L. D. Landau, E. M. Lifshitz, and F. Mechanics, of A Course of Theoretical Physics, 1987.

A. Griffin, W. Wu, and S. Stringari, Hydrodynamic Modes in a Trapped Bose Gas above the Bose-Einstein Transition, Physical Review Letters, vol.78, issue.10, p.1838, 1997.
DOI : 10.1103/PhysRevLett.78.1838

Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Evolution of a Bose gas in anisotropic time-dependent traps, Physical Review A, vol.55, issue.1, p.18, 1997.
DOI : 10.1103/PhysRevA.55.R18

L. Pitaevskii and S. Stringari, Elementary Excitations in Trapped Bose-Einstein Condensed Gases Beyond the Mean-Field Approximation, Physical Review Letters, vol.81, issue.21, p.4541, 1998.
DOI : 10.1103/PhysRevLett.81.4541

S. Stringari, Collective Excitations of a Trapped Bose-Condensed Gas, Physical Review Letters, vol.77, issue.12, p.2360, 1996.
DOI : 10.1103/PhysRevLett.77.2360

L. A. Sidorenkov, M. K. Tey, R. Grimm, Y. Hou, L. Pitaevskii et al., Second sound and the superfluid fraction in a Fermi gas with resonant interactions, Nature, vol.302, issue.7452, p.78, 2013.
DOI : 10.1038/nature12136

E. H. Lieb and W. Liniger, Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, Physical Review, vol.130, issue.4, p.1605, 1963.
DOI : 10.1103/PhysRev.130.1605

A. Minguzzi, P. Vignolo, M. L. Chiofalo, and M. P. Tosi, Hydrodynamic excitations in a spin-polarized Fermi gas under harmonic confinement in one dimension, Physical Review A, vol.64, issue.3, p.33605, 2001.
DOI : 10.1103/PhysRevA.64.033605

C. Menotti and S. Stringari, Collective oscillations of a one-dimensional trapped Bose-Einstein gas, Physical Review A, vol.66, issue.4, p.43610, 2002.
DOI : 10.1103/PhysRevA.66.043610

P. Öhberg and L. Santos, Dynamical Transition from a Quasi-One-Dimensional Bose-Einstein Condensate to a Tonks-Girardeau Gas, Physical Review Letters, vol.89, issue.24, p.240402, 2002.
DOI : 10.1103/PhysRevLett.89.240402

P. Pedri, L. Santos, P. Öhberg, and S. Stringari, Violation of self-similarity in the expansion of a one-dimensional Bose gas, Physical Review A, vol.68, issue.4, p.43601, 2003.
DOI : 10.1103/PhysRevA.68.043601

S. Peotta and M. D. Ventra, Quantum shock waves and population inversion in collisions of ultracold atomic clouds, Physical Review A, vol.89, issue.1, p.13621, 2014.
DOI : 10.1103/PhysRevA.89.013621

A. S. Campbell, D. M. Gangardt, and K. V. Kheruntsyan, Sudden Expansion of a One-Dimensional Bose Gas from Power-Law Traps, Physical Review Letters, vol.114, issue.12, p.125302, 2015.
DOI : 10.1103/PhysRevLett.114.125302

H. Moritz, T. Stöferle, M. Köhl, and T. Esslinger, Exciting Collective Oscillations in a Trapped 1D Gas, Physical Review Letters, vol.91, issue.25, p.250402, 2003.
DOI : 10.1103/PhysRevLett.91.250402

E. Haller, M. Gustavsson, M. J. Mark, J. G. Danzl, G. P. Hart et al., Realization of an Excited, Strongly Correlated Quantum Gas Phase, Science, vol.325, issue.5945, p.1224, 2009.
DOI : 10.1126/science.1175850

B. Fang, G. Carleo, A. Johnson, and I. Bouchoule, Quench-Induced Breathing Mode of One-Dimensional Bose Gases, Physical Review Letters, vol.113, issue.3, p.35301, 2014.
DOI : 10.1103/PhysRevLett.113.035301
URL : https://hal.archives-ouvertes.fr/hal-00916875

M. J. Davis, S. A. Morgan, and K. Burnett, Simulations of Bose Fields at Finite Temperature, Physical Review Letters, vol.87, issue.16, p.160402, 2001.
DOI : 10.1103/PhysRevLett.87.160402

P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques, Advances in Physics, vol.146, issue.5, p.363, 2008.
DOI : 10.1137/030601211

D. S. Petrov, G. V. Shlyapnikov, and J. T. Walraven, Regimes of Quantum Degeneracy in Trapped 1D Gases, Physical Review Letters, vol.85, issue.18, p.3745, 2000.
DOI : 10.1103/PhysRevLett.85.3745

S. Richard, F. Gerbier, J. H. Thywissen, M. Hugbart, P. Bouyer et al., Momentum Spectroscopy of 1D Phase Fluctuations in Bose-Einstein Condensates, Physical Review Letters, vol.91, issue.1, p.10405, 2003.
DOI : 10.1103/PhysRevLett.91.010405
URL : https://hal.archives-ouvertes.fr/hal-00116296

A. H. Van-amerongen, J. J. Van-es, P. Wicke, K. V. Kheruntsyan, and N. J. Van-druten, Yang-Yang Thermodynamics on an Atom Chip, Physical Review Letters, vol.100, issue.9, p.90402, 2008.
DOI : 10.1103/PhysRevLett.100.090402

S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Non-equilibrium coherence dynamics in one-dimensional Bose gases, Nature, vol.78, issue.7160, p.324, 2007.
DOI : 10.1038/nature06149

T. Jacqmin, J. Armijo, T. Berrada, K. V. Kheruntsyan, and I. Bouchoule, Sub-Poissonian Fluctuations in a 1D Bose Gas: From the Quantum Quasicondensate to the Strongly Interacting Regime, Physical Review Letters, vol.106, issue.23, p.230405, 2011.
DOI : 10.1103/PhysRevLett.106.230405
URL : https://hal.archives-ouvertes.fr/hal-00576883

J. Armijo, T. Jacqmin, K. Kheruntsyan, and I. Bouchoule, Mapping out the quasicondensate transition through the dimensional crossover from one to three dimensions, Physical Review A, vol.83, issue.2, p.21605, 2011.
DOI : 10.1103/PhysRevA.83.021605

K. V. Kheruntsyan, D. M. Gangardt, P. D. Drummond, and G. V. Shlyapnikov, Finite-temperature correlations and density profiles of an inhomogeneous interacting one-dimensional Bose gas, Physical Review A, vol.71, issue.5, p.53615, 2005.
DOI : 10.1103/PhysRevA.71.053615

I. Bouchoule, K. V. Kheruntsyan, and G. V. Shlyapnikov, Interaction-induced crossover versus finite-size condensation in a weakly interacting trapped one-dimensional Bose gas, Physical Review A, vol.75, issue.3, p.31606, 2007.
DOI : 10.1103/PhysRevA.75.031606
URL : https://hal.archives-ouvertes.fr/hal-00116204

M. J. Davis, P. B. Blakie, A. H. Van-amerongen, N. J. Van-druten, and K. V. Kheruntsyan, Yang-Yang thermometry and momentum distribution of a trapped one-dimensional Bose gas, Physical Review A, vol.85, issue.3, p.31604, 2012.
DOI : 10.1103/PhysRevA.85.031604

A. Sinatra, Y. Castin, and C. Lobo, A Monte Carlo formulation of the Bogolubov theory, Journal of Modern Optics, vol.75, issue.14-15, p.2629, 2000.
DOI : 10.1103/PhysRevA.58.4824
URL : https://hal.archives-ouvertes.fr/hal-00002454

Y. Castin, R. Dum, E. Mandonnet, A. Minguzzi, and I. Carusotto, Coherence properties of a continuous atom laser, Journal of Modern Optics, vol.15, issue.14-15, p.2671, 2000.
DOI : 10.1103/PhysRevB.6.3409
URL : https://hal.archives-ouvertes.fr/hal-00002120

C. Mora and Y. Castin, Extension of Bogoliubov theory to quasicondensates, Physical Review A, vol.67, issue.5, p.53615, 2003.
DOI : 10.1103/PhysRevA.67.053615
URL : https://hal.archives-ouvertes.fr/hal-00157445

M. A. Cazalilla, Bosonizing one-dimensional cold atomic gases, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.37, issue.7, p.1, 2004.
DOI : 10.1088/0953-4075/37/7/051
URL : http://arxiv.org/abs/cond-mat/0307033

F. D. Haldane, Effective Harmonic-Fluid Approach to Low-Energy Properties of One-Dimensional Quantum Fluids, Physical Review Letters, vol.47, issue.25, p.1840, 1981.
DOI : 10.1103/PhysRevLett.47.1840

A. G. Sykes, D. M. Gangardt, M. J. Davis, K. Viering, M. G. Raizen et al., Spatial Nonlocal Pair Correlations in a Repulsive 1D Bose Gas, Physical Review Letters, vol.100, issue.16, p.160406, 2008.
DOI : 10.1103/PhysRevLett.100.160406

P. Deuar, A. G. Sykes, D. M. Gangardt, M. J. Davis, P. D. Drummond et al., Nonlocal pair correlations in the one-dimensional Bose gas at finite temperature, Physical Review A, vol.79, issue.4, p.43619, 2009.
DOI : 10.1103/PhysRevA.79.043619

N. J. Bouchoule, C. I. Van-druten, and . Westbrook, Sincê ?(x) is a real function in the classical field picture, one should, in fact, expand on real sinusoidal functions; see: I, Atom Chips, 2011.

C. W. Gardiner, J. R. Anglin, and T. I. Fudge, The stochastic Gross-Pitaevskii equation, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.35, issue.6, p.1555, 2002.
DOI : 10.1088/0953-4075/35/6/310
URL : http://arxiv.org/abs/cond-mat/0112129

C. W. Gardiner and M. J. Davis, The stochastic Gross???Pitaevskii equation: II, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.36, issue.23, p.4731, 2003.
DOI : 10.1088/0953-4075/36/23/010

M. J. Davis, R. J. Ballagh, and K. Burnett, Dynamics of thermal Bose fields in the classical limit, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.34, issue.22, p.4487, 2001.
DOI : 10.1088/0953-4075/34/22/316

M. J. Davis, S. A. Morgan, and K. Burnett, Simulations of Bose Fields at Finite Temperature, Physical Review Letters, vol.87, issue.16, p.160402, 2001.
DOI : 10.1103/PhysRevLett.87.160402

. However, the density is overestimated for non-degenerate gases. Consequently, in a harmonic trap and in the absence of an energy cutoff, the classical field predicts tails of the density that decay as 1/|x|, leading to an unphysical divergence of the total atom number

S. J. Rooney, P. B. Blakie, and A. S. Bradley, Numerical method for the stochastic projected Gross-Pitaevskii equation, Physical Review E, vol.89, issue.1, p.13302, 2014.
DOI : 10.1103/PhysRevE.89.013302
URL : http://arxiv.org/abs/1310.0161

A. S. Bradley, P. B. Blakie, and C. W. Gardiner, Properties of the stochastic Gross???Pitaevskii equation: finite temperature Ehrenfest relations and the optimal plane wave representation, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.38, issue.23, p.4259, 2005.
DOI : 10.1088/0953-4075/38/23/008

P. B. Blakie and M. J. Davis, Projected Gross-Pitaevskii equation for harmonically confined Bose gases at finite temperature, Physical Review A, vol.72, issue.6, p.63608, 2005.
DOI : 10.1103/PhysRevA.72.063608

A. Sinatra, C. Lobo, and Y. Castin, The truncated Wigner method for Bose-condensed gases: limits of validity and applications, Journal of Physics B: Atomic, Molecular and Optical Physics, vol.35, issue.17, p.3599, 2002.
DOI : 10.1088/0953-4075/35/17/301

A. Bezett and P. B. Blakie, Projected Gross-Pitaevskii equation theory of finite-temperature collective modes for a trapped Bose gas, Physical Review A, vol.79, issue.2, p.23602, 2009.
DOI : 10.1103/PhysRevA.79.023602

T. P. Simula, M. J. Davis, and P. B. Blakie, Superfluidity of an interacting trapped quasi-two-dimensional Bose gas, Physical Review A, vol.77, issue.2, p.23618, 2008.
DOI : 10.1103/PhysRevA.77.023618

]. I. Bouchoule, M. Arzamasovs, K. V. Kheruntsyan, and D. M. Gangardt, Two-body momentum correlations in a weakly interacting one-dimensional Bose gas, Physical Review A, vol.86, issue.3, p.33626, 2012.
DOI : 10.1103/PhysRevA.86.033626
URL : https://hal.archives-ouvertes.fr/hal-00719723