Explain a KdV soliton through two constructions

Can two very different constructions describe exactly the same wave? In this project you obtain a KdV pulse first from a nonlinear ordinary differential equation and then from a linear integral equation with one exponential kernel. Matching the two identifies what the bound-state eigenvalue determines, what its norming data add, and what a numerical evolution can verify.

Required background. Be able to derive the travelling-wave profile, calculate its conserved integrals, and solve the rank-one reconstruction. The convergence laboratory supplies the executable comparison; you can complete the analytic part without Python.

One positive pulse, two mathematical descriptions

Work with real smooth rapidly decaying fields on the line and

$$u_t+6uu_x+u_{xxx}=0.$$

The target solution has two parameters: inverse width $\kappa\gt0$ and initial center $x_0\in\mathbb R$. Its complete formula is

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

Your task is to obtain this formula by both routes, identify their common data and explain which parts of the argument apply only to this reflectionless one-bound-state family. The Library derivation gives the differential-expression and scattering calculations behind the construction.

Route one: reduce the nonlinear wave equation

Set $u(x,t)=U(\xi)$ with $\xi=x-vt-x_0$. Decay of $U$ and its derivatives makes the integration constant vanish:

$$-vU'+6UU'+U'''=0 \quad\Longrightarrow\quad U''=vU-3U^2.$$

Multiplying by $2U'$ and integrating again gives

$$(U')^2=vU^2-2U^3.$$

A positive pulse has a nonzero turning point $U=v/2$, so $v\gt0$. Write $v=4\kappa^2$. Direct substitution then yields $U=2\kappa^2\operatorname{sech}^2(\kappa\xi)$, with the center absorbing the translation constant. This computation fixes amplitude, speed and width together; they cannot be chosen independently.

For a direct PDE check, use

$$U''=4\kappa^2U-3U^2,\qquad U'''=4\kappa^2U'-6UU',\qquad u_t=-4\kappa^2U'.$$

Their sum in the KdV equation is zero. This is an exact check of the claimed pulse, not a numerical residual. Decaying travelling waves and their speed–amplitude relation are treated by Lax 1968, report pp. 2–3, equations (1.5)–(1.8), PDF; his field is $w=6u$.

Route two: reconstruct from one exponential

Use the Schrödinger operator $L=-\partial_x^2-u$ and reflectionless data with one bound-state eigenvalue $-\kappa^2$. For $y\geq x$ at each fixed time, the right-end reconstruction convention is

$$\begin{aligned} K(x,y,t)+F(x+y,t) &+\int_x^\infty K(x,z,t)F(z+y,t)\,dz=0,\\ F(s,t)&=c(t)e^{-\kappa s},\qquad c(t)\gt0,\\ u(x,t)&=2\frac{d}{dx}K(x,x,t). \end{aligned}$$

The derivative in the last line is the total derivative along the diagonal. The sign is tied to $L=-\partial_x^2-u$. In Aktosun 2009, § VIII, equations (8.1)–(8.3), the Schrödinger potential is $q=-u$, and its reconstruction has the opposite sign.

Try $K(x,y,t)=-A(x,t)e^{-\kappa y}$. Since the remaining integral is elementary,

$$A=\frac{c\,e^{-\kappa x}} {1+\dfrac{c}{2\kappa}e^{-2\kappa x}}, \qquad K(x,x,t)=\frac{-c\,e^{-2\kappa x}} {1+\dfrac{c}{2\kappa}e^{-2\kappa x}}.$$

Define the positive scalar $r=c\,e^{-2\kappa x}/(2\kappa)$. Differentiating $K(x,x,t)=-2\kappa r/(1+r)$ gives

$$u(x,t)=\frac{8\kappa^2r}{(1+r)^2}.$$

The norming coefficient evolves as $c(t)=c_0e^{8\kappa^3t}$ in this convention; see Aktosun 2009, § VII, equation (7.9). Match it to the center by choosing

$$c_0=2\kappa e^{2\kappa x_0}.$$

Then $r=e^{-2\kappa(x-4\kappa^2t-x_0)}$. The identity

$$\operatorname{sech}^2 z=\frac{4e^{-2z}}{(1+e^{-2z})^2}$$

recovers exactly the travelling wave from the first route.

Match the parameters and the information

For the computational example, $\kappa=1$ and $x_0=-2$. The initial spectral data are the bound-state eigenvalue $-1$, zero reflection coefficient, and $c_0=2e^{-4}$. At $t=1$, $c=2e^4$ and the center is $2$.

Quantity	Travelling-wave description	Scattering description
Amplitude and width	$2\kappa^2$ and inverse width $\kappa$	Bound-state eigenvalue $-\kappa^2$
Speed	$4\kappa^2$	Exponential growth rate $8\kappa^3$ of $c(t)$
Initial location	$x_0$	$c_0=2\kappa e^{2\kappa x_0}$
Radiation in this example	Absent from the exact profile	Reflection coefficient zero

The eigenvalue alone does not record position. Translating the pulse changes $c_0$ while leaving $-\kappa^2$ unchanged. Likewise, the three integrals

$$M=4\kappa,\qquad P=\frac{16}{3}\kappa^3,\qquad E=\frac{32}{5}\kappa^5$$

contain no $x_0$. Recover these by substituting $z=\kappa(x-4\kappa^2t-x_0)$ into their defining integrals. A list of invariants can be correct while missing information needed to reconstruct a particular solution.

Compare exact reasoning and executable evidence

The Python experiment, with its inputs, saved results, requirements and notes, performs checks that use different representations:

• It evaluates the hyperbolic and rational reconstruction formulas for $\kappa=0.5,1,1.3$. Their largest discrepancy, normalized by the pulse height, is $3.94\times10^{-16}$.
• It finite-differences the total diagonal derivative of $K$ and compares $2d_xK(x,x)$ with the pulse; the largest normalized discrepancy is $2.26\times10^{-12}$.
• It quadratures the Marchenko integral directly instead of substituting its closed value; the largest residual normalized by $|F(x+y)|$ is $1.74\times10^{-13}$.
• It quadratures $M,P,E$ on an interval extending twenty widths each side of the center; the largest relative discrepancy is $1.15\times10^{-14}$.
• It evolves the PDE by a separate periodic Fourier method and performs independent time, space and domain refinements. The resolved reference has relative final $L^2$ error $1.80\times10^{-10}$.

The laboratory gives the run commands and error definitions. These computations check the implemented formulas and the declared finite-time approximation. Algebra establishes the equality of the two pulse constructions; numerical agreement neither proves a general inverse-scattering theorem nor treats an arbitrary initial profile.

For a useful contrast, translate the same pulse at the wrong speed. Its three line integrals remain correct, while its PDE residual and profile error are large. Include this failed example when explaining why a conservation plot alone cannot validate an evolution.

Exercises

Guided: recover location from norming data

Let $\kappa=1/2$ and $c_0=e^3$. Find the initial center, amplitude, speed and center at $t=2$. Obtain the same final center from $c(2)$.

Hint

Solve $c_0=2\kappa e^{2\kappa x_0}$ for $x_0$. The center at any time is $(2\kappa)^{-1}\log[c(t)/(2\kappa)]$.

Solution

Since $2\kappa=1$, $x_0=3$. The amplitude is $1/2$, the speed is $1$, and the center at $t=2$ is $5$. The growth rate is $8\kappa^3=1$, so $c(2)=e^5$; its logarithm again gives center $5$. The eigenvalue remains $-1/4$ throughout.

Independent: find the missing derivative

A calculation uses $2\partial_xK(x,y)|_{y=x}$ instead of $2d_xK(x,x)$ to reconstruct the field. Show that the two operations differ, and calculate what the incorrect formula gives at the pulse center.

Hint

Use $d_xK(x,x)=K_x(x,x)+K_y(x,x)$. At the center, $r=1$ and $K(x,x)=-\kappa$.

Solution

The kernel has $K_y=-\kappa K$, so at the center $K_y=\kappa^2$. The correct diagonal derivative is $u/2=\kappa^2$, hence $K_x=0$ there. The incorrect reconstruction gives zero where the true pulse reaches $2\kappa^2$. Both derivative labels can look plausible; specifying the diagonal path prevents the error.

Transfer: change the norming evolution

Keep the same eigenvalue and positive $c_0$, but set $c(t)=c_0e^{\alpha t}$ with arbitrary real $\alpha$. Determine the reconstructed pulse speed and its KdV residual. Which value of $\alpha$ makes it a KdV solution, and why can the three conserved integrals not detect a wrong choice?

Hint

Read the center from $r$, then substitute a travelling profile with speed $v=\alpha/(2\kappa)$ into the differential equation.

Solution

The reconstructed profile remains $2\kappa^2\operatorname{sech}^2[\kappa(x-vt-x_0)]$, with $v=\alpha/(2\kappa)$. Its residual is $(4\kappa^2-v)u_x$. Thus $\alpha=8\kappa^3$ is required. Every other choice merely translates the same shape at a wrong speed, so integrals depending only on that shape still remain constant. Spectral data at one time must be paired with the correct time evolution.

Explain the result in your own calculation

Produce a short derivation containing both constructions, the mapping $(\kappa,x_0)\leftrightarrow(-\kappa^2,c_0)$, one direct PDE check, and an interpretation of the independent refinements. Your explanation should identify the role of decay, the diagonal derivative, zero reflection and the finite periodic approximation. Then solve the transfer task without looking at its solution. That changed-setting calculation tests whether you understand the relation between reconstruction and evolution.

Continue with the two-soliton collision when you can match both one-pulse constructions. Its interaction term shows why adding two separately translated solutions does not solve the nonlinear equation.

References

• Aktosun, Tuncay. “Inverse Scattering Transform and the Theory of Solitons.” In Encyclopedia of Complexity and Systems Science, Springer, 2009, pp. 4960–4971. DOI. Author version arXiv:0905.4746v1, HTML. Section and equation locators refer to that version.
• Lax, Peter D. “Integrals of Nonlinear Equations of Evolution and Solitary Waves.” Communications on Pure and Applied Mathematics 21(5), 467–490 (1968). DOI. Open report PDF. The cited page numbers are the report’s printed labels.