August  2013, 7(3): 1051-1074. doi: 10.3934/ipi.2013.7.1051

The Gaussian beam method for the wigner equation with discontinuous potentials

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084

2. 

Department of Mathematics, Institute of Nature Science, and Ministry of Education Key Laboratory in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240, China

3. 

Department of Mathematics, Institute of Natural Sciences, and MOE Key Lab in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240

Received  June 2012 Revised  January 2013 Published  September 2013

For the Wigner equation with discontinuous potentials, a phase space Gaussian beam (PSGB) summation method is proposed in this paper. We first derive the equations satisfied by the parameters for PSGBs and establish the relations for parameters of the Gaussian beams between the physical space (GBs) and the phase space, which motivates an efficient initial data preparation thus a reduced computational cost than previous method in the literature. The method consists of three steps: 1) Decompose the initial value of the wave function into the sum of GBs and use the parameter relations to prepare the initial values of PSGBs; 2) Solve the evolution equations for each PSGB; 3) Sum all the PSGBs to construct the approximate solution of the Wigner equation. Additionally, in order to connect PSGBs at the discontinuous points of the potential, we provide interface conditions for a single phase space Gaussian beam. Numerical examples are given to verify the validity and accuracy of method.
Citation: Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems & Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051
References:
[1]

J. Akian, R. Alexandre and S. Bougacha, A Gaussian beam approach for computing Wigner measures in convex domains,, Kinetic and Related Models, 4 (2011), 589.  doi: 10.3934/krm.2011.4.589.  Google Scholar

[2]

G. Ariel, B. Engquist, N. Tanushev and R. Tsai, Gaussian beam decomposition of high frequency wave fields using expectation-maximization,, Journal of Computational Physics, 230 (2011), 2303.   Google Scholar

[3]

A. Arnold and F. Nier, Numerical analysis of the deterministic particle method applied to the wigner equation,, Mathematics of Computation, 58 (1992), 645.  doi: 10.1090/S0025-5718-1992-1122055-5.  Google Scholar

[4]

A. Arnold and C. Ringhofer, Operator splitting methods applied to spectral discretizations of quantum transport equations,, SIAM J. Numer. Anal., 32 (1995), 1876.  doi: 10.1137/0732084.  Google Scholar

[5]

A. Arnold, H. Lange and P. Zweifel, A discrete-velocity, stationary Wigner equation,, J. Math. Phys., 41 (2000), 7167.  doi: 10.1063/1.1318732.  Google Scholar

[6]

A. Arnold, Mathematical properties of quantum evolution equations,, In, 1946 (2008), 45.   Google Scholar

[7]

W. Bao, S. Jin and P. A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime,, J. Comput. Phys., 175 (2002), 487.  doi: 10.1006/jcph.2001.6956.  Google Scholar

[8]

S. Bougacha, J. Akian and R. Alexandre, Gaussian beam summation for the wave equation in a convex domain,, Commun. Math. Sci., 7 (2009), 973.   Google Scholar

[9]

L. Dieci and T. Eirola, Positive definiteness in the numerical solution of Riccati differential equations,, Numer. Math., 67 (1994), 303.  doi: 10.1007/s002110050030.  Google Scholar

[10]

W. Frensley, Wigner-function model of a resonant-tunneling semiconductor device,, Phys. Rev. B, 36 (1987), 1570.  doi: 10.1103/PhysRevB.36.1570.  Google Scholar

[11]

P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms,, Comm. Pure Appl. Math., 50 (1997), 323.   Google Scholar

[12]

T. Goudon, Analysis of a semidiscrete version of theWigner equation,, SIAM J. Numer. Anal., 40 (2002), 2007.  doi: 10.1137/S0036142901388366.  Google Scholar

[13]

H. Guo and R. Chen, Short-time Chebyshev propagator for the Liouville-von Neumann equation,, Journal of Chemical Physics, 110 (1999), 6626.  doi: 10.1063/1.478570.  Google Scholar

[14]

E. J. Heller, Time-dependent approach to semiclassical dynamics,, The Journal of Chemical Physics, 62 (1975), 1544.  doi: 10.1063/1.430620.  Google Scholar

[15]

I. Horenko, B. Schmidt and C. Schütte, Multdimensional classical Liouville dynamics with quantum initial conditions,, Journal of Chermical Physics, 117 (2002), 4643.  doi: 10.1063/1.1498467.  Google Scholar

[16]

N. Kluksdahl, A. Kriman, D. Ferry and C. Ringhofer, Self-consistent study of the resonant tunneling diode,, Phys. Rev. B., 39 (1989), 7720.  doi: 10.1103/PhysRevB.39.7720.  Google Scholar

[17]

S. Jin, P. Markowich and C. Sparber, Mathematical and computational methods for semiclassical Schrödinger equations,, Acta Numerica, 20 (2011), 121.  doi: 10.1017/S0962492911000031.  Google Scholar

[18]

S. Jin, D. Wei and D. Yin, Gaussian beam methods for the Schrödinger equation with discontinuous potentials,, submit to Journal of Computational and Applied Mathematics., ().   Google Scholar

[19]

S. Jin, H. Wu and X. Yang, Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations,, Communications in Mathematical Sciences, 6 (2008), 995.   Google Scholar

[20]

H. Kosina and M. Nedjalkov, Wigner function-based device modeling,, In, 10 (2006), 731.   Google Scholar

[21]

S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime,, Journal of Computational Physics, 228 (2009), 2951.  doi: 10.1016/j.jcp.2009.01.007.  Google Scholar

[22]

S. Leung, J. Qian and R. Burridge, Eulerian Gaussian beams for high-frequency wave propagation,, Geophysics, 72 (2007), 61.  doi: 10.1190/1.2752136.  Google Scholar

[23]

P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Rev. Mat. Iberoamericana, 9 (1993), 553.  doi: 10.4171/RMI/143.  Google Scholar

[24]

H. Liu, O. Runborg and N. M. Tanushev, Error Estimates for Gaussian Beam Superpositions,, Math. Comp., 82 (2013), 919.   Google Scholar

[25]

P. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[26]

P. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations,, Math. Meth. Appl. Sci., 11 (1989), 459.  doi: 10.1002/mma.1670110404.  Google Scholar

[27]

M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition,, Wave motion, 47 (2010), 421.  doi: 10.1016/j.wavemoti.2010.02.001.  Google Scholar

[28]

M. Nedjalkov, R. Kosik, H. Kosina and S. Selberherr, Wigner transport through tunneling structures-scattering interpretation of the potential operator,, In, (2002), 187.  doi: 10.1109/SISPAD.2002.1034548.  Google Scholar

[29]

M. Nedjalkov, D. Vasileska, D. Ferry, C. Jacoboni, C. Ringhofer, I. Dimov and V. Palanovski, Wigner transport models of the electron-phonon kinetics in quantum wires,, Phys. Rev. B., 74 (2006).  doi: 10.1103/PhysRevB.74.035311.  Google Scholar

[30]

A. Norris, S. White and J. Schrieffer, Gaussian wave packets in inhomogeneous media with curved interfaces,, Pro. R. Soc. Lond. A, 412 (1987), 93.  doi: 10.1098/rspa.1987.0082.  Google Scholar

[31]

M. M. Popov, A new method of computation of wave fields using Gaussian beams,, Wave Motion, 4 (1982), 85.  doi: 10.1016/0165-2125(82)90016-6.  Google Scholar

[32]

J. Qian and L. Ying, Fast Gaussian wavepacket transforms and Gaussian beams for the Schröinger equation,, J. Comp. Phys., 229 (2010), 7848.  doi: 10.1016/j.jcp.2010.06.043.  Google Scholar

[33]

J. Ralston, Gaussian beams and the propagation of singularities,, Studies in PDEs, 23 (1982), 206.   Google Scholar

[34]

U. Ravaioli, M. Osman, W. Pötz, N. Kluksdahl and D. Ferry, Investigation of ballistic transport through resonant-tunnelling quantum wells using Wigner function approach,, Physica B., 134 (1985), 36.  doi: 10.1016/0378-4363(85)90317-1.  Google Scholar

[35]

C. Ringhofer, A spectral method for the numerical simulation of quantum tunneling phenomena,, SIAM J. Numer. Anal., 27 (1990), 32.  doi: 10.1137/0727003.  Google Scholar

[36]

C. Ringhofer, Computational methods for semiclassical and quantum transport in semiconductor devices,, Acta Numerica., 6 (1997), 485.  doi: 10.1017/S0962492900002762.  Google Scholar

[37]

D. Robert, On the Herman-Kluk seimiclassical approximation,, Reviews in Mathematical Physics, 22 (2010), 1123.  doi: 10.1142/S0129055X1000417X.  Google Scholar

[38]

L. Shifren and D. Ferry, A Wigner function based ensemble Monte Carlo approach for accurate incorporation of quantum effects in device simulation,, J. Comp. Electr., 1 (2002), 55.   Google Scholar

[39]

V. Sverdlov, E. Ungersboeck, H. Kosina and S. Selberherr, Current transport models for nanoscale semiconductor devices,, Materials Sci. Engin. R., 58 (2008), 228.  doi: 10.1016/j.mser.2007.11.001.  Google Scholar

[40]

N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields,, J. Comput. Phys., 228 (2009), 8856.  doi: 10.1016/j.jcp.2009.08.028.  Google Scholar

[41]

N. M. Tanushev, J. L. Qian and J. Ralston, Mountain waves and Gaussian beams,, SIAM Multiscale Modeling and Simulation, 6 (2007), 688.  doi: 10.1137/060673667.  Google Scholar

[42]

N. M. Tanushev, Superpositions and higher order Gaussian beams,, Commun. Math. Sci., 6 (2008), 449.   Google Scholar

[43]

E. Wigner, On the quantum correction for the thermodynamic equilibrium,, Phys. Rev., 40 (1932), 749.   Google Scholar

[44]

D. Yin and C. Zheng, Gaussian beam formulation and interface conditions for the one-dimensional linear Schödinger equation,, Wave Motion, 48 (2011), 310.  doi: 10.1016/j.wavemoti.2010.11.006.  Google Scholar

[45]

D. Yin and C. Zheng, Composite coherent states approximation for one-dimensional multi-phased wave functions,, Communications in Computational Physics, 11 (2012), 951.  doi: 10.4208/cicp.101010.250511a.  Google Scholar

show all references

References:
[1]

J. Akian, R. Alexandre and S. Bougacha, A Gaussian beam approach for computing Wigner measures in convex domains,, Kinetic and Related Models, 4 (2011), 589.  doi: 10.3934/krm.2011.4.589.  Google Scholar

[2]

G. Ariel, B. Engquist, N. Tanushev and R. Tsai, Gaussian beam decomposition of high frequency wave fields using expectation-maximization,, Journal of Computational Physics, 230 (2011), 2303.   Google Scholar

[3]

A. Arnold and F. Nier, Numerical analysis of the deterministic particle method applied to the wigner equation,, Mathematics of Computation, 58 (1992), 645.  doi: 10.1090/S0025-5718-1992-1122055-5.  Google Scholar

[4]

A. Arnold and C. Ringhofer, Operator splitting methods applied to spectral discretizations of quantum transport equations,, SIAM J. Numer. Anal., 32 (1995), 1876.  doi: 10.1137/0732084.  Google Scholar

[5]

A. Arnold, H. Lange and P. Zweifel, A discrete-velocity, stationary Wigner equation,, J. Math. Phys., 41 (2000), 7167.  doi: 10.1063/1.1318732.  Google Scholar

[6]

A. Arnold, Mathematical properties of quantum evolution equations,, In, 1946 (2008), 45.   Google Scholar

[7]

W. Bao, S. Jin and P. A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime,, J. Comput. Phys., 175 (2002), 487.  doi: 10.1006/jcph.2001.6956.  Google Scholar

[8]

S. Bougacha, J. Akian and R. Alexandre, Gaussian beam summation for the wave equation in a convex domain,, Commun. Math. Sci., 7 (2009), 973.   Google Scholar

[9]

L. Dieci and T. Eirola, Positive definiteness in the numerical solution of Riccati differential equations,, Numer. Math., 67 (1994), 303.  doi: 10.1007/s002110050030.  Google Scholar

[10]

W. Frensley, Wigner-function model of a resonant-tunneling semiconductor device,, Phys. Rev. B, 36 (1987), 1570.  doi: 10.1103/PhysRevB.36.1570.  Google Scholar

[11]

P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms,, Comm. Pure Appl. Math., 50 (1997), 323.   Google Scholar

[12]

T. Goudon, Analysis of a semidiscrete version of theWigner equation,, SIAM J. Numer. Anal., 40 (2002), 2007.  doi: 10.1137/S0036142901388366.  Google Scholar

[13]

H. Guo and R. Chen, Short-time Chebyshev propagator for the Liouville-von Neumann equation,, Journal of Chemical Physics, 110 (1999), 6626.  doi: 10.1063/1.478570.  Google Scholar

[14]

E. J. Heller, Time-dependent approach to semiclassical dynamics,, The Journal of Chemical Physics, 62 (1975), 1544.  doi: 10.1063/1.430620.  Google Scholar

[15]

I. Horenko, B. Schmidt and C. Schütte, Multdimensional classical Liouville dynamics with quantum initial conditions,, Journal of Chermical Physics, 117 (2002), 4643.  doi: 10.1063/1.1498467.  Google Scholar

[16]

N. Kluksdahl, A. Kriman, D. Ferry and C. Ringhofer, Self-consistent study of the resonant tunneling diode,, Phys. Rev. B., 39 (1989), 7720.  doi: 10.1103/PhysRevB.39.7720.  Google Scholar

[17]

S. Jin, P. Markowich and C. Sparber, Mathematical and computational methods for semiclassical Schrödinger equations,, Acta Numerica, 20 (2011), 121.  doi: 10.1017/S0962492911000031.  Google Scholar

[18]

S. Jin, D. Wei and D. Yin, Gaussian beam methods for the Schrödinger equation with discontinuous potentials,, submit to Journal of Computational and Applied Mathematics., ().   Google Scholar

[19]

S. Jin, H. Wu and X. Yang, Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations,, Communications in Mathematical Sciences, 6 (2008), 995.   Google Scholar

[20]

H. Kosina and M. Nedjalkov, Wigner function-based device modeling,, In, 10 (2006), 731.   Google Scholar

[21]

S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime,, Journal of Computational Physics, 228 (2009), 2951.  doi: 10.1016/j.jcp.2009.01.007.  Google Scholar

[22]

S. Leung, J. Qian and R. Burridge, Eulerian Gaussian beams for high-frequency wave propagation,, Geophysics, 72 (2007), 61.  doi: 10.1190/1.2752136.  Google Scholar

[23]

P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Rev. Mat. Iberoamericana, 9 (1993), 553.  doi: 10.4171/RMI/143.  Google Scholar

[24]

H. Liu, O. Runborg and N. M. Tanushev, Error Estimates for Gaussian Beam Superpositions,, Math. Comp., 82 (2013), 919.   Google Scholar

[25]

P. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[26]

P. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations,, Math. Meth. Appl. Sci., 11 (1989), 459.  doi: 10.1002/mma.1670110404.  Google Scholar

[27]

M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition,, Wave motion, 47 (2010), 421.  doi: 10.1016/j.wavemoti.2010.02.001.  Google Scholar

[28]

M. Nedjalkov, R. Kosik, H. Kosina and S. Selberherr, Wigner transport through tunneling structures-scattering interpretation of the potential operator,, In, (2002), 187.  doi: 10.1109/SISPAD.2002.1034548.  Google Scholar

[29]

M. Nedjalkov, D. Vasileska, D. Ferry, C. Jacoboni, C. Ringhofer, I. Dimov and V. Palanovski, Wigner transport models of the electron-phonon kinetics in quantum wires,, Phys. Rev. B., 74 (2006).  doi: 10.1103/PhysRevB.74.035311.  Google Scholar

[30]

A. Norris, S. White and J. Schrieffer, Gaussian wave packets in inhomogeneous media with curved interfaces,, Pro. R. Soc. Lond. A, 412 (1987), 93.  doi: 10.1098/rspa.1987.0082.  Google Scholar

[31]

M. M. Popov, A new method of computation of wave fields using Gaussian beams,, Wave Motion, 4 (1982), 85.  doi: 10.1016/0165-2125(82)90016-6.  Google Scholar

[32]

J. Qian and L. Ying, Fast Gaussian wavepacket transforms and Gaussian beams for the Schröinger equation,, J. Comp. Phys., 229 (2010), 7848.  doi: 10.1016/j.jcp.2010.06.043.  Google Scholar

[33]

J. Ralston, Gaussian beams and the propagation of singularities,, Studies in PDEs, 23 (1982), 206.   Google Scholar

[34]

U. Ravaioli, M. Osman, W. Pötz, N. Kluksdahl and D. Ferry, Investigation of ballistic transport through resonant-tunnelling quantum wells using Wigner function approach,, Physica B., 134 (1985), 36.  doi: 10.1016/0378-4363(85)90317-1.  Google Scholar

[35]

C. Ringhofer, A spectral method for the numerical simulation of quantum tunneling phenomena,, SIAM J. Numer. Anal., 27 (1990), 32.  doi: 10.1137/0727003.  Google Scholar

[36]

C. Ringhofer, Computational methods for semiclassical and quantum transport in semiconductor devices,, Acta Numerica., 6 (1997), 485.  doi: 10.1017/S0962492900002762.  Google Scholar

[37]

D. Robert, On the Herman-Kluk seimiclassical approximation,, Reviews in Mathematical Physics, 22 (2010), 1123.  doi: 10.1142/S0129055X1000417X.  Google Scholar

[38]

L. Shifren and D. Ferry, A Wigner function based ensemble Monte Carlo approach for accurate incorporation of quantum effects in device simulation,, J. Comp. Electr., 1 (2002), 55.   Google Scholar

[39]

V. Sverdlov, E. Ungersboeck, H. Kosina and S. Selberherr, Current transport models for nanoscale semiconductor devices,, Materials Sci. Engin. R., 58 (2008), 228.  doi: 10.1016/j.mser.2007.11.001.  Google Scholar

[40]

N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields,, J. Comput. Phys., 228 (2009), 8856.  doi: 10.1016/j.jcp.2009.08.028.  Google Scholar

[41]

N. M. Tanushev, J. L. Qian and J. Ralston, Mountain waves and Gaussian beams,, SIAM Multiscale Modeling and Simulation, 6 (2007), 688.  doi: 10.1137/060673667.  Google Scholar

[42]

N. M. Tanushev, Superpositions and higher order Gaussian beams,, Commun. Math. Sci., 6 (2008), 449.   Google Scholar

[43]

E. Wigner, On the quantum correction for the thermodynamic equilibrium,, Phys. Rev., 40 (1932), 749.   Google Scholar

[44]

D. Yin and C. Zheng, Gaussian beam formulation and interface conditions for the one-dimensional linear Schödinger equation,, Wave Motion, 48 (2011), 310.  doi: 10.1016/j.wavemoti.2010.11.006.  Google Scholar

[45]

D. Yin and C. Zheng, Composite coherent states approximation for one-dimensional multi-phased wave functions,, Communications in Computational Physics, 11 (2012), 951.  doi: 10.4208/cicp.101010.250511a.  Google Scholar

[1]

Jean-Luc Akian, Radjesvarane Alexandre, Salma Bougacha. A Gaussian beam approach for computing Wigner measures in convex domains. Kinetic & Related Models, 2011, 4 (3) : 589-631. doi: 10.3934/krm.2011.4.589

[2]

Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871

[3]

Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027

[4]

Wolfgang Wagner. A random cloud model for the Wigner equation. Kinetic & Related Models, 2016, 9 (1) : 217-235. doi: 10.3934/krm.2016.9.217

[5]

Thomas Chen, Ryan Denlinger, Nataša Pavlović. Moments and regularity for a Boltzmann equation via Wigner transform. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4979-5015. doi: 10.3934/dcds.2019204

[6]

Shi Jin, Peng Qi. A hybrid Schrödinger/Gaussian beam solver for quantum barriers and surface hopping. Kinetic & Related Models, 2011, 4 (4) : 1097-1120. doi: 10.3934/krm.2011.4.1097

[7]

Delio Mugnolo. Gaussian estimates for a heat equation on a network. Networks & Heterogeneous Media, 2007, 2 (1) : 55-79. doi: 10.3934/nhm.2007.2.55

[8]

Orazio Muscato, Wolfgang Wagner. A stochastic algorithm without time discretization error for the Wigner equation. Kinetic & Related Models, 2019, 12 (1) : 59-77. doi: 10.3934/krm.2019003

[9]

Pao-Liu Chow. Asymptotic solutions of a nonlinear stochastic beam equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 735-749. doi: 10.3934/dcdsb.2006.6.735

[10]

Maurizio Garrione, Manuel Zamora. Periodic solutions of the Brillouin electron beam focusing equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 961-975. doi: 10.3934/cpaa.2014.13.961

[11]

Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783

[12]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[13]

Zifei Shen, Fashun Gao, Minbo Yang. On critical Choquard equation with potential well. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3567-3593. doi: 10.3934/dcds.2018151

[14]

Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142

[15]

Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92

[16]

Giuseppe Maria Coclite, Lorenzo di Ruvo. Discontinuous solutions for the generalized short pulse equation. Evolution Equations & Control Theory, 2019, 8 (4) : 737-753. doi: 10.3934/eect.2019036

[17]

Zhenguo Liang, Jiansheng Geng. Quasi-periodic solutions for 1D resonant beam equation. Communications on Pure & Applied Analysis, 2006, 5 (4) : 839-853. doi: 10.3934/cpaa.2006.5.839

[18]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171

[19]

Xiaolong Han, Guozhen Lu. Regularity of solutions to an integral equation associated with Bessel potential. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1111-1119. doi: 10.3934/cpaa.2011.10.1111

[20]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]