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Focusing nlkg equation with singular potential
Asymptotics for the modified witham equation
1. | Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043, Japan |
2. | Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico |
${{\partial }_{t}}v+{{\partial }_{x}}\sqrt{{{a}^{2}}-\partial _{x}^{2}v}={{\partial }_{x}}\left( {{v}^{3}} \right),\ \ \left( t,x \right)\in \mathbb{R}\times \mathbb{R},$ |
$\sqrt{a^{2}-\partial _{x}^{2}}$ |
References:
[1] |
M. V. Fedoryuk, Asymptotic Methods in Analysis, in Analysis. I. Integral representations and asymptotic methods, Encyclopaedia of Mathematical Sciences, 13. Springer-Verlag, Berlin, 1989. vi+238 pp. |
[2] |
N. Hayashi and E. Kaikina, Asymptotics for the third-order nonlinear Schrödinger equation in the critical case, to appear in MMAS. |
[3] |
N. Hayashi, J. Mendez-Navarro and P. Naumkin, Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case, submitted to JDE, 2016. |
[4] |
N. Hayashi and P. Naumkin,
Factorization technique for the fourth-order nonlinear Schrödinger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377.
|
[5] |
N. Hayashi and P. I. Naumkin,
Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Analysis, 116 (2015), 112-131.
|
[6] |
N. Hayashi and P. I. Naumkin, Large time asymptotics of solutions for the modified KdV equation with a fifth order dispersive term, submitted to ARMA, 2015. |
[7] |
R. S. Smith,
Nonlinear Kelvin and continental-shelf waves, J. Fluid Mech., 52 (1972), 379-391.
|
[8] |
E. M. Stein and R. Shakarchi, Functional Analysis. Introduction to Further Topics in Analysis, Princeton Lectures in Analysis, 4. Princeton University Press, Princeton, NJ, 2011. ⅹⅷ+423 pp. |
[9] |
G. B. Whitham, Variational methods and applications to water waves, Hyperbolic Equations and Waves (Rencontres, Battelle Res. Inst., Seattle, WA, 1968), Springer, Berlin, 1970, pp. 153-172. |
[10] |
G. B. Whitham, Linear and Nonlinear Waves, Pure Appl. Math., Wiley, New York, 1974. |
show all references
References:
[1] |
M. V. Fedoryuk, Asymptotic Methods in Analysis, in Analysis. I. Integral representations and asymptotic methods, Encyclopaedia of Mathematical Sciences, 13. Springer-Verlag, Berlin, 1989. vi+238 pp. |
[2] |
N. Hayashi and E. Kaikina, Asymptotics for the third-order nonlinear Schrödinger equation in the critical case, to appear in MMAS. |
[3] |
N. Hayashi, J. Mendez-Navarro and P. Naumkin, Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case, submitted to JDE, 2016. |
[4] |
N. Hayashi and P. Naumkin,
Factorization technique for the fourth-order nonlinear Schrödinger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377.
|
[5] |
N. Hayashi and P. I. Naumkin,
Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Analysis, 116 (2015), 112-131.
|
[6] |
N. Hayashi and P. I. Naumkin, Large time asymptotics of solutions for the modified KdV equation with a fifth order dispersive term, submitted to ARMA, 2015. |
[7] |
R. S. Smith,
Nonlinear Kelvin and continental-shelf waves, J. Fluid Mech., 52 (1972), 379-391.
|
[8] |
E. M. Stein and R. Shakarchi, Functional Analysis. Introduction to Further Topics in Analysis, Princeton Lectures in Analysis, 4. Princeton University Press, Princeton, NJ, 2011. ⅹⅷ+423 pp. |
[9] |
G. B. Whitham, Variational methods and applications to water waves, Hyperbolic Equations and Waves (Rencontres, Battelle Res. Inst., Seattle, WA, 1968), Springer, Berlin, 1970, pp. 153-172. |
[10] |
G. B. Whitham, Linear and Nonlinear Waves, Pure Appl. Math., Wiley, New York, 1974. |
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