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April  2021, 20(4): 1447-1478. doi: 10.3934/cpaa.2021028

Asymptotics for the higher-order derivative nonlinear Schrödinger equation

Pavel I. Naumkin 1, and  Isahi Sánchez-Suárez 2,,

1. 

Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, MEXICO
2. 

Universidad Politécnica de Uruapan, CP 60210 Uruapan Michoacán, MEXICO

* Corresponding author

Received  September 2020 Revised  January 2021 Published  April 2021 Early access  March 2021

Fund Project: The work of P. I. N. is partially supported by CONACYT project 283698 and PAPIIT project IN103221

We study the Cauchy problem for the derivative higher-order nonlinear Schrödinger equation
$ \begin{cases} i\partial_{t}v+\dfrac{a}{2}\partial_{x}^{2}v-\dfrac{b}{4}\partial_{x} ^{4}v = \left( \overline{\partial_{x}v}\right) ^{2},\text{ }t>1,\text{ } x\in\mathbb{R},\\ v\left( 1,x\right) = v_{0}\left( x\right) ,\text{ }x\in\mathbb{R}\text{,} \end{cases} $
where
$ a,b>0. $
 Our aim is to prove global existence and calculate the large time asymptotics of solutions. We develop the factorization techniques originated in papers [13,10,12]. Also we follow the method of papers [9,11] to transform the quadratic nonlinearity to critical cubic nonlinearities similarly to the normal forms of Shatah [18].
Keywords: Asymptotics of solutions, derivative nonlinear Schrödinger equation, higher order, factorization techniques, modified scattering.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35Q35.
Citation: Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1447-1478. doi: 10.3934/cpaa.2021028
References:
[1]

M. Ben-Artzi, H. Koch and J. C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Math. Acad. Sci., 330 (2000), 87-92.  doi: 10.1016/S0764-4442(00)00120-8.

[2]

A. P. Calderon and R. Vaillancourt, A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci., 69 (1972), 1185-1187.  doi: 10.1073/pnas.69.5.1185.

[3]

Th. Cazenave, Semilinear Schrödinger equations, Courant Institute of Mathematical Sciences, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[4]

S. Cohn, Resonance and long time existence for the quadratic semilinear Schrödinger equation, Commun. Pure Appl. Math., 45 (1992), 973-1001.  doi: 10.1002/cpa.3160450804.

[5]

R. R. Coifman and Y. Meyer, Au dela des operateurs pseudo-differentiels, Societe Mathematique de France, Paris, 1978.

[6]

H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal., 18 (1975), 115-131.  doi: 10.1016/0022-1236(75)90020-8.

[7]

K. B. Dysthe, Note on a modification to the nonlinear Schrödinger equation for application to deep water waves, Proc. R. Soc. Lond. Ser. A, 369 (1979), 105-114. 

[8]

Y. Fukumoto and H. K. Mofatt, Motion and expansion of a viscous vortex ring: I. A higher-order asymptotic formula for the velocity, J. Fluid. Mech., 417 (2000), 1-45.  doi: 10.1017/S0022112000008995.

[9]

N. Hayashi and P. I. Naumkin, A quadratic nonlinear Schrödinger equation in one space dimension, J. Differ. Equ, 186 (2002), 165-185.  doi: 10.1016/S0022-0396(02)00010-4.

[10]

N. Hayashi and P. I. Naumkin, The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.  doi: 10.1007/s00033-007-7008-8.

[11]

N. Hayashi and P. I. Naumkin, Asymptotic behavior for a quadratic nonlinear Schrödinger equation, Electron. J. Differential Equations, 15 2008, 38 pp.

[12]

N. Hayashi and P.I. Naumkin, On the inhomogeneous fourth-order nonlinear Schrödinger equation, J. Math. Phys., 56 (2015), 25 pp. doi: 10.1063/1.4929657.

[13]

N. Hayashi and T. Ozawa, Scattering theory in the weighted $L^{2}(R^{n})$ spaces for some Sc rödinger equations, Ann. I. H. P. (Phys. Théor.), 48 (1988), 17-37.

[14]

I. L. Hwang, The $L^{2}$ -boundedness of pseudodifferential operators, Trans. Amer. Math. Soc., 302 (1987), 55-76.  doi: 10.2307/2000896.

[15]

V. L. Karpman, Stabilization of soliton instabilities by high-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.  doi: 10.1016/0375-9601(95)00752-0.

[16]

V. L. Karpman and A. G. Shagalov, Stabilitiy of soliton described by nonlinear Schrödinger-type equations with high-order dispersion, Physica D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.

[17]

T. Ozawa, Remarks on quadratic nonlinear Schrödinger equations, Funkcial. Ekvac., 38 (1995), 217-232. 

[18]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Commun. Pure Appl. Math., 38 (1985), 685-696.  doi: 10.1002/cpa.3160380516.

show all references

References:
[1]

M. Ben-Artzi, H. Koch and J. C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Math. Acad. Sci., 330 (2000), 87-92.  doi: 10.1016/S0764-4442(00)00120-8.

[2]

A. P. Calderon and R. Vaillancourt, A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci., 69 (1972), 1185-1187.  doi: 10.1073/pnas.69.5.1185.

[3]

Th. Cazenave, Semilinear Schrödinger equations, Courant Institute of Mathematical Sciences, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[4]

S. Cohn, Resonance and long time existence for the quadratic semilinear Schrödinger equation, Commun. Pure Appl. Math., 45 (1992), 973-1001.  doi: 10.1002/cpa.3160450804.

[5]

R. R. Coifman and Y. Meyer, Au dela des operateurs pseudo-differentiels, Societe Mathematique de France, Paris, 1978.

[6]

H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal., 18 (1975), 115-131.  doi: 10.1016/0022-1236(75)90020-8.

[7]

K. B. Dysthe, Note on a modification to the nonlinear Schrödinger equation for application to deep water waves, Proc. R. Soc. Lond. Ser. A, 369 (1979), 105-114. 

[8]

Y. Fukumoto and H. K. Mofatt, Motion and expansion of a viscous vortex ring: I. A higher-order asymptotic formula for the velocity, J. Fluid. Mech., 417 (2000), 1-45.  doi: 10.1017/S0022112000008995.

[9]

N. Hayashi and P. I. Naumkin, A quadratic nonlinear Schrödinger equation in one space dimension, J. Differ. Equ, 186 (2002), 165-185.  doi: 10.1016/S0022-0396(02)00010-4.

[10]

N. Hayashi and P. I. Naumkin, The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.  doi: 10.1007/s00033-007-7008-8.

[11]

N. Hayashi and P. I. Naumkin, Asymptotic behavior for a quadratic nonlinear Schrödinger equation, Electron. J. Differential Equations, 15 2008, 38 pp.

[12]

N. Hayashi and P.I. Naumkin, On the inhomogeneous fourth-order nonlinear Schrödinger equation, J. Math. Phys., 56 (2015), 25 pp. doi: 10.1063/1.4929657.

[13]

N. Hayashi and T. Ozawa, Scattering theory in the weighted $L^{2}(R^{n})$ spaces for some Sc rödinger equations, Ann. I. H. P. (Phys. Théor.), 48 (1988), 17-37.

[14]

I. L. Hwang, The $L^{2}$ -boundedness of pseudodifferential operators, Trans. Amer. Math. Soc., 302 (1987), 55-76.  doi: 10.2307/2000896.

[15]

V. L. Karpman, Stabilization of soliton instabilities by high-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.  doi: 10.1016/0375-9601(95)00752-0.

[16]

V. L. Karpman and A. G. Shagalov, Stabilitiy of soliton described by nonlinear Schrödinger-type equations with high-order dispersion, Physica D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.

[17]

T. Ozawa, Remarks on quadratic nonlinear Schrödinger equations, Funkcial. Ekvac., 38 (1995), 217-232. 

[18]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Commun. Pure Appl. Math., 38 (1985), 685-696.  doi: 10.1002/cpa.3160380516.

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