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# Nonlinear Schrödinger equation (NLS)

Nonlinear Schrödinger equation
Order2nd
SolvableExactly solvable
The Cauchy problemSolvable by inverse scattering transform
First derived bySchrödinger, 1926

## Definition

$i\frac{\partial a}{\partial t}=3k\frac{{\partial }^{2}a}{\partial {x}^{2}}+ka{|a|}^{2}$

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.

## Solutions

### Traveling wave

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

$i\frac{\partial a}{\partial t}=\frac{{\partial }^{2}a}{\partial {x}^{2}}+\gamma a{|a|}^{2}$

in the form

$a\left(x,t\right)={e}^{ipx-i\chi t}V\left(z\right),\text{ }z=x-{c}_{0}t,$

where $p,\chi$ and ${c}_{0}$ are constants and $V\left(z\right)$ is a new unknown function.

For function $V\left(z\right)$ we get

${V}_{z}^{2}=A+\alpha {V}^{2}-\frac{\gamma }{2}{V}^{4}.$

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 $\alpha >0,$ $\gamma >0$ we have solution $V\left(z\right)$ in form of a solitary wave

$V\left(z\right)=\sqrt{\frac{2\alpha }{\gamma }}\frac{1}{ch\left(\sqrt{\alpha }\left(x-{c}_{0}t\right)\right)}.$

Corresponding solution of the nonlinear Schrödinger equation is

$u\left(x,t\right)={e}^{i\left(px-\chi t\right)}\sqrt{\frac{2\alpha }{\gamma }}\frac{1}{ch\left(\sqrt{\alpha }\left(x-{c}_{0}t\right)\right)},$

where

$p=-\frac{{c}_{0}}{2},\text{ }\chi =\alpha -\frac{{c}_{0}^{2}}{4}.$

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

## Lax pair

The Lax pair for the nonlinear Schrödinger equation

${a}_{t}=i{a}_{xx}-2i{|a|}^{2}a$

is

${\Psi }_{x}=\stackrel{^}{P}\Psi =\left(\begin{array}{cc}-i\lambda & a\\ {a}^{*}& i\lambda \end{array}\right)\Psi ,$

${\Psi }_{t}=\stackrel{^}{Q}\Psi =\left(\begin{array}{cc}i{|q|}^{2}+2i{\lambda }^{2}& -i{q}_{x}+2q\lambda \\ i{q}_{x}^{*}+2{q}^{*}\lambda & -i{|q|}^{2}-2i{\lambda }^{2}\end{array}\right)\Psi ,$

where

$\Psi =\left(\begin{array}{c}{\psi }_{1}\left(x,t\right)\\ {\psi }_{2}\left(x,t\right)\end{array}\right).$

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.

## References

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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.
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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.
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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.