Geodesic
On the~existence of certain elliptic solutions of the~cubically nonlinear Schr\"{o}dinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 1, pp. 32-43

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We consider solutions of the cubically nonlinear Schrödinger equation. For a certain class of solutions of the form $\Psi(t,z)=(f(t,z)+id(z))e^{i\phi(z)}$ with $f,\phi,d\in\mathbb{R}$, we prove that they are nonexistent in the general case $f_z\neq 0$, $f_t\neq 0$, $d_z\neq 0$. In the three nongeneric cases ($f_z\neq 0$), ($f_t\neq 0$, $f_t=0$, $d_z=0$), and ($f_z=0$, $f_t\neq 0$), we present a two-parameter set of solutions, for which we find the constraints specifying real bounded and unbounded solutions.
Keywords: nonlinear Schrödinger equation, Weierstrass elliptic functions, traveling wave. 
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     author = {H. W. Sch\"urmann and V. S. Serov},
     title = {On the~existence of certain elliptic solutions of the~cubically nonlinear {Schr\"{o}dinger} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     volume = {219},
     number = {1},
     year = {2024},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2024_219_1_a3/}
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H. W. Schürmann; V. S. Serov. On the~existence of certain elliptic solutions of the~cubically nonlinear Schr\"{o}dinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 1, pp. 32-43. http://geodesic.mathdoc.fr/item/TMF_2024_219_1_a3/