Geodesic
Asymptotics at~$t\to\infty$ of~the solution to~the Cauchy problem for the Korteweg--de~Vries equation in~the class of~potentials with finite-gap behavior as~$x\to\pm\infty$
Teoretičeskaâ i matematičeskaâ fizika, Tome 78 (1989) no. 3, pp. 345-356

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The Cauchy problem for the KdV equation is considered in the class of functions approaching at $x\to\pm\infty$ two different finite-gap solutions of this equation which correspond to the same Riemann surface. With the help of the inverse scattering method the large time asymptotics of the Cauchy problem solution is considered. 
@article{TMF_1989_78_3_a3,
     author = {R. F. Bikbaev and R. A. Sharipov},
     title = {Asymptotics at~$t\to\infty$ of~the solution to~the {Cauchy} problem for the {Korteweg--de~Vries} equation in~the class of~potentials with finite-gap behavior as~$x\to\pm\infty$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {345--356},
     publisher = {mathdoc},
     volume = {78},
     number = {3},
     year = {1989},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1989_78_3_a3/}
}
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R. F. Bikbaev; R. A. Sharipov. Asymptotics at~$t\to\infty$ of~the solution to~the Cauchy problem for the Korteweg--de~Vries equation in~the class of~potentials with finite-gap behavior as~$x\to\pm\infty$. Teoretičeskaâ i matematičeskaâ fizika, Tome 78 (1989) no. 3, pp. 345-356. http://geodesic.mathdoc.fr/item/TMF_1989_78_3_a3/