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A generalization of [[nonlinear Schroedinger equation]] with |
A generalization of [[nonlinear Schroedinger equation]] with |
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additional quintic nonlinerity and a nonlinear dispersive term |
additional quintic nonlinerity and a nonlinear dispersive term |
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was proposed in <ref name={{kundu|1984}} /> |
was proposed in <ref name={{kundu|1984}} /> in the form |
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in the form |
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:<math> \psi_t+\psi_{xx} -2 \beta |\psi|^2q+\delta^2|\psi|^4\psi \pm 2i |
:<math> \psi_t+\psi_{xx} -2 \beta |\psi|^2q+\delta^2|\psi|^4\psi \pm 2i |
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\delta (| \psi|^2)_x \psi=0, ...... (3) </math> |
\delta (| \psi|^2)_x \psi=0, ...... (3) </math> |
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which may be obtained from the '''Kundu Equation''' (2), when restricted to |
which may be obtained from the '''Kundu Equation''' (2), when restricted to |
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<math> |
<math>\alpha =0<math>. |
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The same equation, limited further to the particular case |
The same equation, limited further to the particular case |
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<math>\beta =0 |
<math>\beta =0,<math> |
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was introduced later as [[Eckhaus |
was introduced later as [[Eckhaus equation]], following which equation (3) is |
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equation]], following which equation (3) is |
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presently known as the ''' Kundu-Ekchaus''' eqution. |
presently known as the ''' Kundu-Ekchaus''' eqution. |
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The ''' Kundu-Ekchaus''' equation can be reduced |
The ''' Kundu-Ekchaus''' equation can be reduced |
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its generalizations and |
its generalizations and |
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link with other equations have been studied. |
link with other equations have been studied. |
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In particular, relationship of ''' Kundu-Ekchaus''' equation |
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with the Johnson's hydrodynamic equation near criticality is |
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established<ref name={{kundu|1987}} > |
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{{citation | first=A. |last=Kundu |title= Exact solutions in higher order nonlinear equations |
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gauge transformation |
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| journal= Physica D | volume=25 | year=1987 |
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|pages=399-406 }} |
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</ref> |
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, its [[discretizations]] <ref name={{levi|2009}} > |
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{{citation | first1=D. |last1=Levi| first2=C. |last2=Scimiterna |title= |
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The Kundu–Eckhaus equation and its discretizations |
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| journal= J. Phys. A | volume= | year=2009 |
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|pages= }} |
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</ref> |
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, reduction via [[Lie |
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symmetry]] <ref name={{tooman|2013}} > |
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{{citation | first1= |last1=Toomanian| first2= |last2=Asadi |title= |
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Reductions for Kundu-Eckhaus equation via Lie symmetry analysis |
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| journal= Math. Sciences | volume=7 | year=2013 |
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|pages=50 }} |
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</ref> , complex structure via [[Bernoulli subequation]] |
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<ref name={{beok|2015}} > |
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{{citation | first1=H. M. |last1= Beokonus| first2=Q. H. |last2= Bulut |title= |
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On the complex structure of Kundu-Eckhaus equation via Bernoulli subequation |
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fungtion method |
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| journal= Waves in Random and Complex Media | volume= 28 Aug. | year=2015 |
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|pages= }} |
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</ref> , |
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bright and dark |
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Revision as of 06:54, 25 April 2016
Kundu-Eckhaus Equation
A generalization of nonlinear Schroedinger equation with additional quintic nonlinerity and a nonlinear dispersive term was proposed in [1] in the form
which may be obtained from the Kundu Equation (2), when restricted to <math>\alpha =0<math>. The same equation, limited further to the particular case <math>\beta =0,<math> was introduced later as Eckhaus equation, following which equation (3) is presently known as the Kundu-Ekchaus eqution. The Kundu-Ekchaus equation can be reduced to the nonlinear Schroedinger equation through a nonlinear transformation of the field and known therefore to be gauge equivalent integrable systems, since they are equivalent under the gauge transformation.
Properties and Applications
the Kundu-Ekchaus equation is asociated with a Lax pair, higher conserved quantity, exact soliton solution, rogue wave solution etc. Over the years various aspects of this equation, its generalizations and link with other equations have been studied. In particular, relationship of Kundu-Ekchaus equation with the Johnson's hydrodynamic equation near criticality is established[2] , its discretizations [3] , reduction via [[Lie symmetry]] [4] , complex structure via Bernoulli subequation [5] , bright and dark
- ^ Cite error: The named reference
Template:Kunduwas invoked but never defined (see the help page). - ^
Kundu, A. (1987), "Exact solutions in higher order nonlinear equations
gauge transformation", Physica D, 25: 399–406
{{citation}}: line feed character in|title=at position 54 (help) - ^ Levi, D.; Scimiterna, C. (2009), "The Kundu–Eckhaus equation and its discretizations", J. Phys. A
- ^ Toomanian; Asadi (2013), "Reductions for Kundu-Eckhaus equation via Lie symmetry analysis", Math. Sciences, 7: 50
- ^
Beokonus, H. M.; Bulut, Q. H. (2015), "On the complex structure of Kundu-Eckhaus equation via Bernoulli subequation
fungtion method", Waves in Random and Complex Media, 28 Aug.
{{citation}}: line feed character in|title=at position 79 (help)