In mathematics, the Novikov–Veselov equation (or Veselov–Novikov equation) is a natural (2+1)-dimensional analogue of the Korteweg–de Vries (KdV) equation. Unlike another (2+1)-dimensional analogue of KdV, the Kadomtsev–Petviashvili equation, it is integrable via the inverse scattering transform for the 2-dimensional stationary Schrödinger equation. Similarly, the Korteweg–de Vries equation is integrable via the inverse scattering transform for the 1-dimensional Schrödinger equation. The equation is named after S.P. Novikov and A.P. Veselov who published it in .
The Novikov–Veselov equation is most commonly written as
where
v=v(x1,x2,t),
w=w(x1,x2,t)
\Re
\partial=
| 1 | |
| 2 |
(
| \partial | |
| x1 |
-i
| \partial | |
| x2 |
), \partial=
| 1 | |
| 2 |
(
| \partial | |
| x1 |
+i
| \partial | |
| x2 |
).
The function
v
w
v
E
L\psi=E\psi, L=-\Delta+v(x,t), \Delta=
| 2 | |
| \partial | |
| x1 |
+
| 2. | |
| \partial | |
| x2 |
When the functions
v
w
v=v(x1,t)
w=w(x1,t)
E\to\pminfty
The inverse scattering transform method for solving nonlinear partial differential equations (PDEs) begins with the discovery of C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, who demonstrated that the Korteweg–de Vries equation can be integrated via the inverse scattering problem for the 1-dimensional stationary Schrödinger equation. The algebraic nature of this discovery was revealed by Lax who showed that the Korteweg–de Vries equation can be written in the following operator form (the so-called Lax pair):
where
L=-
| 2 | |
| \partial | |
| x |
+v(x,t)
A=
| 3 | |
| \partial | |
| x |
+
| 3 | |
| 4 |
(v(x,t)\partialx+\partialxv(x,t))
[ ⋅ , ⋅ ]
\begin{align} &L\psi=λ\psi,\\ &\psi=A\psi \end{align}
for all values of
λ
Afterwards, a representation of the form was found for many other physically interesting nonlinear equations, like the Kadomtsev–Petviashvili equation, sine-Gordon equation, nonlinear Schrödinger equation and others. This led to an extensive development of the theory of inverse scattering transform for integrating nonlinear partial differential equations.
When trying to generalize representation to two dimensions, one obtains that it holds only for trivial cases (operators
L
A
B
L
or, equivalently, to search for the condition of compatibility of the equations
\begin{align} &L\psi=λ\psi,\\ &\psi=A\psi \end{align}
at one fixed value of parameter
λ
Representation for the 2-dimensional Schrödinger operator
L
The dispersionless version of the Novikov–Veselov equation was derived in a model of nonlinear geometrical optics .
The behavior of solutions to the Novikov–Veselov equation depends essentially on the regularity of the scattering data for this solution. If the scattering data are regular, then the solution vanishes uniformly with time. If the scattering data have singularities, then the solution may develop solitons. For example, the scattering data of the Grinevich–Zakharov soliton solutions of the Novikov–Veselov equation have singular points.
Solitons are traditionally a key object of study in the theory of nonlinear integrable equations. The solitons of the Novikov–Veselov equation at positive energy are transparent potentials, similarly to the one-dimensional case (in which solitons are reflectionless potentials). However, unlike the one-dimensional case where there exist well-known exponentially decaying solitons, the Novikov–Veselov equation (at least at non-zero energy) does not possess exponentially localized solitons .