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Averaging principles for the Swift-Hohenberg equation
Multi-spikes solutions for a system of coupled elliptic equations with quadratic nonlinearity
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
$ \begin{equation*} \begin{cases} -\varepsilon^{2}\Delta v+P(x)v = \mu vw, &x\in{\mathbb{R}}^{N},\\ -\varepsilon^{2}\Delta w+Q(x)w = \frac{\mu}{2} v^{2}+\gamma w^{2}, &x\in{\mathbb{R}}^{N}, \end{cases} \end{equation*} $ |
$ P(x) $ |
$ Q(x) $ |
$ x_{0} $ |
$ 1\leq k\leq N+1 $ |
$ k $ |
$ x_{0} $ |
$ \varepsilon $ |
References:
[1] |
A. Ambrosetti, M. Badiale and S. Cingolani,
Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300.
doi: 10.1007/s002050050067. |
[2] |
A. Ambrosetti and E. Colorado,
Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.
doi: 10.1016/j.crma.2006.01.024. |
[3] |
A. Ambrosetti and E. Colorado,
Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[4] |
A. Ambrosetti, E. Colorado and D. Ruiz,
Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 30 (2007), 85-112.
doi: 10.1007/s00526-006-0079-0. |
[5] |
T. Bartsch, N. Dancer and Z. Q. Wang,
A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[6] |
A. V. Buryak and Y. S. Kivshar,
Solitons due to second harmonic generation, Phys. Lett. A, 197 (1995), 407-412.
doi: 10.1016/0375-9601(94)00989-3. |
[7] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[8] |
A. V. Buryak, P. Di Trapani, D. V. Skryabin and S. Trillo,
Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep., 370 (2002), 63-235.
doi: 10.1016/S0370-1573(02)00196-5. |
[9] |
T. Bartsch and Z. Q. Wang,
Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207.
|
[10] |
T. Bartsch, Z. Q. Wang and J. Wei,
Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[11] |
T. Cazenave, Semilinear Schrödinger Equations, vol.10 of Courant Lecture Notes in Mathematics, New York University Courant Insitute of Mathematical Sciences, New York, 2003.
doi: 10.1090/cln/010. |
[12] |
D. Cao, E. S. Noussair and S. Yan,
Solutions with multiple "peaks" for nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 235-264.
doi: 10.1017/S030821050002134X. |
[13] |
M. Conti, S. Terracini and G. Verzini,
Nehari's problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 871-888.
doi: 10.1016/S0294-1449(02)00104-X. |
[14] |
M. Del Pino and P. Felmer,
Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 127-149.
doi: 10.1016/S0294-1449(97)89296-7. |
[15] |
E. N. Dancer and S. Yan,
Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262.
doi: 10.2140/pjm.1999.189.241. |
[16] |
E. N. Dancer, J. Wei and T. Weth,
A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[17] |
C. Gui and J. Wei,
Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.
doi: 10.1016/S0022-0396(99)80016-3. |
[18] |
C. Gui and J. Wei,
On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math., 52 (2000), 522-538.
doi: 10.4153/CJM-2000-024-x. |
[19] |
Y. Li,
On a singularly perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980.
|
[20] |
Y. Li and L. Nirenberg,
The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490.
doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.3.CO;2-Q. |
[21] |
C. Lin, W. Ni and I. Takagi,
Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[22] |
Z. Liu and Z. Q. Wang,
Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.
doi: 10.1515/ans-2010-0109. |
[23] |
E. Montefusco, B. Pellacci and M. Squassina,
Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.
doi: 10.4171/JEMS/103. |
[24] |
B. Noris, H. Tavares, S. Terracini and G. Verzini,
Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.
doi: 10.1002/cpa.20309. |
[25] |
E. S. Noussair and S. Yan,
On positive multipeak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227.
doi: 10.1112/S002461070000898X. |
[26] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
|
[27] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[28] |
C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, vol.39 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999. |
[29] |
R. Tian and Z. Q. Wang,
Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
|
[30] |
J. Wei and T. Weth,
Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106.
doi: 10.1007/s00205-008-0121-9. |
[31] |
C. Wang and J. Zhou, Infinitely many solitary waves due to the second-harmonic generation in quadratic media, bo be appeared in Acta Math. Sci. Ser. B (Engl. Ed.) (2020, no.1). |
[32] |
A. C. Yew,
Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations, 173 (2001), 92-137.
doi: 10.1006/jdeq.2000.3922. |
[33] |
A. C. Yew, A. R. Champneys and P. J. McKenna,
Multiple solitary waves due to second-harmonic generation in quadratic media, J. Nonlinear Sci., 9 (1999), 33-52.
doi: 10.1007/s003329900063. |
[34] |
L. Zhao, F. Zhao and J. Shi,
Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calc. Var. Partial Differential Equations, 54 (2015), 2657-2691.
doi: 10.1007/s00526-015-0879-1. |
show all references
References:
[1] |
A. Ambrosetti, M. Badiale and S. Cingolani,
Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300.
doi: 10.1007/s002050050067. |
[2] |
A. Ambrosetti and E. Colorado,
Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.
doi: 10.1016/j.crma.2006.01.024. |
[3] |
A. Ambrosetti and E. Colorado,
Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[4] |
A. Ambrosetti, E. Colorado and D. Ruiz,
Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 30 (2007), 85-112.
doi: 10.1007/s00526-006-0079-0. |
[5] |
T. Bartsch, N. Dancer and Z. Q. Wang,
A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[6] |
A. V. Buryak and Y. S. Kivshar,
Solitons due to second harmonic generation, Phys. Lett. A, 197 (1995), 407-412.
doi: 10.1016/0375-9601(94)00989-3. |
[7] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[8] |
A. V. Buryak, P. Di Trapani, D. V. Skryabin and S. Trillo,
Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep., 370 (2002), 63-235.
doi: 10.1016/S0370-1573(02)00196-5. |
[9] |
T. Bartsch and Z. Q. Wang,
Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207.
|
[10] |
T. Bartsch, Z. Q. Wang and J. Wei,
Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.
doi: 10.1007/s11784-007-0033-6. |
[11] |
T. Cazenave, Semilinear Schrödinger Equations, vol.10 of Courant Lecture Notes in Mathematics, New York University Courant Insitute of Mathematical Sciences, New York, 2003.
doi: 10.1090/cln/010. |
[12] |
D. Cao, E. S. Noussair and S. Yan,
Solutions with multiple "peaks" for nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 235-264.
doi: 10.1017/S030821050002134X. |
[13] |
M. Conti, S. Terracini and G. Verzini,
Nehari's problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 871-888.
doi: 10.1016/S0294-1449(02)00104-X. |
[14] |
M. Del Pino and P. Felmer,
Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 127-149.
doi: 10.1016/S0294-1449(97)89296-7. |
[15] |
E. N. Dancer and S. Yan,
Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262.
doi: 10.2140/pjm.1999.189.241. |
[16] |
E. N. Dancer, J. Wei and T. Weth,
A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[17] |
C. Gui and J. Wei,
Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.
doi: 10.1016/S0022-0396(99)80016-3. |
[18] |
C. Gui and J. Wei,
On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math., 52 (2000), 522-538.
doi: 10.4153/CJM-2000-024-x. |
[19] |
Y. Li,
On a singularly perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980.
|
[20] |
Y. Li and L. Nirenberg,
The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490.
doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.3.CO;2-Q. |
[21] |
C. Lin, W. Ni and I. Takagi,
Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[22] |
Z. Liu and Z. Q. Wang,
Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.
doi: 10.1515/ans-2010-0109. |
[23] |
E. Montefusco, B. Pellacci and M. Squassina,
Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.
doi: 10.4171/JEMS/103. |
[24] |
B. Noris, H. Tavares, S. Terracini and G. Verzini,
Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.
doi: 10.1002/cpa.20309. |
[25] |
E. S. Noussair and S. Yan,
On positive multipeak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227.
doi: 10.1112/S002461070000898X. |
[26] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
|
[27] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[28] |
C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, vol.39 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999. |
[29] |
R. Tian and Z. Q. Wang,
Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
|
[30] |
J. Wei and T. Weth,
Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106.
doi: 10.1007/s00205-008-0121-9. |
[31] |
C. Wang and J. Zhou, Infinitely many solitary waves due to the second-harmonic generation in quadratic media, bo be appeared in Acta Math. Sci. Ser. B (Engl. Ed.) (2020, no.1). |
[32] |
A. C. Yew,
Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations, 173 (2001), 92-137.
doi: 10.1006/jdeq.2000.3922. |
[33] |
A. C. Yew, A. R. Champneys and P. J. McKenna,
Multiple solitary waves due to second-harmonic generation in quadratic media, J. Nonlinear Sci., 9 (1999), 33-52.
doi: 10.1007/s003329900063. |
[34] |
L. Zhao, F. Zhao and J. Shi,
Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calc. Var. Partial Differential Equations, 54 (2015), 2657-2691.
doi: 10.1007/s00526-015-0879-1. |
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