Inverse scattering transform

The inverse scattering transform arose from studying solitary waves. J.S. Russell described a "wave of translation" or "solitary wave" occurring in shallow water.^[5] First J.V. Boussinesq and later D. Korteweg and G. deVries discovered the Korteweg-deVries (KdV) equation, a nonlinear partial differential equation describing these waves.^[5] Later, N. Zabusky and M. Kruskal, using numerical methods for investigating the Fermi–Pasta–Ulam–Tsingou problem, found that solitary waves had the elastic properties of colliding particles; the waves' initial and ultimate amplitudes and velocities remained unchanged after wave collisions.^[5] These particle-like waves are called solitons and arise in nonlinear equations because of a weak balance between dispersive and nonlinear effects.^[5]

Gardner, Greene, Kruskal and Miura introduced the inverse scattering transform for solving the Korteweg–de Vries equation.^[6] Lax, Ablowitz, Kaup, Newell, and Segur generalized this approach which led to solving other nonlinear equations including the nonlinear Schrödinger equation, sine-Gordon equation, modified Korteweg–De Vries equation, Kadomtsev–Petviashvili equation, the Ishimori equation, Toda lattice equation, and the Dym equation.^[5][7][8] This approach has also been applied to different types of nonlinear equations including differential-difference, partial difference, multidimensional equations and fractional integrable nonlinear systems.^[5]

The independent variables are a spatial variable ${\displaystyle x}$ and a time variable ${\displaystyle t}$. Subscripts or differential operators (${\textstyle \partial _{x},\partial _{t}}$) indicate differentiation. The function ${\displaystyle u(x,t)}$ is a solution of a nonlinear partial differential equation, ${\textstyle u_{t}+N(u)=0}$, with initial condition (value) ${\textstyle u(x,0)}$.^[2]: 72

The differential equation's solution meets the integrability and Fadeev conditions:^[2]: 40

Integrability condition:${\displaystyle \int _{-\infty }^{\infty }\ |u(x)|\ dx\ <\infty }$

Fadeev condition: ${\displaystyle \int _{-\infty }^{\infty }\ (1+|x|)|u(x)|\ dx\ <\infty }$

The Lax differential operators, ${\textstyle L}$ and ${\textstyle M}$, are linear ordinary differential operators with coefficients that may contain the function ${\textstyle u(x,t)}$ or its derivatives. The self-adjoint operator ${\textstyle L}$ has a time derivative ${\textstyle L_{t}}$ and generates a eigenvalue (spectral) equation with eigenfunctions ${\textstyle \psi }$ and time-constant eigenvalues (spectral parameters) ${\textstyle \lambda }$.^{[1]: 4963 [2]: 98}

${\displaystyle L(\psi )=\lambda \psi ,\ }$ and ${\textstyle \ L_{t}(\psi ){\overset {def}{=}}(L(\psi ))_{t}-L(\psi _{t})}$

The operator ${\textstyle M}$ describes how the eigenfunctions evolve over time, and generates a new eigenfunction ${\textstyle {\widetilde {\psi }}}$ of operator ${\textstyle L}$ from eigenfunction ${\textstyle \psi }$ of ${\textstyle L}$.^[1]: 4963

${\displaystyle {\widetilde {\psi }}=\psi _{t}-M(\psi )\ }$

The Lax operators combine to form a multiplicative operator, not a differential operator, of the eigenfunctions ${\textstyle \psi }$.^[1]: 4963

${\displaystyle (L_{t}+LM-ML)\psi =0}$

The Lax operators are chosen to make the multiplicative operator equal to the nonlinear differential equation.^[1]: 4963

${\displaystyle L_{t}+LM-ML=u_{t}+N(u)=0}$

The AKNS differential operators, developed by Ablowitz, Kaup, Newell, and Segur, are an alternative to the Lax differential operators and achieve a similar result.^{[1]: 4964 [9][10]}

The direct scattering transform generates initial scattering data; this may include the reflection coefficients, transmission coefficient, eigenvalue data, and normalization constants of the eigenfunction solutions for this differential equation.^{[2]: 39–48}

${\displaystyle L(\psi )=\lambda \psi }$

The equations describing how scattering data evolves over time occur as solutions to a 1st order linear ordinary differential equation with respect to time. Using varying approaches, this first order linear differential equation may arise from the linear differential operators (Lax pair, AKNS pair), a combination of the linear differential operators and the nonlinear differential equation, or through additional substitution, integration or differentiation operations. Spatially asymptotic equations (${\textstyle x\to \pm \infty }$) simplify solving these differential equations.^{[1]: 4967–4968 [2]: 68–72 [6]}

The Marchenko equation combines the scattering data into a linear Fredholm integral equation. The solution to this integral equation leads to the solution, u(x,t), of the nonlinear differential equation.^{[2]: 48–57}

The nonlinear differential Korteweg–De Vries equation is ^[11]: 4

${\displaystyle u_{t}-6uu_{x}+u_{xxx}=0}$

The Lax operators are:^{[2]: 97–102}

${\displaystyle L=-\partial _{x}^{2}+u(x,t)\ }$ and ${\textstyle \ M=-4\partial _{x}^{3}+6u\partial _{x}+3u_{x}}$

The multiplicative operator is:

${\displaystyle L_{t}+LM-ML=u_{t}-6uu_{x}+u_{xxx}=0}$

The solutions to this differential equation

${\textstyle L(\psi )=-\psi _{xx}+u(x,0)\psi =\lambda \psi }$

may include scattering solutions with a continuous range of eigenvalues (continuous spectrum) and bound-state solutions with discrete eigenvalues (discrete spectrum). The scattering data includes transmission coefficients ${\textstyle T(k,0)}$, left reflection coefficient ${\textstyle R_{L}(k,0)}$, right reflection coefficient ${\textstyle R_{R}(k,0)}$, discrete eigenvalues ${\textstyle -\kappa _{1}^{2},\ldots ,-\kappa _{N}^{2}}$, and left and right bound-state normalization (norming) constants.^[1]: 4960

${\displaystyle c(0)_{Lj}=\left(\int _{-\infty }^{\infty }\ \psi _{L}^{2}(ik_{j},x,0)\ dx\right)^{-1/2}\ j=1,\dots ,N}$

${\displaystyle c(0)_{Rj}=\left(\int _{-\infty }^{\infty }\ \psi _{R}^{2}(ik_{j},x,0)\ dx\right)^{-1/2}\ j=1,\dots ,N}$

The spatially asymptotic left ${\textstyle \psi _{L}(k,x,t)}$ and right ${\textstyle \psi _{R}(k,x,t)}$ Jost functions simplify this step.^{[1]: 4965–4966}

${\displaystyle {\begin{aligned}\psi _{L}(x,k,t)&=e^{ikx}+o(1),\ x\to +\infty \\\psi _{L}(x,k,t)&={\frac {e^{ikx}}{T(k,t)}}+{\frac {R_{L}(k,t)e^{-ikx}}{T(k,t)}}+o(1),\ x\to -\infty \\\psi _{R}(x,k,t)&={\frac {e^{-ikx}}{T(k,t)}}+{\frac {R_{R}(k,t)e^{ikx}}{T(k,t)}}+o(1),\ x\to +\infty \\\psi _{R}(x,k,t)&=e^{-ikx}+o(1),\ x\to -\infty \\\end{aligned}}}$

The dependency constants ${\textstyle \gamma _{j}(t)}$ relate the right and left Jost functions and right and left normalization constants.^{[1]: 4965–4966}

${\displaystyle \gamma _{j}(t)={\frac {\psi _{L}(x,i\kappa _{j},t)}{\psi _{R}(x,i\kappa _{j},t)}}=(-1)^{N-j}{\frac {c_{Rj}(t)}{c_{Lj}(t)}}}$

The Lax ${\textstyle M}$ differential operator generates an eigenfunction which can be expressed as a time-dependent linear combination of other eigenfunctions.^[1]: 4967

${\displaystyle \partial _{t}\psi _{L}(k,x,t)-M\psi _{L}(x,k,t)=a_{L}(k,t)\psi _{L}(x,k,t)+b_{L}(k,t)\psi _{R}(x,k,t)}$

${\displaystyle \partial _{t}\psi _{R}(k,x,t)-M\psi _{R}(x,k,t)=a_{R}(k,t)\psi _{L}(x,k,t)+b_{R}(k,t)\psi _{R}(x,k,t)}$

The solutions to these differential equations, determined using scattering and bound-state spatially asymptotic Jost functions, indicate a time-constant transmission coefficient ${\textstyle T(k,t)}$, but time-dependent reflection coefficients and normalization coefficients.^{[1]: 4967–4968}

${\displaystyle {\begin{aligned}R_{L}(k,t)&=R_{L}(k,0)e^{-i8k^{3}t}\\R_{R}(k,t)&=R_{R}(k,0)e^{+i8k^{3}t}\\c_{Lj}(t)&=c_{Lj}(0)e^{+4\kappa _{j}^{3}t},\ j=1,\ldots ,N\\c_{Rj}(t)&=c_{Rj}(0)e^{-4\kappa _{j}^{3}t},\ j=1,\ldots ,N\end{aligned}}}$

The Marchenko kernel is ${\textstyle F(x,t)}$.^{[1]: 4968–4969}

${\displaystyle F(x,t){\overset {def}{=}}{\frac {1}{2\pi }}\int _{-\infty }^{\infty }R_{R}(k,t)e^{ikx}\ dk+\sum _{j=1}^{N}c(t)_{Lj}^{2}e^{-\kappa _{j}x}}$

The Marchenko integral equation is a linear integral equation solved for ${\textstyle K(x,y,t)}$.^{[1]: 4968–4969}

${\displaystyle K(x,z,t)+F(x+z,t)+\int _{x}^{\infty }K(x,y,t)F(y+z,t)\ dy=0}$

The solution to the Marchenko equation, ${\textstyle K(x,y,t)}$, generates the solution ${\textstyle u(x,t)}$ to the nonlinear partial differential equation.^[1]: 4969

${\displaystyle u(x,t)=-2{\frac {\partial K(x,x,t)}{\partial x}}}$

• Korteweg–de Vries equation
• nonlinear Schrödinger equation
• Camassa-Holm equation
• Sine-Gordon equation
• Toda lattice
• Ishimori equation
• Dym equation

• Quantum inverse scattering method
• Integrable system