=0$	Positive pulse moving left
$w(x,t)=u(-x,t)$	$w_t-6w w_x-w_{xxx}=0$	Positive pulse moving left
$r(x,t)=u(-x,-t)$	$r_t+6r r_x+r_{xxx}=0$	Positive pulse moving right

For example, $q_t=-u_t$, $q q_x=u u_x$ and $q_{xxx}=-u_{xxx}$. All three terms in the transformed equation therefore acquire the common factor $-1$. Changing the sign of the nonlinear term while keeping the same positive profile and the same direction of motion does not perform this conversion.

Coefficients, scales and a moving frame

Consider a physical or differently normalized field $v(x,t)$ satisfying

$$v_t+c_0v_x+a v v_x+b v_{xxx}=0, \qquad a b\ne0,$$

where $a,b,c_0$ are constant. For any positive length $\ell$, define

$$X=\frac{x-c_0t}{\ell},\qquad T=\frac{bt}{\ell^3},\qquad v(x,t)=\frac{6b}{a\ell^2}U(X,T).$$

The chain rule reduces the equation to $U_T+6U U_X+U_{XXX}=0$. The scale of time includes the sign of $b$: for $b\lt0$, increasing physical time corresponds to decreasing $T$. The amplitude scale can also be negative. Neither sign should be silently replaced by an absolute value.

A localized pulse with speed $c_0+s$ in the original coordinates is

$$v(x,t)=\frac{3s}{a}\operatorname{sech}^2\!\left[ \frac12\sqrt{\frac{s}{b}}\bigl(x-(c_0+s)t-x_0\bigr) \right],\qquad \frac{s}{b}\gt0.$$

To check it, integrate the travelling equation once with zero background:

$$bV''=sV-\frac a2V^2.$$

The coefficient of $V$ fixes the width and the coefficient of $V^2$ fixes the peak. The condition $s/b\gt0$ makes the exponential tails real. The sign of the elevation is the sign of $s/a$; a “positive soliton” is therefore convention dependent. The cases $a=0$ or $b=0$ are different equations and cannot be reached by dividing through this normalization.

When $x$ retains a length dimension in the normalized equation, consistency assigns $[u]=\mathrm{length}^{-2}$, $[t]=\mathrm{length}^{3}$ and $[\kappa]=\mathrm{length}^{-1}$. Fully dimensionless variables are also possible. A dimensional laboratory field needs the coefficients and conversion factors above before numerical speeds can be compared.

Constant backgrounds and periodic domains

Let $w$ solve the normalized zero-background equation. For a real constant $u_b$,

$$u(x,t)=u_b+w(x-6u_b t,t)$$

solves the same KdV equation. Indeed, the extra contribution $-6u_bw_x$ from $u_t$ cancels the contribution $6u_bw_x$ from the nonlinear term. A pulse on this background travels at $6u_b+4\kappa^2$.

This transformation changes the boundary conditions. For $u_b\ne0$, $\int_{\mathbb R}u\,dx$ is not a finite mass integral. Background-subtracted quantities must be defined before they are used, and the asymptotic spectral operator becomes $-\partial_x^2-u_b$. The continuum threshold shifts from $0$ to $-u_b$, and the pulse eigenvalue becomes $-u_b-\kappa^2$.

On a circle, the field and its needed derivatives must agree at the endpoints. A single $\operatorname{sech}^2$ pulse restricted to a long interval is not exactly periodic. Its tails can be small enough for a controlled numerical comparison, as tested in the convergence laboratory, but that does not turn the line scattering problem into the periodic spectral problem.

The auxiliary operator and its normalization

For the positive-pulse field use

$$L=-\partial_x^2-u, \qquad A=-4\partial_x^3-6u\partial_x-3u_x.$$

The last term in $A$ multiplies by $-3u_x$; it is not an instruction to differentiate everything to its right. The Library derivation verifies the differential-expression identity $L_t=[A,L]$. For a fixed bounded real decaying pulse, $L$ acts on $L^2(\mathbb R)$ with domain $H^2(\mathbb R)$. Compatibility of differential expressions alone is not a proof of every operator-domain assertion about a time-dependent evolution.

In the negative-well convention $q=-u$, write $L=-\partial_x^2+q$. A bound state has eigenvalue $\lambda=-\kappa^2$, whereas a scattering state has $\lambda=k^2$ with real $k$. The wave number $k$ of this auxiliary equation is not the propagation speed or frequency of the nonlinear pulse.

At a fixed time, normalize the right-end Jost solution by

$$f_+(k,x)e^{-ikx}\longrightarrow1 \quad\text{as }x\longrightarrow+\infty.$$

For a pulse centered at $a$, an explicit solution is

$$f_+(k,x)=e^{ikx} \frac{k+i\kappa\tanh[\kappa(x-a)]}{k+i\kappa}.$$

At $x\to-\infty$ its coefficient of $e^{ikx}$ is $(k-i\kappa)/(k+i\kappa)$, and there is no $e^{-ikx}$ term. For real $k\gt0$, a wave with unit amplitude incident from the left consequently has

$$R(k)=0,\qquad T(k)=\frac{k+i\kappa}{k-i\kappa}.$$

The reciprocal appears if one reports the coefficient in the right-normalized Jost solution instead of the transmission amplitude. State the normalization before comparing these formulas.

For the bound state, define a positive norming constant by

$$\gamma^{-1}=\left\|f_+(i\kappa,\cdot)\right\|_{L^2(\mathbb R)}, \qquad c=\gamma^2.$$

For this pulse, $f_+(i\kappa,x)=\tfrac12e^{-\kappa a}\operatorname{sech}[\kappa(x-a)]$, so $c=2\kappa e^{2\kappa a}$. As $a=x_0+4\kappa^2t$, this gives $c(t)=c(0)e^{8\kappa^3t}$ and $\gamma(t)=\gamma(0)e^{4\kappa^3t}$. The distinction between a norming constant and its square explains a common factor-of-two discrepancy in time exponents. The normalization and evolution agree with Grunert and Teschl 2009, § 2, eqs. (2.10) and (2.13), author PDF p. 5.

The reconstruction lesson uses $F(s,t)=c(t)e^{-\kappa s}$ and a kernel integrated from $x$ to $+\infty$. Its recovery formula is $u(x,t)=2\,dK(x,x,t)/dx$, with a total derivative along the diagonal. An author using the potential $q=-u$ has the opposite recovery sign. A kernel normalized at the other spatial end also requires changed limits and exponentials; a label such as “left” or “right” is insufficient to translate it.

Linear dispersion and nonlinear pulse speed

Linearizing about zero and inserting $e^{i(kx-\omega t)}$ gives

$$\omega=-k^3,\qquad v_{\mathrm{phase}}=-k^2,\qquad v_{\mathrm{group}}=-3k^2.$$

Thus small linear wave packets move left in this convention, while the positive nonlinear pulse moves right. This is consistent: the pulse balances dispersion with a finite nonlinear term. Linearization about $u_b$ instead gives $\omega=6u_bk-k^3$. These are physical field-wave dispersion formulas, separate from the auxiliary Schrödinger spectral parameter.

References

• Grunert, Katrin, and Gerald Teschl. “Long-time asymptotics for the Korteweg–de Vries equation via nonlinear steepest descent.” Mathematical Physics, Analysis and Geometry 12 (2009), 287–324. DOI. Open author PDF. Locators above use the author PDF’s printed pages, not the journal pagination.