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On concentration of semi-classical solitary waves for a generalized Kadomtsev-Petviashvili equation
a. | School of Mathematics, Jilin University, Changchun 130012, China |
b. | School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China |
c. | State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, China |
The present paper is concerned with semi-classical solitary wave solutions of a generalized Kadomtsev-Petviashvili equation in $\mathbb{R}^{2}$. Parameter $\varepsilon$ and potential $V(x,y)$ are included in the problem. The existence of the least energy solution is established for all $\varepsilon>0$ small. Moreover, we point out that these solutions converge to a least energy solution of the associated limit problem and concentrate to the minimum point of the potential as $\varepsilon \to 0$.
References:
[1] |
M. J. Ablowitz and P. A. Clarkson,
Solitons, Nonlinear Evolution Equations and Inverse Scattering, volume 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511623998. |
[2] |
M. J. Ablowitz and H. Segur,
Solitons and the Inverse Scattering Transform, volume 4 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa. , 1981. |
[3] |
D. Boris, T. Grava and C. Klein,
On critical behaviour in generalized Kadomtsev-Petviashvili equations, Physica D: Nonlinear Phenomena, 333 (2016), 157-170.
doi: 10.1016/j.physd.2016.01.011. |
[4] |
A. De Bouard and J.-C. Saut,
Solitary waves of generalized Kadomtsev-Petviashvili equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 14 (1997), 211-236.
doi: 10.1016/S0294-1449(97)80145-X. |
[5] |
J. Bourgain,
On the cauchy problem for the Kadomstev-Petviashvili equation, Geometric and Functional Analysis, 3 (1993), 315-341.
doi: 10.1007/BF01896259. |
[6] |
M. del Pino and P. L. Felmer,
Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[7] |
W. Ding and W. Ni,
On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283-308.
doi: 10.1007/BF00282336. |
[8] |
Y. Ding,
Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations, 249 (2010), 1015-1034.
doi: 10.1016/j.jde.2010.03.022. |
[9] |
Y. Ding and X. Liu,
Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differential Equations, 252 (2012), 4962-4987.
doi: 10.1016/j.jde.2012.01.023. |
[10] |
Y. Ding and B. Ruf,
Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal., 44 (2012), 3755-3785.
doi: 10.1137/110850670. |
[11] |
Y. Ding and T. Xu,
Localized concentration of semi-classical states for nonlinear Dirac equations, Arch. Ration. Mech. Anal., 216 (2015), 415-447.
doi: 10.1007/s00205-014-0811-4. |
[12] |
B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Soviet Physics Doklady, 15 (1970), 539. Google Scholar |
[13] |
C. Klein and J.-C. Saut,
A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Phys. D, 295/296 (2015), 46-65.
doi: 10.1016/j.physd.2014.12.004. |
[14] |
Y. Liu,
Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation, Trans. Amer. Math. Soc., 353 (2001), 191-208.
doi: 10.1090/S0002-9947-00-02465-X. |
[15] |
Y. Liu,
Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions, J. Differential Equations, 180 (2002), 153-170.
doi: 10.1006/jdeq.2001.4054. |
[16] |
Y. Liu and X.-P. Wang, Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation, Comm. Math. Phys., 183 (1997), 253-266. Google Scholar |
[17] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Global well-posedness for the KP-Ⅰ equation, Math. Ann., 324 (2002), 255-275.
doi: 10.1007/s00208-002-0338-0. |
[18] |
L. Molinet, J.-C. Saut and N. Tzvetkov,
Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-Ⅰ equation, Duke Math. J., 115 (2002), 353-384.
doi: 10.1215/S0012-7094-02-11525-7. |
[19] |
L. Molinet, J.C. Saut and N. Tzvetkov,
Remarks on the mass constraint for KP-type equations, SIAM J. Math. Anal., 39 (2007), 627-641.
doi: 10.1137/060654256. |
[20] |
J.-C. Saut,
Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J., 42 (1993), 1011-1026.
doi: 10.1512/iumj.1993.42.42047. |
[21] |
S. Ukai,
Local solutions of the Kadomtsev-Petviashvili equation, J. Fac. Sci. Univ. Tokyo, Sect. IA Math, 36 (1989), 193-209.
|
[22] |
X. Wang,
On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[23] |
M. Willem and Z.-Q. Wang, A multiplicity result for the generalized Kadomtsev-Petviashvili equation, Journal of the Juliusz Schauder Center, 7 (1996), 261-270. Google Scholar |
[24] |
M. Willem,
Minimax Theorems, volume 24. Birkhäuser Boston, Inc. , Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[25] |
W. Zou,
Solitary waves of the generalized Kadomtsev-Petviashvili equations, Appl. Math. Lett., 15 (2002), 35-39.
doi: 10.1016/S0893-9659(01)00089-1. |
show all references
References:
[1] |
M. J. Ablowitz and P. A. Clarkson,
Solitons, Nonlinear Evolution Equations and Inverse Scattering, volume 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511623998. |
[2] |
M. J. Ablowitz and H. Segur,
Solitons and the Inverse Scattering Transform, volume 4 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa. , 1981. |
[3] |
D. Boris, T. Grava and C. Klein,
On critical behaviour in generalized Kadomtsev-Petviashvili equations, Physica D: Nonlinear Phenomena, 333 (2016), 157-170.
doi: 10.1016/j.physd.2016.01.011. |
[4] |
A. De Bouard and J.-C. Saut,
Solitary waves of generalized Kadomtsev-Petviashvili equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 14 (1997), 211-236.
doi: 10.1016/S0294-1449(97)80145-X. |
[5] |
J. Bourgain,
On the cauchy problem for the Kadomstev-Petviashvili equation, Geometric and Functional Analysis, 3 (1993), 315-341.
doi: 10.1007/BF01896259. |
[6] |
M. del Pino and P. L. Felmer,
Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[7] |
W. Ding and W. Ni,
On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283-308.
doi: 10.1007/BF00282336. |
[8] |
Y. Ding,
Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations, 249 (2010), 1015-1034.
doi: 10.1016/j.jde.2010.03.022. |
[9] |
Y. Ding and X. Liu,
Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differential Equations, 252 (2012), 4962-4987.
doi: 10.1016/j.jde.2012.01.023. |
[10] |
Y. Ding and B. Ruf,
Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal., 44 (2012), 3755-3785.
doi: 10.1137/110850670. |
[11] |
Y. Ding and T. Xu,
Localized concentration of semi-classical states for nonlinear Dirac equations, Arch. Ration. Mech. Anal., 216 (2015), 415-447.
doi: 10.1007/s00205-014-0811-4. |
[12] |
B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Soviet Physics Doklady, 15 (1970), 539. Google Scholar |
[13] |
C. Klein and J.-C. Saut,
A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Phys. D, 295/296 (2015), 46-65.
doi: 10.1016/j.physd.2014.12.004. |
[14] |
Y. Liu,
Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation, Trans. Amer. Math. Soc., 353 (2001), 191-208.
doi: 10.1090/S0002-9947-00-02465-X. |
[15] |
Y. Liu,
Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions, J. Differential Equations, 180 (2002), 153-170.
doi: 10.1006/jdeq.2001.4054. |
[16] |
Y. Liu and X.-P. Wang, Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation, Comm. Math. Phys., 183 (1997), 253-266. Google Scholar |
[17] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Global well-posedness for the KP-Ⅰ equation, Math. Ann., 324 (2002), 255-275.
doi: 10.1007/s00208-002-0338-0. |
[18] |
L. Molinet, J.-C. Saut and N. Tzvetkov,
Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-Ⅰ equation, Duke Math. J., 115 (2002), 353-384.
doi: 10.1215/S0012-7094-02-11525-7. |
[19] |
L. Molinet, J.C. Saut and N. Tzvetkov,
Remarks on the mass constraint for KP-type equations, SIAM J. Math. Anal., 39 (2007), 627-641.
doi: 10.1137/060654256. |
[20] |
J.-C. Saut,
Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J., 42 (1993), 1011-1026.
doi: 10.1512/iumj.1993.42.42047. |
[21] |
S. Ukai,
Local solutions of the Kadomtsev-Petviashvili equation, J. Fac. Sci. Univ. Tokyo, Sect. IA Math, 36 (1989), 193-209.
|
[22] |
X. Wang,
On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[23] |
M. Willem and Z.-Q. Wang, A multiplicity result for the generalized Kadomtsev-Petviashvili equation, Journal of the Juliusz Schauder Center, 7 (1996), 261-270. Google Scholar |
[24] |
M. Willem,
Minimax Theorems, volume 24. Birkhäuser Boston, Inc. , Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[25] |
W. Zou,
Solitary waves of the generalized Kadomtsev-Petviashvili equations, Appl. Math. Lett., 15 (2002), 35-39.
doi: 10.1016/S0893-9659(01)00089-1. |
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