TY - JOUR

T1 - Large time asymptotics for the fractional modified Korteweg-de Vries equation with α∈ (2 , 4)

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

AU - Sánchez-Suárez, Isahi

N1 - Funding Information:
The work of N.H. is partially supported by JSPS KAKENHI Grant Numbers JP20K03680, JP19H05597. The work of P.I.N. is partially supported by CONACYT project 283698 and PAPIIT project IN103221. We would like to thank unknown referees for their valuable comments and suggestions.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/12

Y1 - 2022/12

N2 - We study the large time asymptotics for solutions to the Cauchy problem for the fractional modified Korteweg-de Vries equation {∂tw+1α|∂x|α-1∂xw=∂x(w3),t>0,x∈R,w(0,x)=w0(x),x∈R,where α∈ (2 , 4) , and |∂x|α=F-1|ξ|αF is the fractional derivative. The case of α= 3 corresponds to the classical modified KdV equation. In the case of α= 2 it is the modified Benjamin-Ono equation. Our aim is to find the large time asymptotic formulas for the solutions of the Cauchy problem for the fractional modified KdV equation. We develop the method based on the factorization techniques which was started in papers Hayashi, N., Naumkin, P.I. (Z. Angew. Math. Phys.) 59, 1002–1028 (2008), Hayashi, N., Naumkin, P.I. (SUT J. Math.) 52, 49–95 (2016) Hayashi, N., Ozawa, T.: (Ann. I.H.P. (Phys. Théor.)) 48, 17-37 (1988), Naumkin, P.I. (J. Differential Equations) 269(7), 5701–5729 (2020). Also we apply the known results on the L2 - boundedness of pseudodifferential operators.

AB - We study the large time asymptotics for solutions to the Cauchy problem for the fractional modified Korteweg-de Vries equation {∂tw+1α|∂x|α-1∂xw=∂x(w3),t>0,x∈R,w(0,x)=w0(x),x∈R,where α∈ (2 , 4) , and |∂x|α=F-1|ξ|αF is the fractional derivative. The case of α= 3 corresponds to the classical modified KdV equation. In the case of α= 2 it is the modified Benjamin-Ono equation. Our aim is to find the large time asymptotic formulas for the solutions of the Cauchy problem for the fractional modified KdV equation. We develop the method based on the factorization techniques which was started in papers Hayashi, N., Naumkin, P.I. (Z. Angew. Math. Phys.) 59, 1002–1028 (2008), Hayashi, N., Naumkin, P.I. (SUT J. Math.) 52, 49–95 (2016) Hayashi, N., Ozawa, T.: (Ann. I.H.P. (Phys. Théor.)) 48, 17-37 (1988), Naumkin, P.I. (J. Differential Equations) 269(7), 5701–5729 (2020). Also we apply the known results on the L2 - boundedness of pseudodifferential operators.

KW - Asymptotics for large time

KW - Fractional modified KdV equation

KW - Modified scattering

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U2 - 10.1007/s42985-022-00206-y

DO - 10.1007/s42985-022-00206-y

M3 - Article

AN - SCOPUS:85140208909

SN - 2662-2963

VL - 3

JO - Partial Differential Equations and Applications

JF - Partial Differential Equations and Applications

IS - 6

M1 - 76

ER -