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November  2020, 40(11): 6135-6158. doi: 10.3934/dcds.2020273

Multitransition solutions for a generalized Frenkel-Kontorova model

Wen-Long Li 1,, and  Xiaojun Cui 2,

1. 

School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Wen-Long Li

Received  September 2019 Revised  May 2020 Published  July 2020

We study a generalized Frenkel-Kontorova model. Using minimal and Birkhoff solutions as building blocks, we construct a lot of homoclinic solutions and heteroclinic solutions for this generalized Frenkel-Kontorova model under gap conditions. These new solutions are not minimal and Birkhoff any more. We use constrained minimization method to prove our results.

Keywords: Multitransition solutions, Moser-Bangert theory, Frenkel-Kontorova model, constrained minimization method.
Mathematics Subject Classification: 70G75, 74G22, 74G35.
Citation: Wen-Long Li, Xiaojun Cui. Multitransition solutions for a generalized Frenkel-Kontorova model. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6135-6158. doi: 10.3934/dcds.2020273
References:
[1]

V. Bangert, The existence of gaps in minimal foliations, Aequationes Mathematicae, 34 (1987), 153-166.  doi: 10.1007/BF01830667.

[2]

V. Bangert, A uniqueness theorem for Zn-periodic variational problems, Comment. Math. Helv., 62 (1987), 511-531. 

[3]

V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 95-138.  doi: 10.1016/S0294-1449(16)30328-6.

[4]

U. Bessi, Slope-changing solutions of elliptic problems on $\bf R^n$, Nonlinear Anal., 68 (2008), 3923-3947.  doi: 10.1016/j.na.2007.04.031.

[5]

L. A. Caffarelli and R. de la Llave, Planelike minimizers in periodic media, Comm. Pure Appl. Math., 54 (2001), 1403-1441.  doi: 10.1002/cpa.10008.

[6]

L. A. Caffarelli and R. de la Llave, Interfaces of ground states in Ising models with periodic coefficients, J. Stat. Phys., 118 (2005), 687-719.  doi: 10.1007/s10955-004-8825-1.

[7]

M. Cozzi, S. Dipierro and E. Valdinoci, Planelike interfaces in long-range Ising models and connections with nonlocal minimal surfaces, J. Stat. Phys., 167 (2017), 1401-1451.  doi: 10.1007/s10955-017-1783-1.

[8]

R. de La Llave and E. Valdinoci, Critical points inside the gaps of ground state laminations for some models in statistical mechanics, J. Stat. Phys., 129 (2007), 81-119.  doi: 10.1007/s10955-007-9345-6.

[9]

R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309-1344.  doi: 10.1016/j.anihpc.2008.11.002.

[10]

R. de La Llave and E. Valdinoci, Ground states and critical points for Aubry-Mather theory in statistical mechanics, J. Nonlinear Sci., 20 (2010), 153-218.  doi: 10.1007/s00332-009-9055-0.

[11]

W.-L. Li and X. Cui, Heteroclinic solutions for a Frenkel-Kontorova model by minimization methods of Rabinowitz and Stredulinsky, J. Differential Equations, 268 (2020), 1106-1155.  doi: 10.1016/j.jde.2019.08.048.

[12]

J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.  doi: 10.5802/aif.1377.

[13]

X.-Q. Miao, W.-X. Qin and Y.-N. Wang, Secondary invariants of Birkhoff minimizers and heteroclinic orbits, J. Differential Equations, 260 (2016), 1522-1557.  doi: 10.1016/j.jde.2015.09.039.

[14]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272.  doi: 10.1016/S0294-1449(16)30387-0.

[15]

B. Mramor and B. Rink, Ghost circles in lattice Aubry-Mather theory, J. Differential Equations, 252 (2012), 3163-3208.  doi: 10.1016/j.jde.2011.11.023.

[16]

P. H. Rabinowitz, Single and multitransition solutions for a family of semilinear elliptic PDE's, Milan J. Math., 79 (2011), 113-127.  doi: 10.1007/s00032-011-0139-6.

[17]

P. H. Rabinowitz and Ed Stredulinsky, Mixed states for an Allen-Cahn type equation, Comm. Pure Appl. Math., 56 (2003), 1078-1134.  doi: 10.1002/cpa.10087.

[18]

P. H. Rabinowitz and Ed Stredulinsky, Mixed states for an Allen-Cahn type equation, II, Calc. Var. Partial Differential Equation, 21 (2004), 157-207.  doi: 10.1007/s00526-003-0251-8.

[19]

P. H. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 673-688.  doi: 10.1016/j.anihpc.2003.10.002.

[20]

P. H. Rabinowitz and Ed Stredulinsky, On some results of Moser and of Bangert, II, Adv. Nonlinear Stud., 4 (2004), 377-396.  doi: 10.1515/ans-2004-0402.

[21]

P. H. Rabinowitz and Ed Stredulinsky, Infinite transition solutions for a class of Allen-Cahn model equations, J. Fixed Point Theory Appl., 4 (2008), 247-262.  doi: 10.1007/s11784-008-0091-4.

[22]

P. H. Rabinowitz and Ed Stredulinsky, On a class of infinite transition solutions for an Allen-Cahn model equation, Discrete Contin. Dyn. Syst., 21 (2008), 319-332.  doi: 10.3934/dcds.2008.21.319.

[23]

P. H. Rabinowitz and E. W. Stredulinsky, Extensions of Moser-Bangert Theory: Locally Minimal Solutions, Progress in Nonlinear Differential Equations and their Applications, 81, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8117-3.

[24]

M. Torres, Plane-like minimal surfaces in periodic media with exclusions, SIAM J. Math. Anal., 36 (2004), 523-551.  doi: 10.1137/S0036141001399970.

[25]

E. Valdinoci, Plane-like minimizers in periodic media: jet flows and Ginzburg-Landau-type functionals, J. Reine Angew. Math., 574 (2004), 147-185.  doi: 10.1515/crll.2004.068.

show all references

References:
[1]

V. Bangert, The existence of gaps in minimal foliations, Aequationes Mathematicae, 34 (1987), 153-166.  doi: 10.1007/BF01830667.

[2]

V. Bangert, A uniqueness theorem for Zn-periodic variational problems, Comment. Math. Helv., 62 (1987), 511-531. 

[3]

V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 95-138.  doi: 10.1016/S0294-1449(16)30328-6.

[4]

U. Bessi, Slope-changing solutions of elliptic problems on $\bf R^n$, Nonlinear Anal., 68 (2008), 3923-3947.  doi: 10.1016/j.na.2007.04.031.

[5]

L. A. Caffarelli and R. de la Llave, Planelike minimizers in periodic media, Comm. Pure Appl. Math., 54 (2001), 1403-1441.  doi: 10.1002/cpa.10008.

[6]

L. A. Caffarelli and R. de la Llave, Interfaces of ground states in Ising models with periodic coefficients, J. Stat. Phys., 118 (2005), 687-719.  doi: 10.1007/s10955-004-8825-1.

[7]

M. Cozzi, S. Dipierro and E. Valdinoci, Planelike interfaces in long-range Ising models and connections with nonlocal minimal surfaces, J. Stat. Phys., 167 (2017), 1401-1451.  doi: 10.1007/s10955-017-1783-1.

[8]

R. de La Llave and E. Valdinoci, Critical points inside the gaps of ground state laminations for some models in statistical mechanics, J. Stat. Phys., 129 (2007), 81-119.  doi: 10.1007/s10955-007-9345-6.

[9]

R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309-1344.  doi: 10.1016/j.anihpc.2008.11.002.

[10]

R. de La Llave and E. Valdinoci, Ground states and critical points for Aubry-Mather theory in statistical mechanics, J. Nonlinear Sci., 20 (2010), 153-218.  doi: 10.1007/s00332-009-9055-0.

[11]

W.-L. Li and X. Cui, Heteroclinic solutions for a Frenkel-Kontorova model by minimization methods of Rabinowitz and Stredulinsky, J. Differential Equations, 268 (2020), 1106-1155.  doi: 10.1016/j.jde.2019.08.048.

[12]

J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.  doi: 10.5802/aif.1377.

[13]

X.-Q. Miao, W.-X. Qin and Y.-N. Wang, Secondary invariants of Birkhoff minimizers and heteroclinic orbits, J. Differential Equations, 260 (2016), 1522-1557.  doi: 10.1016/j.jde.2015.09.039.

[14]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272.  doi: 10.1016/S0294-1449(16)30387-0.

[15]

B. Mramor and B. Rink, Ghost circles in lattice Aubry-Mather theory, J. Differential Equations, 252 (2012), 3163-3208.  doi: 10.1016/j.jde.2011.11.023.

[16]

P. H. Rabinowitz, Single and multitransition solutions for a family of semilinear elliptic PDE's, Milan J. Math., 79 (2011), 113-127.  doi: 10.1007/s00032-011-0139-6.

[17]

P. H. Rabinowitz and Ed Stredulinsky, Mixed states for an Allen-Cahn type equation, Comm. Pure Appl. Math., 56 (2003), 1078-1134.  doi: 10.1002/cpa.10087.

[18]

P. H. Rabinowitz and Ed Stredulinsky, Mixed states for an Allen-Cahn type equation, II, Calc. Var. Partial Differential Equation, 21 (2004), 157-207.  doi: 10.1007/s00526-003-0251-8.

[19]

P. H. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 673-688.  doi: 10.1016/j.anihpc.2003.10.002.

[20]

P. H. Rabinowitz and Ed Stredulinsky, On some results of Moser and of Bangert, II, Adv. Nonlinear Stud., 4 (2004), 377-396.  doi: 10.1515/ans-2004-0402.

[21]

P. H. Rabinowitz and Ed Stredulinsky, Infinite transition solutions for a class of Allen-Cahn model equations, J. Fixed Point Theory Appl., 4 (2008), 247-262.  doi: 10.1007/s11784-008-0091-4.

[22]

P. H. Rabinowitz and Ed Stredulinsky, On a class of infinite transition solutions for an Allen-Cahn model equation, Discrete Contin. Dyn. Syst., 21 (2008), 319-332.  doi: 10.3934/dcds.2008.21.319.

[23]

P. H. Rabinowitz and E. W. Stredulinsky, Extensions of Moser-Bangert Theory: Locally Minimal Solutions, Progress in Nonlinear Differential Equations and their Applications, 81, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8117-3.

[24]

M. Torres, Plane-like minimal surfaces in periodic media with exclusions, SIAM J. Math. Anal., 36 (2004), 523-551.  doi: 10.1137/S0036141001399970.

[25]

E. Valdinoci, Plane-like minimizers in periodic media: jet flows and Ginzburg-Landau-type functionals, J. Reine Angew. Math., 574 (2004), 147-185.  doi: 10.1515/crll.2004.068.

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