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October  2017, 10(5): 1095-1106. doi: 10.3934/dcdss.2017059

On concentration of semi-classical solitary waves for a generalized Kadomtsev-Petviashvili equation

Yuanhong Wei a,, , Yong Li a,b,c, and  Xue Yang a,b,

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

* Corresponding author: Yuanhong Wei

Received  November 2016 Revised  November 2016 Published  June 2017

Fund Project: Y. Wei is supported by NSFC(grant No. 11301209). Y. Li is supported by National Basic Research Program of China (grant No. 2013CB834100), NSFC(grant No. 11571065) and NSFC(grant No. 11171132). X. Yang is supported by NSFC(grant No. 11201173).

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 $\varepsilon \to 0$.

Keywords: Concentration, semi-classical, solitary wave, Kadomtsev-Petviashvili equation, critical point.
Mathematics Subject Classification: Primary: 35A15; Secondary: 35Q53.
Citation: Yuanhong Wei, Yong Li, Xue Yang. On concentration of semi-classical solitary waves for a generalized Kadomtsev-Petviashvili equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1095-1106. doi: 10.3934/dcdss.2017059
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.

[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. 

[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. 

[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.

[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. 

[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. 

[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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