A table of solutions of the one-dimensional Burgers equation

Authors:
Edward R. Benton and George W. Platzman

Journal:
Quart. Appl. Math. **30** (1972), 195-212

MSC:
Primary 35Q99

DOI:
https://doi.org/10.1090/qam/306736

MathSciNet review:
306736

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Abstract: The literature relating to the one-dimensional Burgers equation is surveyed. About thirty-five distinct solutions of this equation are classified in tabular form. The physically interesting cases are illustrated by means of isochronal graphs.

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*Nonlinear partial differential equations in engineering*, Academic Press, New York-London, 1965. MR **0210342**
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*Lossless propagation of one-dimensional, finite amplitude sound waves*, J. Math. Anal. Appl. **10** (1965), 116–173. MR **170575**, DOI https://doi.org/10.1016/0022-247X%2865%2990153-8
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E. R. Benton, *Some new exact, viscous, nonsteady solutions of Burgers’ equation*, Phys. Fluids **9**, 1247–48 (1966)
---, *Solutions illustrating the decay of dissipation layers in Burgers’ nonlinear diffusion equation*, Phys. Fluids **10**, 2113–19 (1967)
D. T. Blackstock, *Thermoviscous attenuation of plane, periodic, finite-amplitude sound waves*, J. Acoust. Soc. Amer. **36**, 534–542 (1964)
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*Connection between the Fay and Fubini solutions for plane sound waves of finite amplitude*, J. Acoust. Soc. Amer. **39** (1966), 1019–1026. MR **198801**, DOI https://doi.org/10.1121/1.1909986
J. M. Burgers, *Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion*, Trans. Roy. Neth. Acad. Sci. Amsterdam, **17**, 1–53 (1939)
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*Application of a model system to illustrate some points of the statistical theory of free turbulence*, Nederl. Akad. Wetensch., Proc. **43** (1940), 2–12. MR **1147**
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*The formation of vortex sheets in a simplified type of turbulent motion*, Nederl. Akad. Wetensch., Proc. **53** (1950), 122–133. MR **34671**
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*Correlation problems in a one-dimensional model of turbulence. I*, Nederl. Akad. Wetensch., Proc. **53** (1950), 247–260. MR **35581**
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*Statistical problems connected with the solution of a nonlinear partial differential equation*, Nonlinear Problems of Engineering, Academic Press, New York, 1964, pp. 123–137. MR **0175430**
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*The partial differential equation $u_t+uu_x=\mu u_{xx}$*, Comm. Pure Appl. Math. **3** (1950), 201–230. MR **47234**, DOI https://doi.org/10.1002/cpa.3160030302
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*The Cameron-Martin-Wiener method in turbulence and in Burgers’ model: General formulae, and application to late decay*, J. Fluid Mech. **41** (1970), 593–618. MR **278617**, DOI https://doi.org/10.1017/S0022112070000770
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*Frequency spectrum of finite amplitude ultrasonic waves in liquids*, Phys. Fluids **3** (1960), 346–352. MR **121003**, DOI https://doi.org/10.1063/1.1706039
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*On periodic solutions of Burgers’ equation*, J. Appl. Math. Mech. **25** (1962), 1597–1607. MR **0185286**, DOI https://doi.org/10.1016/0021-8928%2862%2990138-7
R. H. Kraichnan, *Langrangian-history statistical theory for Burgers’ equation*, Phys. Fluids **11**, 265–277 (1968)
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*Wiener-Hermite expansion in model turbulence at large Reynolds numbers*, Phys. Fluids **7** (1964), 1178–1190. MR **167097**, DOI https://doi.org/10.1063/1.1711359
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J. S. Mendousse, *Nonlinear dissipative distortion of progressive sound waves at moderate amplitude*, J. Acoust. Soc. Amer. **25**, 51–54 (1953)
C. N. K. Mooers, *Gerstner wave’s Fourier decomposition and related identities*, J. Geophys. Res. **73**, 5843–5847 (1968)
Y. Ogura, *A note on the energy transfer in Burgers’ model of turbulence*, 75th Anniversary Volume of the Journal of the Meteorological Society of Japan, 1957, pp. 92–94
S. A. Orszag and L. R. Bissonnette, *Dynamical properties of truncated Wiener-Hermite expansions*, Phys. Fluids **10**, 2603–2613 (1967)
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I. Rudnick, *On the attenuation of a repeated sawtooth shock wave*, J. Acoust. Soc. Amer. **25**, 1012–1013 (1953)
P. G. Saffman, *Lectures on homogeneous turbulence*, in *Topics in nonlinear physics* (N. J. Zabusky, editor), Springer-Verlag New York, 1968, pp. 485–614
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A. Siegel, T. Imamura and W. C. Meecham, *Wiener-Hermite functional expansion in turbulence with the Burgers model*, Phys. Fluids **6**, 1519–1521 (1963)
- Armand Siegel, Tsutomu Imamura, and William C. Meecham,
*Wiener-Hermite expansion in model turbulence in the late decay stage*, J. Mathematical Phys. **6** (1965), 707–721. MR **175174**, DOI https://doi.org/10.1063/1.1704328
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*Finite amplitude acoustic waves in a relaxing medium*, Soviet Physics Acoust. **8** (1962), 170–175. MR **0153253**
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*Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation*, J. Mathematical Phys. **10** (1969), 536–539. MR **271526**, DOI https://doi.org/10.1063/1.1664873
T. Tatsumi, *Nonlinear wave expansion for turbulence in the Burgers model of a fluid*, Phys. Fluids **12** (Part II), II 258–II 264 (1969)
G. I. Taylor, *The conditions necessary for discontinuous motion in gases*, Proc. Roy. Soc. **A84**, 371–377 (1910)
B. van der Pol, *On a non-linear partial differential equation satisfied by the logarithm of the Jacobean theta-functions, with arithmetical applications*, Proc. Acad. Sci. Amsterdam **A13**, 261–284 (1951)
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*INITIAL VALUE PROBLEMS ASSOCIATED WITH BURGERS’ EQUATION*, ProQuest LLC, Ann Arbor, MI, 1968. Thesis (S.C.D.C.)–Washington University in St. Louis. MR **2617602**
J. J. Walton, *Integration of the Lagrangian-history approximation to Burgers’ equation*, Phys. Fluids **13**, 1634–1635 (1970)
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*The mean pressure and velocity in a plane acoustic wave in a gas*, J. Acoust. Soc. Amer. **22** (1950), 319–327. MR **36644**, DOI https://doi.org/10.1121/1.1906606
N. J. Zabusky, *Phenomena associated with the oscillations of a nonlinear model string: The problem of Fermi, Pasta, and Ulam*, in *Proceedings of the conference on mathematical models in the physical sciences* (S. Drobot, editor), Prentice Hall, New York, 1963, pp. 99–133
P. A. Blythe, *Non-linear wave propagation in a relaxing gas*, J. Fluid Mech. **37**, 31–50 (1969)
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*On the near-equilibrium and near-frozen regions in an expansion wave in a relaxing gas*, J. Fluid Mech. **19** (1964), 81–102. MR **163556**, DOI https://doi.org/10.1017/S0022112064000556
W. Lick, *Nonlinear wave propagation in fluids*, in *Annual reviews of fluid mechanics* **2**, (M. van Dyke, W. G. Vincenti, and J. V. Wehausen, editors), Annual Reviews Inc., Palo Alto, California, 1970, pp. 113–136
J. P. Moran and S. F. Shen, *On the formation of weak plane shock waves by impulsive motion of a piston*, J. Fluid Mech. **25**, 705–718 (1966)
M. Morduchow and A. J. Paullay, *Stability of normal shock waves with viscosity and heat conduction*, Phys. Fluids **14**, 323–331 (1971)

W. F. Ames, *Nonlinear partial differential equations in engineering*, Academic Press, New York, 1965
E. D. Banta, *Lossless propagation of one-dimensional finite amplitude sound waves*, J. Math. Anal. Appl. **10**, 166–173 (1965)
J. Bass, *Les fonctions pseudo-aléatoires*, C. R. Acad. Sci. Paris **C1-III**, 1–69 (1962)
H. Bateman, *Some recent researches on the motion of fluids*, Mon. Weather Rev. **43**, 163–170 (1915)
R. Bellman, S. P. Azen, and J. M. Richardson, *On new and direct computational approaches to some mathematical models of turbulence*, Quart. Appl. Math. **23**, 55–67 (1965)
E. R. Benton, *Some new exact, viscous, nonsteady solutions of Burgers’ equation*, Phys. Fluids **9**, 1247–48 (1966)
---, *Solutions illustrating the decay of dissipation layers in Burgers’ nonlinear diffusion equation*, Phys. Fluids **10**, 2113–19 (1967)
D. T. Blackstock, *Thermoviscous attenuation of plane, periodic, finite-amplitude sound waves*, J. Acoust. Soc. Amer. **36**, 534–542 (1964)
---, *Connection between the Fay and Fubini solutions for plane sounds waves of finite amplitude*, J. Acoust. Soc. Amer. **39**, 1019–1026 (1966)
J. M. Burgers, *Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion*, Trans. Roy. Neth. Acad. Sci. Amsterdam, **17**, 1–53 (1939)
---, *Application of a model system to illustrate some points of the statistical theory of free turbulence*, Proc. Roy. Neth. Acad. Sci. Amsterdam **43**, 2–12 (1940)
---, *A mathematical model illustrating the theory of turbulence*, in *Advances in applied mechanics* (R. von Mises and T. von Kármán, editors) **1**, Academic Press, New York, 1948, pp. 171–199
---, *The formation of vortex sheets in a simplified type of turbulent motion*, Proc. Roy. Neth. Acad. Sci. Amsterdam **53**, 122–133 (1950)
---, *Correlation problems in a one-dimensional model of turbulence*, Proc. Roy. Neth. Acad. Sci. Amsterdam **53**, 247–260, 393–406, 718–742 (1950)
---, *Statistical problems connected with the solution of a simple non-linear partial differential equation*, Proc. Roy. Neth. Acad. Sci. Amsterdam **B57**, 45–72, 159–169, 403–433 (1954)
---, *An approximate equation for the correlation function connected with a non-linear problem*, in *Proceedings of the eighth international congress for applied mechanics* (University of Istanbul, Turkey) **2**, 89–103 (1955)
---, *A model for one-dimensional compressible turbulence with two sets of characteristics*, Proc. Roy. Neth. Acad. Sci. Amsterdam **B58**, 1–18 (1955)
---, *Statistical problems connected with the solution of a nonlinear partial differential equation*, in *Nonlinear problems of engineering* (W. F. Ames, editor), Academic Press, New York, 1964, pp. 123–137
---, *Functions and integrals connected with solutions of the diffusion or heat flow equation*, Tech. Note BN-398, The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, 96 pages (1965)
C.-W. Chu, *A class of reducible systems of quasi-linear partial differential equations*, Quart. Appl. Math. **23**, 257–278 (1965)
J. D. Cole, *On a quasi-linear parabolic equation occurring in aerodynamics*, Quart. Appl. Math. **9**, 225–236 (1951)
D. H. Cooper, *Integrated treatment of tracing and tracking error*, J. Audio Eng. Soc. **12**, 2–7 (1964)
S. Crow and G. Canavan, *Relationship between a Wiener-Hermite expansion and an energy cascade*, J. Fluid Mech. **41**, 387–403 (1970)
R. D. Fay, *Plane sound waves of finite amplitude*, J. Acoust. Soc. Amer. **3**,222–241 (1931)
E. Fubini-Ghiron, *Anomalie nella propagazione di onde acustiche di grande ampiezza*, Alta Frequenza **4**, 530–581 (1935)
A. Giorgini, *A numerical experiment on a turbulence model*, in *Developments in mechanics*, Johnson Publishing Co., 1968, pp. 1379–1408
Z. A. Goldberg, *Finite-amplitude waves in magnetohydrodynamics*, Soviet Physics JETP **15**, 179–181 (1962)
L. E. Hargrove, *Fourier series for the finite amplitude sound waveform in a dissipationless medium*, J. Acoust. Soc. Amer. **32**, 511–512 (1960)
W. D. Hayes, *The basic theory of gasdynamic discontinuities*, Chapter D in *Fundamentals of gas dynamics* (H. W. Emmons, editior), Princeton Univ. Press, Princeton, 1958, pp. 416–481
E. Hopf, *The partial differential equation* ${u_t} + u{u_x} = \mu {u_{xx.}}$. Comm. Pure Appl. Math. **3**, 201–230 (1950)
I. Hosokawa and K. Yamamoto, *Numerical study of the Burgers model of turbulence based on the characteristic functional formalism*, Phys. Fluids **13**, 1683–1692 (1970)
D. T. Jeng, R. Foerster, S. Haaland and W. C. Meecham, *Statistical initial-value problem for Burgers’ model equation of turbulence*, Phys. Fluids **9**, 2114–2120 (1966)
W.-H Kahng and A. Siegel, *The Cameron-Martin-Wiener method in turbulence and in Burgers’ model: general formulae, and application to late decay*, J. Fluid Mech. **41**, 593–618 (1970)
W. Keck and R. T. Beyer, *Frequency spectrum of finite amplitude ultrasonic waves in liquids*, Phys. Fluids **3**, 346–352 (1960)
N. N. Kochina, *On periodic solutions of Burgers’ equation*, J. Appl. Math, and Mech. **25**, 1597–1607 (1961)
R. H. Kraichnan, *Langrangian-history statistical theory for Burgers’ equation*, Phys. Fluids **11**, 265–277 (1968)
M. D. Kruskal and N. J. Zabusky, *Stroboscopic-perturbation procedure for treating a class of nonlinear wave equations*, J. Math. Phys. **5**, 231–244 (1964)
P. A. Lagerstrom, J. D. Cole and L. Trilling, *Problems in the theory of viscous compressible fluids*, Calif. Inst. Tech., 232 pages (1949)
W. Lick, *The propagation of disturbances on glaciers*, J. Geophys. Res. **75**, 2189–2197 (1970)
M. J. Lighthill, *Viscosity effects in sound waves of finite amplitude*, in *Surveys in mechanics* (G. K. Batchelor and R. M. Davies, editors), Cambridge Univ. Press, Cambridge, 1956, pp. 250–351
W. C. Meecham and A. Siegel, *Wiener-Hermite expansion in model turbulence at large Reynolds numbers*, Phys. Fluids **7**, 1178–1190 (1964)
---, and M.-Y. Su, *Prediction of equilibrium properties for nearly normal model turbulence*, Phys. Fluids **12**, 1582–1591 (1968)
J. S. Mendousse, *Nonlinear dissipative distortion of progressive sound waves at moderate amplitude*, J. Acoust. Soc. Amer. **25**, 51–54 (1953)
C. N. K. Mooers, *Gerstner wave’s Fourier decomposition and related identities*, J. Geophys. Res. **73**, 5843–5847 (1968)
Y. Ogura, *A note on the energy transfer in Burgers’ model of turbulence*, 75th Anniversary Volume of the Journal of the Meteorological Society of Japan, 1957, pp. 92–94
S. A. Orszag and L. R. Bissonnette, *Dynamical properties of truncated Wiener-Hermite expansions*, Phys. Fluids **10**, 2603–2613 (1967)
G. W. Platzman, *An exact integral of complete spectral equations for unsteady one-dimensional flow*, Tellus **16**, 422–431 (1964)
L. A. Pospelov, *Propagation of finite-amplitude elastic waves*, Soviet Physics Acoust. **11**, 302–304 (1966)
W. H. Reid, *On the transfer of energy in Burgers’ model of turbulence*, Appl. Sci. Res. **A6**, 85–91 (1956)
E. Y. Rodin, *Propagation of waves of finite amplitude in thermoviscous media*, NASA CR-643, 82 pages(1966)
---, *A Riccati solution for Burgers’ equation*, Quart. Appl. Math. **27**, 541–545 (1970)
---, *On some approximate and exact solutions of boundary value problems for Burgers’ equation*, J. Math. Anal. Appl. **30**, 401–414 (1970)
I. Rudnick, *On the attenuation of a repeated sawtooth shock wave*, J. Acoust. Soc. Amer. **25**, 1012–1013 (1953)
P. G. Saffman, *Lectures on homogeneous turbulence*, in *Topics in nonlinear physics* (N. J. Zabusky, editor), Springer-Verlag New York, 1968, pp. 485–614
M. E. Shvets and V. P. Meleshko, *Numerical algorithm of a solution of the system of equations of hydrodynamics of the atmosphere*, Izv. Acad. Sci. USSR Atmospher. Ocean. Phys. **1**, 519–520 (1965)
A. Siegel, T. Imamura and W. C. Meecham, *Wiener-Hermite functional expansion in turbulence with the Burgers model*, Phys. Fluids **6**, 1519–1521 (1963)
---, *Wiener-Hermite expansion in model turbulence in the late decay stage*, J. Math. Phys. **6**, 707–721 (1965)
S. I. Soluyan and R. V. Khokhlov, *Decay of finite-amplitude acoustic waves in a dissipative medium* (in Russian), Vestnik Moskov. Univ. Ser. III Fiz. Astronam. **3**, 52–61 (1961)
C. H. Su and C. S. Gardner, *Korteweg-de Vries equation and generalizations III: Derivation of the Korteweg-de Vries equation and Burgers equation*, J. Math. Phys. **10**, 536–539 (1969)
T. Tatsumi, *Nonlinear wave expansion for turbulence in the Burgers model of a fluid*, Phys. Fluids **12** (Part II), II 258–II 264 (1969)
G. I. Taylor, *The conditions necessary for discontinuous motion in gases*, Proc. Roy. Soc. **A84**, 371–377 (1910)
B. van der Pol, *On a non-linear partial differential equation satisfied by the logarithm of the Jacobean theta-functions, with arithmetical applications*, Proc. Acad. Sci. Amsterdam **A13**, 261–284 (1951)
R. A. Walsh, *Initial value problems associated with Burgers’ equation* (Sc. D. Thesis), Report No. AM-68-2, Washington University, St. Louis, Mo., 107 pages (1968)
J. J. Walton, *Integration of the Lagrangian-history approximation to Burgers’ equation*, Phys. Fluids **13**, 1634–1635 (1970)
P. J. Westervelt, *The mean pressure and velocity in a plane acoustic wave in a gas*, J. Acoust. Soc. Amer. **22**, 319–327 (1950)
N. J. Zabusky, *Phenomena associated with the oscillations of a nonlinear model string: The problem of Fermi, Pasta, and Ulam*, in *Proceedings of the conference on mathematical models in the physical sciences* (S. Drobot, editor), Prentice Hall, New York, 1963, pp. 99–133
P. A. Blythe, *Non-linear wave propagation in a relaxing gas*, J. Fluid Mech. **37**, 31–50 (1969)
J. G. Jones, *On the near-equilibrium and near-frozen regions in an expansion wave in a relaxing gas*, J. Fluid Mech. **19**, 81–102 (1964)
W. Lick, *Nonlinear wave propagation in fluids*, in *Annual reviews of fluid mechanics* **2**, (M. van Dyke, W. G. Vincenti, and J. V. Wehausen, editors), Annual Reviews Inc., Palo Alto, California, 1970, pp. 113–136
J. P. Moran and S. F. Shen, *On the formation of weak plane shock waves by impulsive motion of a piston*, J. Fluid Mech. **25**, 705–718 (1966)
M. Morduchow and A. J. Paullay, *Stability of normal shock waves with viscosity and heat conduction*, Phys. Fluids **14**, 323–331 (1971)

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