Geodesic
Madelung fluid description of the~generalized derivative nonlinear Schr\"odinger equation: Special solutions and their stability
Teoretičeskaâ i matematičeskaâ fizika, Tome 160 (2009) no. 1, pp. 229-239

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A correspondence between the families of generalized nonlinear Schrödinger (NLS) equations and generalized KdV equations was recently found using a Madelung fluid description. We similarly consider a special derivative NLS equation. We find a number of solitary waves and periodic solutions (expressed in terms of elliptic Jacobi functions) for a motion with a stationary profile current velocity. We study the stability of a bright solitary wave (ground state) by conjecturing that the Vakhitov–Kolokolov criterion is applicable.
Keywords: nonlinear partial differential equation, generalized nonlinear Schrödinger equation, generalized Korteweg–de Vries equation, Madelung fluid description. 
@article{TMF_2009_160_1_a21,
     author = {A. Visinescu and D. Grecu and R. Fedele and S. De Nicola},
     title = {Madelung fluid description of the~generalized derivative nonlinear {Schr\"odinger} equation: {Special} solutions and their stability},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {229--239},
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     volume = {160},
     number = {1},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2009_160_1_a21/}
}
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A. Visinescu; D. Grecu; R. Fedele; S. De Nicola. Madelung fluid description of the~generalized derivative nonlinear Schr\"odinger equation: Special solutions and their stability. Teoretičeskaâ i matematičeskaâ fizika, Tome 160 (2009) no. 1, pp. 229-239. http://geodesic.mathdoc.fr/item/TMF_2009_160_1_a21/