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Nonlinear Schrödinger equation (NLS)

Nonlinear Schrödinger equation
SolvableExactly solvable
The Cauchy problemSolvable by inverse scattering transform
First derived bySchrödinger, 1926



i a t =3k 2 a x 2 +ka | a | 2

Structure of this equation is similar to the structure of the Schrödinger equation with potential | a | 2 , that's why it is called nonlinear Schrödinger equation.


Traveling wave

Let us look for the simplest solutions of the nonlinear Schrödinger equation

i a t = 2 a x 2 +γa | a | 2

in the form

a( x,t )= e ipxiχt V( z ),z=x c 0 t,

where p,χ and c 0 are constants and V(z) is a new unknown function.

For function V(z) we get

V z 2 =A+α V 2 γ 2 V 4 .

Solutions of the obtained equation can be expressed via the Jacobi elliptic functions.

Group soliton

In limiting case of A=0 and under a condition α>0, γ>0 we have solution V(z) in form of a solitary wave

V( z )= 2α γ 1 ch( α ( x c 0 t ) ) .

Corresponding solution of the nonlinear Schrödinger equation is

u( x,t )= e i( pxχt ) 2α γ 1 ch( α ( x c 0 t ) ) ,


p= c 0 2 ,χ=α c 0 2 4 .

Solitons of the nonlinear Schrödinger equation are called group solitons or somitems bending solitons.

Lax pair

The Lax pair for the nonlinear Schrödinger equation

a t =i a xx 2i | a | 2 a


Ψ x = P ^ Ψ=( iλ a a * iλ )Ψ,

Ψ t = Q ^ Ψ=( i | q | 2 +2i λ 2 i q x +2qλ i q x * +2 q * λ i | q | 2 2i λ 2 )Ψ,


Ψ=( ψ 1 (x,t) ψ 2 (x,t) ).

This Lax pair was first given by Ablowitz, Kaup, Newell, and Segur (Ablowitz, et al., 1974).

The Cauchy problem

The Cauchy problem for the nonlinear Schrödinger equation can be solved using the inverse scattering transform.

Applications and connections

The nonlinear Schrödinger equation is used as a model in fields as wide as:
  • fiber optics
  • water waves
and more.

See also


  1. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H. The inverse scattering transform-Fourier analysis for nonlinear problems // Stud. Appl. Math., 1974. 53:4. Pp.249–315.
  2. Ablowitz M.J., Segur H. Solitons and the inverse scattering transform // Society for Industrial and Applied Mathematics (SIAM Studies in Applied Mathematics, No. 4), Philadelphia, PA, 1981. — 434 p.
  3. Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.C. Solitons and nonlinear wave equations // Academic Press, New York, 1982. — 630 p.
  4. Filippov A.T. The versatile soliton // Birkhäuser, Boston, 2000. — 261 p.
  5. Kudryashov N.A. Analytical theory of nonlinear differential equations (in Russian) // 2nd ed., Institute of Computer Investigation, Moscow-Izhevsk, 2004. — 360 p.
  6. Kudryashov N.A. Methods of nonlinear mathematical physics (in Russian) // Moscow Engineering Physics Institute, Moscow, 2008. — 352 p.
  7. Lonngren K., Scott A. (Editors) Solitons in action // Academic Press, New York, 1978. — 300 p.
  8. Maymistov A.I. Optical solitons (in Russian) // Soros Educational Journal, 1999. 11. Pp.97-102.
  9. Newell A.C. Solitons in mathematics and physics // Society for Industrial and Applied Mathematics, Pennsylvania, 1985. — 244 p.
  10. Schrödinger E. An undulatory theory of the mechanics of atoms and molecules // Phys. Rev., 1926. 28:6. Pp.1049–1070. DOI:10.1103/PhysRev.28.1049
  11. Whitham G. Linear and nonlinear waves // Wiley-Interscience, 1999. — 660 p.
  12. Zakharov V.E., Shabat A.B. Precise theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in a nonlinear medium (in Russian) // Zh. Eksp. Teor. Fiz., 1971. 61. Pp.118-134.

External links

  1. Nonlinear Schrödinger equation in Wikipedia, the free encyclopedia
  2. Schrödinger Equation in Wolfram MathWorld
  3. Nonlinear Schrödinger equation in EqWorld, the world of mathematical equations

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