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History

The inverse scattering transform arose from studying solitary waves. J.S. Russell described a "wave of translation" or "solitary wave" occurring in shallow water.⁹ 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.¹⁰ 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.¹¹ These particle-like waves are called solitons and arise in nonlinear equations because of a weak balance between dispersive and nonlinear effects.¹²

Gardner, Greene, Kruskal and Miura introduced the inverse scattering transform for solving the Korteweg–de Vries equation.¹³ 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.¹⁴¹⁵¹⁶ This approach has also been applied to different types of nonlinear equations including differential-difference, partial difference, multidimensional equations and fractional integrable nonlinear systems.¹⁷

Description

Nonlinear partial differential equation

The independent variables are a spatial variable x {\displaystyle x} and a time variable t {\displaystyle t} . Subscripts or differential operators ( ∂ x , ∂ t {\textstyle \partial _{x},\partial _{t}} ) indicate differentiation. The function u ( x , t ) {\displaystyle u(x,t)} is a solution of a nonlinear partial differential equation, u t + N ( u ) = 0 {\textstyle u_{t}+N(u)=0} , with initial condition (value) u ( x , 0 ) {\textstyle u(x,0)} .¹⁸: 72 

Requirements

The differential equation's solution meets the integrability and Fadeev conditions:¹⁹: 40 

Integrability condition: ∫ − ∞ ∞   | u ( x ) |   d x   < ∞ {\displaystyle \int _{-\infty }^{\infty }\ |u(x)|\ dx\ <\infty } Fadeev condition: ∫ − ∞ ∞   ( 1 + | x | ) | u ( x ) |   d x   < ∞ {\displaystyle \int _{-\infty }^{\infty }\ (1+|x|)|u(x)|\ dx\ <\infty }

Differential operator pair

The Lax differential operators, L {\textstyle L} and M {\textstyle M} , are linear ordinary differential operators with coefficients that may contain the function u ( x , t ) {\textstyle u(x,t)} or its derivatives. The self-adjoint operator L {\textstyle L} has a time derivative L t {\textstyle L_{t}} and generates a eigenvalue (spectral) equation with eigenfunctions ψ {\textstyle \psi } and time-constant eigenvalues (spectral parameters) λ {\textstyle \lambda } .²⁰: 4963 ²¹: 98 

L ( ψ ) = λ ψ ,   {\displaystyle L(\psi )=\lambda \psi ,\ } and   L t ( ψ ) = d e f ( L ( ψ ) ) t − L ( ψ t ) {\textstyle \ L_{t}(\psi ){\overset {def}{=}}(L(\psi ))_{t}-L(\psi _{t})}

The operator M {\textstyle M} describes how the eigenfunctions evolve over time, and generates a new eigenfunction ψ ~ {\textstyle {\widetilde {\psi }}} of operator L {\textstyle L} from eigenfunction ψ {\textstyle \psi } of L {\textstyle L} .²²: 4963 

ψ ~ = ψ t − M ( ψ )   {\displaystyle {\widetilde {\psi }}=\psi _{t}-M(\psi )\ }

The Lax operators combine to form a multiplicative operator, not a differential operator, of the eigenfuctions ψ {\textstyle \psi } .²³: 4963 

( L t + L M − M L ) ψ = 0 {\displaystyle (L_{t}+LM-ML)\psi =0}

The Lax operators are chosen to make the multiplicative operator equal to the nonlinear differential equation.²⁴: 4963 

L t + L M − M L = u t + N ( u ) = 0 {\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.²⁵: 4964 ²⁶²⁷

Direct scattering transform

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.²⁸: 39–48 

L ( ψ ) = λ ψ {\displaystyle L(\psi )=\lambda \psi }

Scattering data time evolution

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 ( x → ± ∞ {\textstyle x\to \pm \infty } ) simplify solving these differential equations.²⁹: 4967–4968 ³⁰: 68–72 ³¹

Inverse scattering transform

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.³²: 48–57 

Example: Korteweg–De Vries equation

The nonlinear differential Korteweg–De Vries equation is ³³: 4 

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

Lax operators

The Lax operators are:³⁴: 97–102 

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

The multiplicative operator is:

L t + L M − M L = u t − 6 u u x + u x x x = 0 {\displaystyle L_{t}+LM-ML=u_{t}-6uu_{x}+u_{xxx}=0}

Direct scattering transform

The solutions to this differential equation

L ( ψ ) = − ψ x x + u ( x , 0 ) ψ = λ ψ {\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 T ( k , 0 ) {\textstyle T(k,0)} , left reflection coefficient R L ( k , 0 ) {\textstyle R_{L}(k,0)} , right reflection coefficient R R ( k , 0 ) {\textstyle R_{R}(k,0)} , discrete eigenvalues − κ 1 2 , … , − κ N 2 {\textstyle -\kappa _{1}^{2},\ldots ,-\kappa _{N}^{2}} , and left and right bound-state normalization (norming) constants.³⁵: 4960 

c ( 0 ) L j = ( ∫ − ∞ ∞   ψ L 2 ( i k j , x , 0 )   d x ) − 1 / 2   j = 1 , … , N {\displaystyle c(0)_{Lj}=\left(\int _{-\infty }^{\infty }\ \psi _{L}^{2}(ik_{j},x,0)\ dx\right)^{-1/2}\ j=1,\dots ,N} c ( 0 ) R j = ( ∫ − ∞ ∞   ψ R 2 ( i k j , x , 0 )   d x ) − 1 / 2   j = 1 , … , N {\displaystyle c(0)_{Rj}=\left(\int _{-\infty }^{\infty }\ \psi _{R}^{2}(ik_{j},x,0)\ dx\right)^{-1/2}\ j=1,\dots ,N}

Scattering data time evolution

The spatially asymptotic left ψ L ( k , x , t ) {\textstyle \psi _{L}(k,x,t)} and right ψ R ( k , x , t ) {\textstyle \psi _{R}(k,x,t)} Jost functions simplify this step.³⁶: 4965–4966 

ψ L ( x , k , t ) = e i k x + o ( 1 ) ,   x → + ∞ ψ L ( x , k , t ) = e i k x T ( k , t ) + R L ( k , t ) e − i k x T ( k , t ) + o ( 1 ) ,   x → − ∞ ψ R ( x , k , t ) = e − i k x T ( k , t ) + R R ( k , t ) e i k x T ( k , t ) + o ( 1 ) ,   x → + ∞ ψ R ( x , k , t ) = e − i k x + o ( 1 ) ,   x → − ∞ {\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 γ j ( t ) {\textstyle \gamma _{j}(t)} relate the right and left Jost functions and right and left normalization constants.³⁷: 4965–4966 

γ j ( t ) = ψ L ( x , i κ j , t ) ψ R ( x , i κ j , t ) = ( − 1 ) N − j c R j ( t ) c L j ( t ) {\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 M {\textstyle M} differential operator generates an eigenfunction which can be expressed as a time-dependent linear combination of other eigenfunctions.³⁸: 4967 

∂ t ψ L ( k , x , t ) − M ψ L ( x , k , t ) = a L ( k , t ) ψ L ( x , k , t ) + b L ( k , t ) ψ R ( x , k , t ) {\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)} ∂ t ψ R ( k , x , t ) − M ψ R ( x , k , t ) = a R ( k , t ) ψ L ( x , k , t ) + b R ( k , t ) ψ 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 T ( k , t ) {\textstyle T(k,t)} , but time-dependent reflection coefficients and normalization coefficients.³⁹: 4967–4968 

R L ( k , t ) = R L ( k , 0 ) e − i 8 k 3 t R R ( k , t ) = R R ( k , 0 ) e + i 8 k 3 t c L j ( t ) = c L j ( 0 ) e + 4 κ j 3 t ,   j = 1 , … , N c R j ( t ) = c R j ( 0 ) e − 4 κ j 3 t ,   j = 1 , … , N {\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}}}

Inverse scattering transform

The Marchenko kernel is F ( x , t ) {\textstyle F(x,t)} .⁴⁰: 4968–4969 

F ( x , t ) = d e f 1 2 π ∫ − ∞ ∞ R R ( k , t ) e i k x   d k + ∑ j = 1 N c ( t ) L j 2 e − κ j x {\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 K ( x , y , t ) {\textstyle K(x,y,t)} .⁴¹: 4968–4969 

K ( x , z , t ) + F ( x + z , t ) + ∫ x ∞ K ( x , y , t ) F ( y + z , t )   d y = 0 {\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, K ( x , y , t ) {\textstyle K(x,y,t)} , generates the solution u ( x , t ) {\textstyle u(x,t)} to the nonlinear partial differential equation.⁴²: 4969 

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

Examples of integrable equations

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

See also

• Quantum inverse scattering method
• Integrable system

Citations

• Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H. (1973). "Method for Solving the Sine-Gordon Equation". Physical Review Letters. 30 (25): 1262–1264. Bibcode:1973PhRvL..30.1262A. doi:10.1103/PhysRevLett.30.1262.

• Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H. (1974). "The Inverse Scattering Transform—Fourier Analysis for Nonlinear Problems". Studies in Applied Mathematics. 53 (4): 249–315. doi:10.1002/sapm1974534249.

• Ablowitz, Mark J.; Segur, Harvey (1981). Solitons and the Inverse Scattering Transform. SIAM. ISBN 978-0-89871-477-7.

• Ablowitz, Mark J.; Fokas, A. S. (2003). Complex Variables: Introduction and Applications. Cambridge University Press. pp. 604–620. ISBN 978-0-521-53429-1.

• Ablowitz, Mark J. (2023). "Nonlinear waves and the Inverse Scattering Transform". Optik. 278: 170710. Bibcode:2023Optik.27870710A. doi:10.1016/j.ijleo.2023.170710.

• Aktosun, Tuncay (2009). "Inverse Scattering Transform and the Theory of Solitons". Encyclopedia of Complexity and Systems Science. Springer. pp. 4960–4971. doi:10.1007/978-0-387-30440-3_295. ISBN 978-0-387-30440-3.

• Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0.

• Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. (1967). "Method for Solving the Korteweg-deVries Equation". Physical Review Letters. 19 (19): 1095–1097. Bibcode:1967PhRvL..19.1095G. doi:10.1103/PhysRevLett.19.1095.

• Konopelchenko, B.G.; Dubrowsky, V.G. (1991). "Localized solitons for the Ishimori equation". In Sattinger, David H.; Tracy, C.A.; Venakides, Stephanos (eds.). Inverse Scattering and Applications. American Mathematical Soc. pp. 77–90. ISBN 978-0-8218-5129-6.

• Oono, H. (1996). "N-Soliton solution of Harry Dym equation by inverse scattering method.". In Alfinito, E.; Boiti, M.; Martina, L. (eds.). Nonlinear Physics: Theory and Experiment. World Scientific Publishing Company Pte Limited. pp. 241–248. ISBN 978-981-02-2559-9.

• Osborne, A. R. (1995). "Soliton physics and the periodic inverse scattering transform". Physica D: Nonlinear Phenomena. 86 (1): 81–89. doi:10.1016/0167-2789(95)00089-M. ISSN 0167-2789.

Further reading

• Ablowitz, Mark J.; Clarkson, P. A. (12 December 1991). Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press. ISBN 978-0-521-38730-9.

• Bullough, R. K.; Caudrey, P. J. (11 November 2013). Solitons. Springer Science & Business Media. ISBN 978-3-642-81448-8.

• Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. (1974), "Korteweg-deVries equation and generalization. VI. Methods for exact solution.", Comm. Pure Appl. Math., 27: 97–133, doi:10.1002/cpa.3160270108, MR 0336122

• Gelʹfand, Izrailʹ Moiseevich (1955). On the Determination of a Differential Equation from Its Spectral Function. American Mathematical Society. p. 253-304.

• Marchenko, Vladimir A. (1986). Sturm-Liouville Operators and Applications. Operator Theory: Advances and Applications. Vol. 22. Basel: Birkhäuser. doi:10.1007/978-3-0348-5485-6. ISBN 978-3-0348-5486-3.

• Shaw, J. K. (1 May 2004). Mathematical Principles of Optical Fiber Communication. SIAM. ISBN 978-0-89871-556-9.

External links

• "Introductory mathematical paper on IST" (PDF). (300 KiB)
• Inverse Scattering Transform and the Theory of Solitons

References

1. Aktosun 2009. - Aktosun, Tuncay (2009). "Inverse Scattering Transform and the Theory of Solitons". Encyclopedia of Complexity and Systems Science. Springer. pp. 4960–4971. doi:10.1007/978-0-387-30440-3_295. ISBN 978-0-387-30440-3. https://link.springer.com/referenceworkentry/10.1007/978-0-387-30440-3_295 ↩

2. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

3. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

4. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

5. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

6. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

7. Ablowitz & Fokas 2003, pp. 604–620. - Ablowitz, Mark J.; Fokas, A. S. (2003). Complex Variables: Introduction and Applications. Cambridge University Press. pp. 604–620. ISBN 978-0-521-53429-1. https://books.google.com/books?id=SFqbV3i3hO0C ↩

8. Osborne 1995. - Osborne, A. R. (1995). "Soliton physics and the periodic inverse scattering transform". Physica D: Nonlinear Phenomena. 86 (1): 81–89. doi:10.1016/0167-2789(95)00089-M. ISSN 0167-2789. https://www.sciencedirect.com/science/article/abs/pii/016727899500089M ↩

9. Ablowitz 2023. - Ablowitz, Mark J. (2023). "Nonlinear waves and the Inverse Scattering Transform". Optik. 278: 170710. Bibcode:2023Optik.27870710A. doi:10.1016/j.ijleo.2023.170710. https://www.sciencedirect.com/science/article/pii/S0030402623002061 ↩

10. Ablowitz 2023. - Ablowitz, Mark J. (2023). "Nonlinear waves and the Inverse Scattering Transform". Optik. 278: 170710. Bibcode:2023Optik.27870710A. doi:10.1016/j.ijleo.2023.170710. https://www.sciencedirect.com/science/article/pii/S0030402623002061 ↩

11. Ablowitz 2023. - Ablowitz, Mark J. (2023). "Nonlinear waves and the Inverse Scattering Transform". Optik. 278: 170710. Bibcode:2023Optik.27870710A. doi:10.1016/j.ijleo.2023.170710. https://www.sciencedirect.com/science/article/pii/S0030402623002061 ↩

12. Ablowitz 2023. - Ablowitz, Mark J. (2023). "Nonlinear waves and the Inverse Scattering Transform". Optik. 278: 170710. Bibcode:2023Optik.27870710A. doi:10.1016/j.ijleo.2023.170710. https://www.sciencedirect.com/science/article/pii/S0030402623002061 ↩

13. Gardner et al. 1967. - Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. (1967). "Method for Solving the Korteweg-deVries Equation". Physical Review Letters. 19 (19): 1095–1097. Bibcode:1967PhRvL..19.1095G. doi:10.1103/PhysRevLett.19.1095. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.19.1095 ↩

14. Ablowitz 2023. - Ablowitz, Mark J. (2023). "Nonlinear waves and the Inverse Scattering Transform". Optik. 278: 170710. Bibcode:2023Optik.27870710A. doi:10.1016/j.ijleo.2023.170710. https://www.sciencedirect.com/science/article/pii/S0030402623002061 ↩

15. Konopelchenko & Dubrowsky 1991. - Konopelchenko, B.G.; Dubrowsky, V.G. (1991). "Localized solitons for the Ishimori equation". In Sattinger, David H.; Tracy, C.A.; Venakides, Stephanos (eds.). Inverse Scattering and Applications. American Mathematical Soc. pp. 77–90. ISBN 978-0-8218-5129-6. https://books.google.com/books?id=yTYcCAAAQBAJ ↩

16. Oono 1996. - Oono, H. (1996). "N-Soliton solution of Harry Dym equation by inverse scattering method.". In Alfinito, E.; Boiti, M.; Martina, L. (eds.). Nonlinear Physics: Theory and Experiment. World Scientific Publishing Company Pte Limited. pp. 241–248. ISBN 978-981-02-2559-9. https://books.google.com/books?id=35EfzQEACAAJ ↩

17. Ablowitz 2023. - Ablowitz, Mark J. (2023). "Nonlinear waves and the Inverse Scattering Transform". Optik. 278: 170710. Bibcode:2023Optik.27870710A. doi:10.1016/j.ijleo.2023.170710. https://www.sciencedirect.com/science/article/pii/S0030402623002061 ↩

18. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

19. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

20. Aktosun 2009. - Aktosun, Tuncay (2009). "Inverse Scattering Transform and the Theory of Solitons". Encyclopedia of Complexity and Systems Science. Springer. pp. 4960–4971. doi:10.1007/978-0-387-30440-3_295. ISBN 978-0-387-30440-3. https://link.springer.com/referenceworkentry/10.1007/978-0-387-30440-3_295 ↩

21. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

22. Aktosun 2009. - Aktosun, Tuncay (2009). "Inverse Scattering Transform and the Theory of Solitons". Encyclopedia of Complexity and Systems Science. Springer. pp. 4960–4971. doi:10.1007/978-0-387-30440-3_295. ISBN 978-0-387-30440-3. https://link.springer.com/referenceworkentry/10.1007/978-0-387-30440-3_295 ↩

23. Aktosun 2009. - Aktosun, Tuncay (2009). "Inverse Scattering Transform and the Theory of Solitons". Encyclopedia of Complexity and Systems Science. Springer. pp. 4960–4971. doi:10.1007/978-0-387-30440-3_295. ISBN 978-0-387-30440-3. https://link.springer.com/referenceworkentry/10.1007/978-0-387-30440-3_295 ↩

24. Aktosun 2009. - Aktosun, Tuncay (2009). "Inverse Scattering Transform and the Theory of Solitons". Encyclopedia of Complexity and Systems Science. Springer. pp. 4960–4971. doi:10.1007/978-0-387-30440-3_295. ISBN 978-0-387-30440-3. https://link.springer.com/referenceworkentry/10.1007/978-0-387-30440-3_295 ↩

25. Aktosun 2009. - Aktosun, Tuncay (2009). "Inverse Scattering Transform and the Theory of Solitons". Encyclopedia of Complexity and Systems Science. Springer. pp. 4960–4971. doi:10.1007/978-0-387-30440-3_295. ISBN 978-0-387-30440-3. https://link.springer.com/referenceworkentry/10.1007/978-0-387-30440-3_295 ↩

26. Ablowitz et al. 1973. - Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H. (1973). "Method for Solving the Sine-Gordon Equation". Physical Review Letters. 30 (25): 1262–1264. Bibcode:1973PhRvL..30.1262A. doi:10.1103/PhysRevLett.30.1262. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1262 ↩

27. Ablowitz et al. 1974. - Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H. (1974). "The Inverse Scattering Transform—Fourier Analysis for Nonlinear Problems". Studies in Applied Mathematics. 53 (4): 249–315. doi:10.1002/sapm1974534249. https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm1974534249 ↩

28. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

29. Aktosun 2009. - Aktosun, Tuncay (2009). "Inverse Scattering Transform and the Theory of Solitons". Encyclopedia of Complexity and Systems Science. Springer. pp. 4960–4971. doi:10.1007/978-0-387-30440-3_295. ISBN 978-0-387-30440-3. https://link.springer.com/referenceworkentry/10.1007/978-0-387-30440-3_295 ↩

30. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

31. Gardner et al. 1967. - Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. (1967). "Method for Solving the Korteweg-deVries Equation". Physical Review Letters. 19 (19): 1095–1097. Bibcode:1967PhRvL..19.1095G. doi:10.1103/PhysRevLett.19.1095. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.19.1095 ↩

32. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

33. Ablowitz & Segur 1981. - Ablowitz, Mark J.; Segur, Harvey (1981). Solitons and the Inverse Scattering Transform. SIAM. ISBN 978-0-89871-477-7. https://books.google.com/books?id=Bzu4XAUpFZUC ↩

34. Drazin & Johnson 1989. - Drazin, P. G.; Johnson, R. S. (1989). Solitons: An Introduction. Cambridge University Press. ISBN 978-0-521-33655-0. https://books.google.com/books?id=HPmbIDk2u-gC ↩

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