Geodesic
Cauchy problem for a~nonlinear Schr\"odinger equation with a~large initial gradient in the~weakly dispersive limit
Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 1, pp. 3-11

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We consider the Cauchy problem for the cubic nonlinear Schrödinger equation with a large gradient of the initial function and a small dispersion parameter. The renormalization method is used to construct an asymptotic solution in the explicit form of integral convolution. An asymptotic analogue of the renormalization group property is established under scaling transformations determined by the dispersion parameter. In the case of a negative focusing coefficient, a clarifying expression is obtained for the asymptotic solution in terms of known elliptic special functions.
Keywords: cubic nonlinear Schrödinger equation, Cauchy problem, renormalization, asymptotic solution, elliptic functions. 
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     author = {S. V. Zakharov},
     title = {Cauchy problem for a~nonlinear {Schr\"odinger} equation with a~large initial gradient in the~weakly dispersive limit},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {3--11},
     publisher = {mathdoc},
     volume = {219},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2024_219_1_a0/}
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S. V. Zakharov. Cauchy problem for a~nonlinear Schr\"odinger equation with a~large initial gradient in the~weakly dispersive limit. Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 1, pp. 3-11. http://geodesic.mathdoc.fr/item/TMF_2024_219_1_a0/