Gardner equation: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 05:15, 15 June 2024

The Gardner equation is an integrable nonlinear partial differential equation introduced by the mathematician Clifford Gardner in 1968 to generalize KdV equation and modified KdV equation. The Gardner equation has applications in hydrodynamics, plasma physics and quantum field theory^[1]

{\frac {\partial u}{\partial t}}-(6\varepsilon ^{2}u^{2}+6u){\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial x^{3}}}=0,

where $\varepsilon$ is an arbitrary real parameter.

Notes

^ Shingareva & Lizárraga-Celaya 2011, pp. 13, 51.

References

Shingareva, Inna; Lizárraga-Celaya, Carlos (2011). Solving Nonlinear Partial Differential Equations with Maple and Mathematica. Wien ; New York: Springer Science & Business Media. ISBN 978-3-7091-0517-7. OCLC 751824407.

This article about theoretical physics is a stub. You can help Wikipedia by expanding it.

[FOOTNOTEShingarevaLizárraga-Celaya201113,_51-1] Shingareva & Lizárraga-Celaya 2011, pp. 13, 51.

[1]

@@ Line 1: / Line 1: @@
-The '''Gardner equation''' is an [[integrable]] [[nonlinear partial differential equation]] introduced by the mathematician [[Clifford Gardner]] in 1968 to generalize [[KdV equation]] and [[Unnormalized modified KdV equation|modified KdV equation]]. The Gardner equation has applications in [[hydrodynamics]], [[plasma physics]] and [[quantum field theory]]<ref>Inna Shingareva, Carlos Lizárraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple p13 Springer</ref>
+The '''Gardner equation''' is an [[integrable]] [[nonlinear partial differential equation]] introduced by the mathematician [[Clifford Gardner]] in 1968 to generalize [[KdV equation]] and [[Unnormalized modified KdV equation|modified KdV equation]]. The Gardner equation has applications in [[hydrodynamics]], [[plasma physics]] and [[quantum field theory]]{{sfn | Shingareva | Lizárraga-Celaya | 2011 | pp=13,51}}
-: <math>\frac{\partial u}{\partial  t}+(2 a u-3 b u^2)\frac{\partial u}{\partial x }+\frac{\partial^3 u}{\partial  x^3}=0,</math>
+: <math>\frac{\partial u}{\partial  t}-(6 \varepsilon^2 u^2 + 6u) \frac{\partial u}{\partial x }+\frac{\partial^3 u}{\partial  x^3}=0,</math>
-where <math>a</math> and <math>b</math> are constants.
+where <math>\varepsilon</math> is an arbitrary real parameter.
+==See also==
+*[[Korteweg–de Vries equation]]
+==Notes==
+{{reflist}}
 ==References==
+* {{cite book | last=Shingareva | first=Inna | last2=Lizárraga-Celaya | first2=Carlos | title=Solving Nonlinear Partial Differential Equations with Maple and Mathematica | publisher=Springer Science & Business Media | publication-place=Wien ; New York | year=2011 | isbn=978-3-7091-0517-7 | oclc=751824407}}
-<references/>
-#Graham W. Griffiths William E. Shiesser, ''Traveling Wave Analysis of Partial Differential Equations'', Academy Press
-# Richard H. Enns George C. McCGuire, ''Nonlinear Physics'', Birkhauser,1997
-#Inna Shingareva, Carlos Lizárraga-Celaya, ''Solving Nonlinear Partial Differential Equations with Maple'', Springer.
-#Eryk Infeld and George Rowlands, ''Nonlinear Waves, Solitons and Chaos'', Cambridge 2000
-#Saber Elaydi, ''An Introduction to Difference Equations'', Springer 2000
-#Dongming Wang, ''Elimination Practice'', Imperial College Press 2004
-# David Betounes, ''Partial Differential Equations for Computational Science: With Maple and Vector Analysis''. Springer, 1998 {{ISBN|9780387983004}}
-# George Articolo, ''Partial Differential Equations & Boundary Value Problems with Maple V'', Academic Press 1998 {{ISBN|9780120644759}}
 [[Category:Nonlinear partial differential equations]]