July 2018, 17(4): 1573-1594. doi: 10.3934/cpaa.2018075

On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system

1. 

Fak. Mathematik, University of Vienna, Oskar MorgensternPlatz 1, A-1090 Wien, Austria

2. 

Wolfgang Pauli Institute c/o Fak. Math. Univ. Vienna, Oskar MorgensternPlatz 1, A-1090 Wien, Austria

3. 

Laboratoire de Mathématiques, UMR 8628, Université Paris-Saclay, Paris-Sud and CNRS, F-91405 Orsay, France

Received  March 2017 Revised  February 2018 Published  April 2018

We address various issues concerning the Cauchy problem for the Zakharov-Rubenchik system(known as the Benney-Roskes system in water waves theory), which models the interaction of short and long waves in many physical situations. Motivated by the transverse stability/instability of the one-dimensional solitary wave (line solitary), we study the Cauchy problem in the background of a line solitary wave.

Citation: Hung Luong, Norbert J. Mauser, Jean-Claude Saut. On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1573-1594. doi: 10.3934/cpaa.2018075
References:
[1]

M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech., 92 (1979), 691-715.

[2]

H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal., 76 (1988), 183-210.

[3]

H. Added and S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 551-554.

[4]

I. BejenaruS. HerrJ. Holmer and D. Tataru, On the 2D Zakharov system with L2-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089.

[5]

D. J. Benney and A. C. Newell, The propagation of nonlinear envelopes, J. Math. and Phys., 46 (1967), 133-139.

[6]

D. J. Benney and G. J. Roskes, Waves instabilities, Stud. Appl. Math., 48 (1969), 377-385.

[7]

J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202.

[8]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, IRMN, 11 (1996), 515-546.

[9]

R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Partial Differential Equations, 17 (1992), 967-988.

[10]

R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann.Inst. Poincaré H., Phys.Théor., 58 (1993), 85-104.

[11]

T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems, Asymptotic Analysis, 31 (2002), 69-91.

[12]

T. Colin and D. Lannes, Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems, Disc. Cont. Dyn. Systems, 11 (2004), 83-100.

[13]

J.C. Cordero Ceballos, Supersonic limit for the Zakharov-Rubenchik system, J. Diff. Eq., 261 (2016), 5260-5288.

[14]

A. Davey and K. Stewartson, One three-dimensional packets of water waves, Proc. Roy. Soc. Lond. A, 338 (1974), 101-110.

[15]

V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), 703-714.

[16]

J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506.

[17]

J.-M. Ghidaglia and J.-C. Saut, Non existence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations, J. Nonlinear Sci., 6 (1996), 139-145.

[18]

J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Analysis, 151 (1997), 384-436.

[19]

L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. Ⅰ, Comm. Math. Phys., 160 (1994), 173-215.

[20]

L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. Ⅱ, Comm. Math. Phys., 160 (1994), 349-389.

[21]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., XLI (1988), 891-907.

[22]

C. KenigG. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct; Anal., 127 (1995), 204-234.

[23]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.

[24]

D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal., 232 (2006), 495-539.

[25]

D. Lannes, Water Waves: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, vol 188 (2013), AMS, Providence.

[26]

F. Linares and C. Matheus, Well-posedness for the 1-D Zakharov system, Advances in Diff. Equations, 14 (2009), 261-288.

[27]

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1989.

[28]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences volume 53, Springer-Verlag, New-York, 1984.

[29]

T. Mizumachi, Stability of line solitons for the KP-Ⅱ equation in ${{\mathbb{R}}^{2}}$, Memoirs of the AMS, vol. 238, number 1125, (2015).

[30]

T. Mizumachi and N. Tzvetkov, Stability of the line soliton of the KP-Ⅱ equation under periodic transverse perturbations, Math. Ann., 352 (2012), 659-690.

[31]

C. Obrecht, Thèse de Doctorat, Université Paris-Sud (2015) and article in preparation.

[32]

M. Ohta, Stability and instability of standing waves for the generalized Davey-Stewartson system, Diff. Int. Eq., 8 (1995), 1775-1788.

[33]

M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. Poincaré H., Phys. Théor., 62 (1995), 69-80.

[34]

M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized DaveyStewartson system, Ann. Inst. Poincaré H., Phys. Théor., 63 (1995), 111-117.

[35]

F. Oliveira, Stability of the solitons for the one-dimensional Zakharov-Rubenchik system, Physica D, 175 (2003), 220-240.

[36]

F. Oliveira, Adiabatic limit of the Zakharov-Rubenchik system, Reports on Mathematical Physics, 61 (2008), 13-27.

[37]

T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., Kyoto University. Research Institute for Mathematical Sciences. Publications, 28 (1992), 329-361.

[38]

T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Proc. Jap. Acad. A, 67 (1991), 113-116.

[39]

T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Diff. Int. Eq., 5 (1992), 721-745.

[40]

T. PassotP.-L. Sulem and C. Sulem, Generalization of acoustic fronts by focusing ave packets, Physica D, 94 (1996), 168-187.

[41]

G. Ponce and J.-C. Saut, Well-posedness for the Benney-Roskes-Zakharov-Rubenchik system, Discrete Cont. Dynamical Systems, 13 (2005), 811-825.

[42]

F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), 257-388.

[43]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 477-496.

[44]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's, J. Math. Pures et Appl., 80 (2008), 550-590.

[45]

S. H. Schochet and M.I. Weinstein, The nonlinear Schrödinger limit of the Zakharov governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580.

[46]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, Applied Mathematical Sciences 139 New York, Berlin, 1999.

[47]

C. Sulem and P.-L. Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C.R. Ac. Sci. Paris Sér. A-B, 289 (1979), A173-A176.

[48]

H. Takaoka, Well-posedness for the Zakharov system with the periodic boundary condition, Diff. and Int. equations, 6 (1999), 789-810.

[49]

N. Tzvetkov, Low regularity solutions for a generalized Zakharov system, Diff. and Int. Equations, 13 (2000), 423-440.

[50]

V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.

[51]

V. E. Zakharov, Weakly nonlinear waves on the surface of an ideal finite depth fluid, Amer. Math. Soc. Transl., 182 (1998), 167-197.

[52]

V. E. Zakharov and E. A. Kuznetsov, Hamiltonian formalism for nonlinear waves, PhysicsUspekhi, 40 (1997), 1087-1116.

[53]

V. E. Zakharov and A. M. Rubenchik, Nonlinear interaction of high-frequency and low frequency waves, Prikl. Mat. Techn. Phys., 5 (1972), 84-98.

[54]

Xiaofei Zhao and Ziyi Li, Numerical methods and simulations for the dynamics of onedimensional Zakharov-Rubenchik equations, J. Sci. Comput., 59 (2014), 412-438.

show all references

References:
[1]

M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech., 92 (1979), 691-715.

[2]

H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal., 76 (1988), 183-210.

[3]

H. Added and S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 551-554.

[4]

I. BejenaruS. HerrJ. Holmer and D. Tataru, On the 2D Zakharov system with L2-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089.

[5]

D. J. Benney and A. C. Newell, The propagation of nonlinear envelopes, J. Math. and Phys., 46 (1967), 133-139.

[6]

D. J. Benney and G. J. Roskes, Waves instabilities, Stud. Appl. Math., 48 (1969), 377-385.

[7]

J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202.

[8]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, IRMN, 11 (1996), 515-546.

[9]

R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Partial Differential Equations, 17 (1992), 967-988.

[10]

R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann.Inst. Poincaré H., Phys.Théor., 58 (1993), 85-104.

[11]

T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems, Asymptotic Analysis, 31 (2002), 69-91.

[12]

T. Colin and D. Lannes, Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems, Disc. Cont. Dyn. Systems, 11 (2004), 83-100.

[13]

J.C. Cordero Ceballos, Supersonic limit for the Zakharov-Rubenchik system, J. Diff. Eq., 261 (2016), 5260-5288.

[14]

A. Davey and K. Stewartson, One three-dimensional packets of water waves, Proc. Roy. Soc. Lond. A, 338 (1974), 101-110.

[15]

V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), 703-714.

[16]

J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506.

[17]

J.-M. Ghidaglia and J.-C. Saut, Non existence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations, J. Nonlinear Sci., 6 (1996), 139-145.

[18]

J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Analysis, 151 (1997), 384-436.

[19]

L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. Ⅰ, Comm. Math. Phys., 160 (1994), 173-215.

[20]

L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. Ⅱ, Comm. Math. Phys., 160 (1994), 349-389.

[21]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., XLI (1988), 891-907.

[22]

C. KenigG. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct; Anal., 127 (1995), 204-234.

[23]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.

[24]

D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal., 232 (2006), 495-539.

[25]

D. Lannes, Water Waves: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, vol 188 (2013), AMS, Providence.

[26]

F. Linares and C. Matheus, Well-posedness for the 1-D Zakharov system, Advances in Diff. Equations, 14 (2009), 261-288.

[27]

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1989.

[28]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences volume 53, Springer-Verlag, New-York, 1984.

[29]

T. Mizumachi, Stability of line solitons for the KP-Ⅱ equation in ${{\mathbb{R}}^{2}}$, Memoirs of the AMS, vol. 238, number 1125, (2015).

[30]

T. Mizumachi and N. Tzvetkov, Stability of the line soliton of the KP-Ⅱ equation under periodic transverse perturbations, Math. Ann., 352 (2012), 659-690.

[31]

C. Obrecht, Thèse de Doctorat, Université Paris-Sud (2015) and article in preparation.

[32]

M. Ohta, Stability and instability of standing waves for the generalized Davey-Stewartson system, Diff. Int. Eq., 8 (1995), 1775-1788.

[33]

M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. Poincaré H., Phys. Théor., 62 (1995), 69-80.

[34]

M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized DaveyStewartson system, Ann. Inst. Poincaré H., Phys. Théor., 63 (1995), 111-117.

[35]

F. Oliveira, Stability of the solitons for the one-dimensional Zakharov-Rubenchik system, Physica D, 175 (2003), 220-240.

[36]

F. Oliveira, Adiabatic limit of the Zakharov-Rubenchik system, Reports on Mathematical Physics, 61 (2008), 13-27.

[37]

T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., Kyoto University. Research Institute for Mathematical Sciences. Publications, 28 (1992), 329-361.

[38]

T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Proc. Jap. Acad. A, 67 (1991), 113-116.

[39]

T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Diff. Int. Eq., 5 (1992), 721-745.

[40]

T. PassotP.-L. Sulem and C. Sulem, Generalization of acoustic fronts by focusing ave packets, Physica D, 94 (1996), 168-187.

[41]

G. Ponce and J.-C. Saut, Well-posedness for the Benney-Roskes-Zakharov-Rubenchik system, Discrete Cont. Dynamical Systems, 13 (2005), 811-825.

[42]

F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), 257-388.

[43]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 477-496.

[44]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's, J. Math. Pures et Appl., 80 (2008), 550-590.

[45]

S. H. Schochet and M.I. Weinstein, The nonlinear Schrödinger limit of the Zakharov governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580.

[46]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, Applied Mathematical Sciences 139 New York, Berlin, 1999.

[47]

C. Sulem and P.-L. Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C.R. Ac. Sci. Paris Sér. A-B, 289 (1979), A173-A176.

[48]

H. Takaoka, Well-posedness for the Zakharov system with the periodic boundary condition, Diff. and Int. equations, 6 (1999), 789-810.

[49]

N. Tzvetkov, Low regularity solutions for a generalized Zakharov system, Diff. and Int. Equations, 13 (2000), 423-440.

[50]

V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.

[51]

V. E. Zakharov, Weakly nonlinear waves on the surface of an ideal finite depth fluid, Amer. Math. Soc. Transl., 182 (1998), 167-197.

[52]

V. E. Zakharov and E. A. Kuznetsov, Hamiltonian formalism for nonlinear waves, PhysicsUspekhi, 40 (1997), 1087-1116.

[53]

V. E. Zakharov and A. M. Rubenchik, Nonlinear interaction of high-frequency and low frequency waves, Prikl. Mat. Techn. Phys., 5 (1972), 84-98.

[54]

Xiaofei Zhao and Ziyi Li, Numerical methods and simulations for the dynamics of onedimensional Zakharov-Rubenchik equations, J. Sci. Comput., 59 (2014), 412-438.

[1]

Jerry L. Bona, Henrik Kalisch. Models for internal waves in deep water. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 1-20. doi: 10.3934/dcds.2000.6.1

[2]

Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735

[3]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062

[4]

Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239

[5]

Anca-Voichita Matioc. On particle trajectories in linear deep-water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1537-1547. doi: 10.3934/cpaa.2012.11.1537

[6]

Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593

[7]

Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541

[8]

Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701

[9]

Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997

[10]

Mats Ehrnström. Deep-water waves with vorticity: symmetry and rotational behaviour. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 483-491. doi: 10.3934/dcds.2007.19.483

[11]

Charles L. Epstein, Leslie Greengard, Thomas Hagstrom. On the stability of time-domain integral equations for acoustic wave propagation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4367-4382. doi: 10.3934/dcds.2016.36.4367

[12]

Stéphane Junca, Bruno Lombard. Stability of neutral delay differential equations modeling wave propagation in cracked media. Conference Publications, 2015, 2015 (special) : 678-685. doi: 10.3934/proc.2015.0678

[13]

M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215

[14]

Takashi Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Conference Publications, 2009, 2009 (Special) : 592-601. doi: 10.3934/proc.2009.2009.592

[15]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126

[16]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469

[17]

Chang-Yeol Jung, Alex Mahalov. Wave propagation in random waveguides. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 147-159. doi: 10.3934/dcds.2010.28.147

[18]

L’ubomír Baňas, Amy Novick-Cohen, Robert Nürnberg. The degenerate and non-degenerate deep quench obstacle problem: A numerical comparison. Networks & Heterogeneous Media, 2013, 8 (1) : 37-64. doi: 10.3934/nhm.2013.8.37

[19]

Yacine Chitour, Guilherme Mazanti, Mario Sigalotti. Stability of non-autonomous difference equations with applications to transport and wave propagation on networks. Networks & Heterogeneous Media, 2016, 11 (4) : 563-601. doi: 10.3934/nhm.2016010

[20]

Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 599-628. doi: 10.3934/dcds.2013.33.599

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (29)
  • HTML views (166)
  • Cited by (0)

Other articles
by authors

[Back to Top]