Inverse scattering transformation

The inverse scattering transformation (English Inverse Scattering Transformation , abbreviated to IST ) is a method for the exact solution of initial value problems of certain nonlinear partial differential equations (evolution equations) like the Korteweg-de-Vries equation  (KdV), which describe solitons .

history

The inverse scattering transformation was published in 1967 and 1974 by Robert Miura , Martin Kruskal , Clifford Gardner and John Greene (GGKM). Substantial contributions also made

• 1968 Peter Lax , who provided an operator formulation of the GGKM method (with Lax pairs and the Lax equation )
• Early 1970s Vladimir Sakharov and Alexei Shabat (application to the non-linear Schrödinger equation , IST in several spatial dimensions)
• 1974 Mark J. Ablowitz , David J. Kaup , Alan C. Newell and Harvey Segur (summary of the IST as Fourier analysis for nonlinear problems), whereby they also coined the term "inverse scattering transformation".

application

Other equations that could be solved exactly with the inverse scattering transformation are

• the equation of the Toda grid ( Hermann Flaschka )
• the Kadomtsev-Petviashvili equation (KP equation; Zakharov, Shabat)
• the Sinus Gordon equation
• the Benjamin-Ono equation (an integro-differential equation that describes certain water waves ).

The examples originally considered were one-dimensional in space, but there is also IST for multi-dimensional problems such as the KP equation; the Benjamin-Ono equation occupies a position between one-dimensional and multi-dimensional IST schemes.

formulation

The nonlinear evolution equation (NL) is given by:

${\displaystyle u_{t}=K(u)}$

for the function with the initial value (it is also assumed that the (soliton) solutions sought drop off sufficiently for large distances). Subscripts are partial derivatives. It is important that the right-hand side nonlinearities are functions of (and its spatial derivatives). ${\displaystyle u(t,x)}$ ${\displaystyle u(t=0,x)=u_{0}(x)}$ ${\displaystyle u}$

In the KdV equation, for example

${\displaystyle u_{t}=6uu_{x}-u_{xxx}=K(u)}$

In the case of the IST, a linear ordinary differential equation assigned to the NL (here abbreviated as LODE) is considered, which depends on a time-independent spectral parameter and in which the sought solution of the NL enters as a potential . The LODE describes a scattering problem with scattering data given by the spectral parameter (the solution consists of a finite number of bound states and the continuous spectrum ), the reflection coefficient and the normalization constants. Determine the scattering data solution of the scattering problem for the scattering data developed by to (using a linear ordinary differential equation) and then solves the inverse scattering problem (with Marchenko method or Marchenko integral equation, sometimes in addition to Israel Gelfand and Boris Levitan named ), that is, the reconstruction from . This is then the NL solution that they are looking for. ${\displaystyle \lambda }$ ${\displaystyle u(t,x)}$ ${\displaystyle S(\lambda ,t=0)}$ ${\displaystyle t=0}$ ${\displaystyle t=0}$ ${\displaystyle t}$ ${\displaystyle u(x,t)}$ ${\displaystyle S(\lambda ,t)}$

In the case of the KdV equation, the associated LODE is the Schrödinger equation:

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

To get the belonging to the NL LODE, the Lax method is mostly used where it comes down to it, the NL as Lax pair with two linear operators , to formulate: ${\displaystyle L(t)}$ ${\displaystyle B(t)}$

${\displaystyle L\psi =\lambda \psi }$ (Equation 1)

and

${\displaystyle {\frac {dL}{dt}}=[B,L]}$ (the Lax equation)

${\displaystyle [B,L]=BL-LB}$is the commutator of the two operators. It describes the time development of : ${\displaystyle B}$ ${\displaystyle \psi }$

${\displaystyle {\frac {d\psi (t)}{dt}}=B(t)\psi (t)}$ (Equation 2)

The second equation in the Lax pair, the Lax equation, ensures that the spectral parameter and the whole spectrum in general is independent of time (an important point for the application of the IST), the problem is isospectral in time. The Lax equation also corresponds to the original NL after inserting B, L. ${\displaystyle \lambda }$

The Lax pair for the KdV equation is:

${\displaystyle L=-\partial _{x}^{2}+u\,}$

${\displaystyle B=-4\partial _{x}^{3}+3(u\partial _{x}+\partial _{x}(u\cdot ))\,}$

${\displaystyle L}$is of the Sturm-Liouville type and is self-adjoint , B is crooked-adjoint. Insertion into the Lax equation gives the KdV equation.

The ACTUAL then consists of the solution of equation 1 for the scatter data at the time , the time development of the scatter data with equation 2 at the time and the inverse transformation from the scatter data at the time to the potential at the time . ${\displaystyle t=0}$ ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle u}$ ${\displaystyle t}$

In the KdV equation, the solitons result from the finite number of bound states of the scattering problem (the continuous spectrum provides radiation that decays over time). The existence of an infinite number of conservation quantities for the KdV can be derived from the ACTUAL or from the fact that the spectrum of the LODE is independent of time , which ensures the exact integrability .

All previously known nonlinear evolution equations that can be solved with the IST allow a reduction to a group of nonlinear ordinary differential equations , the Painlevé equations . This observation is used as a test for the application of the IST (Painlevé test), and there is a presumption by M. Ablowitz, A. Ramani and H. Segur that this is always the case.

literature

• M. Ablowitz, H. Segur: Solitons and the Inverse Scattering Transform, SIAM 1981
• M. Ablowitz, P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.

Web links

• Roger Grimshaw, Korteweg-de-Vries equation, Encycl. Nonlinear Science, pdf

Individual evidence

1. ↑ Gardner, Greene, Kruskal, Miura, Method for Solving the Korteweg-deVries Equation , Physical Review Letters, Volume 19, 1967, pp. 1095-1097
2. Jump up ↑ Gardner, Greene, Kruskal, Miura, Korteweg-de Vries equation and generalizations VI. Methods for exact solution , Communications on Pure and applied mathematics. Volume 27, 1974, pp. 97-133
3. ^ Lax, Integrals of nonlinear equations of evolutions, Comm. Pure Appl. Math., Vol. 21, 1968, pp. 467-490
4. ^ VE Zakharov, AB Shabat: Exact Theory of Two-Dimensional Self-Focusing and One-Dimensional Self-Modulation of Waves in Nonlinear Media , Soviet Phys. JETP, Vol. 34, 1972, pp. 62-69.
5. ↑ Zakharov, Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem , Funct. Anal. Appl., Vol. 8, 1974, pp. 226-235
6. ↑ Ablowitz, Kaup, Newell, Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., Vol. 53, 1974, pp. 249-315
7. ↑ Ablowitz, Science Citation Classics, 1982, PDF file
8. ↑ MJ Ablowitz, DJ Kaup, AC Newell, H. Segur, Method for Solving the Sine-Gordon Equation, Phys. Rev. Lett., Vol. 30, 1973, pp. 1262-1264
9. ↑ Similar to the elementary quantum mechanical scattering theory
10. ↑ To prove this, differentiate equation (1) according to time and use equation (2). The Lax equation is obtained as a condition for the disappearance of the time derivative of . ${\displaystyle \lambda }$
11. ↑ This leads to, as Lax showed in 1968, that the time evolution operator (solution of equation 2 with ) is unitary and , with the consequence that the whole spectrum of is time invariant. ${\displaystyle U(t)}$ ${\displaystyle U(0)=1}$ ${\displaystyle L(0)=U^{-1}(t)L(t)U(t)}$ ${\displaystyle L}$
12. ↑ M. Ablowitz, A. Ramani, H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of P-type, 2 parts, J. Math. Phys., Volume 21, 1980, pp. 715-721, 1006 -1015