A table of solutions of the one-dimensional Burgers equation

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. Introduction and survey of literature. The quasilinear parabolic equation now known as the "one-dimensional Burgers equation," (du/dt) + u(du/dx) = v(d2u/dx), (1) first appeared in a paper by Bateman [4], who derived two of the essentially steady solutions (1.3 and 1.5 of our Table). It is a special case of some mathematical models of turbulence introduced about thirty years ago by J. M. Burgers [10], [11]. The distinctive feature of (1) is that it is the simplest mathematical formulation of the competition between convection and diffusion. It thus offers a relatively convenient means of studying not only turbulence but also the distortion caused by laminar transport of momentum in an otherwise symmetric disturbance and the decay of dissipation layers formed thereby. Moreover, the transformation u = -(2v/6)(dd/dx) (2) relates u{x, t) and d(x, t) so that if 6 is a solution of the linear diffusion equation (86/dt) = v(d'20/dx2), (3) then u is a solution of the quasilinear Burgers equation (1). Conversely, if u is a solution of (1) then 9 from (2) is a solution of (3), apart from an arbitrary time-dependent multiplicative factor which is irrelevant in (2). In connection with the Burgers equation, transformation (2) appears first in a technical report by Lagerstrom, Cole, and Trilling [38, especially Appendix B], and was published by Cole [21]. At about the same time it was discovered independently by Hopf [30] and also—in the context of the similarity solution u = t~l/2S(z), z = (4vt)~1/2x—by Burgers [14, p. 250]. The similarity form of the Burgers equation—the quasilinear ordinary differential equation for S(z)—is a Riccati equation [51], and can thus be regarded as a basis for motivating transformation (2) inasmuch as (2) is a standard means of linearizing the Riccati equation. More general hydrodynamical applications of this transformation have been discussed by Ames [1, chapter 2], Chu [20], and Shvets and Meleshko [55]. Received February 24, 1971. 196 EDWARD R. BENTON AND GEORGE W. PLATZMAN Another virtue of (1) is that although it appears to lack a pressure gradient term, it is in fact a deductive approximation for propagation of one-dimensional disturbances of moderate amplitude in a uniform diffusive compressible medium, if the variables are interpreted in a particular way. In this "acoustic" analogy for an ideal gas, the u, v and x, t of (1) are, respectively, Kt + i)u', I [si'' + v'. + (t lVL ^ x' — a0t', t', where u' is the fluid velocity, v' and v[ are respectively the kinematic shear and bulk viscosities, k' is the thermal diffusivity, 7 the specific-heat ratio (assumed constant), x' is an axis fixed in the undisturbed medium, i! is the actual time, and a0 is the speed of a linear sound wave. According to this analogy, t in (1) is the actual time and x is position in a coordinate frame moving with speed a0 relative to the undisturbed medium; thus du/dt in (1) expresses the relatively slow wave-form distortion caused by convection and diffusion, rather than the comparatively fast local changes associated with ordinary propagation. The restriction to moderate amplitude is necessary because the analogy depends upon approximations valid only for small Mach number of the disturbance. Moreover, the analogy is derivable only when the propagation is unidirectional, as typically for a "simple" wave or disturbance that moves into an infinite resting medium. Subject to the same restriction, an alternative statement of the acoustic analogy is to take u, v and x, t in (1) as [43], -1(7 + IK, *[V + V. + (7 iy], (5) a0t' — x', x'/a0. The first term of (1) then expresses spatial variation in the translating frame, variation which is equivalent to the temporal changes of the first analogy and corresponds to the same effects of convection and diffusion. Analogy (4) is convenient for initial-value problems, since when t' = 0 we have (x, t) = (x', 0) in this analogy; whereas (5) is convenient for boundary-value problems, since then (x, t) = (a0t', 0) when x' = 0. With varying degrees of rigor the derivation of the one-dimensional Burgers equation from the fundamental gas-dynamic equations under the restrictions inherent in the acoustic analogy has been accomplished for viscous, non-conducting ideal gases by Lagerstrom, Cole, and Trilling [38, Appendix B], and by Su and Gardner [59]; for viscous, nonconducting fluids with quadratic dependence of pressure on rate of expansion by Mendousse [43]; for viscous conducting ideal gases by Lighthill [40] and by Soluyan and Khokhlov [58]; and for viscous, conducting fluids of general equation of state by Hayes [29]. The equation also describes finite-amplitude transverse hydromagnetic waves [27], longitudinal elastic waves in an isotropic solid [48], and disturbances on glaciers [39]. An equation related very simply to the Burgers equation arises in a problem of number theory [62], In a remarkable series of papers extending over many years, Burgers (see references) studied statistical and spectral aspects of the equation (and related systems of equations) when initial conditions are given stochastically. Various aspects of the energy spectrum have also been investigated by Reid [49], Ogura [45] and Tatsumi [60]. More recently, the new deductive theories of turbulence have been "tested" on the Burgers model SOLUTIONS OF THE ONE-DIMENSIONAL BURGERS EQUATION 197 (see [23], [32], [33], [36], [41], [42], [46], [56], [57]) and various numerical experiments have been made on the Burgers equation (for example, see [5], [26], [31]). Saffman [54] has questioned the basis of the Kolmogorov law by using results derived from the Burgers model. The Burgers equation gives an analytic framework for a second-order theory of finite-amplitude dissipative sound propagation [8], [9], [34], [35], [43], [58], It has been used in discussions of shock structure in a Navier-Stokes fluid principally by Lagerstrom, Cole, and Trilling [38], by Lighthill [40] and by Hayes [29], In the notable work of Lighthill, the conflict between the steepening effect of nonlinear convection and the broadening trend of dissipation is made especially clear; this dual process is the essence of the Burgers equation. One of the most interesting solutions of the Burgers equation, the only known exact time-dependent spectral solution (2.6 in the Table and Figure 9), appears first in a paper by Fay [24] where it was derived in the acoustic framework but without the aid of the Burgers equation and with the role of space and time inverted as in (5) (as pointed out by Rudnick [53], there is a minor error in Fay's Eq. (14): the correct numerical factor is 2, not 8). The Fay series was re-discovered by Cole [21] as an approximate solution of the Burgers equation for a sinusoidal initial condition, and by Benton [6], [7] as an exact solution. The relation of Fay's solution to the corresponding in viscid spectral solution of (1), u(x, t) = —2 ^ (nt)~lJn(nt) sin nx, 71= 1 is thoroughly discussed by Blackstock [9] in connection with the sound field generated by sinusoidal motion of a one-dimensional piston (see also [2], [34], [52], [65]). This inviscid solution is known as the Fubini solution in the acoustics literature because of the work of Fubini-Ghiron [25]. It has been rediscovered by workers in several different fields (see [22], [28], [37], [44], [47], [66]). Description of table. The correspondence between (1) and (3) through (2) makes it easy to construct exact solutions of (1) by starting from solutions of (3). Although the general solution of (3) is known for arbitrary initial conditions, and the transformation (2) is trivial to perform, not all special solutions are physically "interesting." It seems worthwhile, therefore, to call attention to those that are. The purpose of the following Table is to present a list of such solutions, arranged in a somewhat systematic way, as a possible aid in further investigations of the Burgers equation. Some new solutions are included, but we have primarily aimed at collecting and organizing numerous results scattered through a somewhat diffuse literature. Eqs. (1, 2, 3) are invariant to a shift of origin x — x0 —> x, t — t0 —> t; u —> u, d —» d, (6) where x0 and t0 are arbitrary, independent constants. They are also invariant under a change of scale: x/a —> x, t/a—*t; alt —> u, /3d —» 0, (7) where a and (i are arbitrary, independent scale factors. Special cases of (7) that are useful in constructing the Table are — x—> x, t —> t; —u —> u (7a) 198 EDWARD R. BENTON AND GEORGE W. PLATZMAN which reverses the direction of the z-axis, and ix —> x, — t —> t; —iu —» u (7b) which rotates the z-axis 90 degrees in the complex z-plane. The third important invariance transformation, x — Ut •—> x, t —> t) u — U ■—> Uj d exp (U(x — Ut)/2v) —> 6, (8) represents translation of the reference frame at the constant speed U (Galilean invariance). In the Table we use nondimensional variables obtained by making the substitutions x/L —> x, vt/L2 —» t) uL/v —> u. (9) The insertion of v here formally has the effect v —> 1 in (1, 2, 3), whereas the length scale enters as in (7) and thus does not alter the equations. It should be noted that in many cases L is not an external parameter (for example, see Note 1.3). If two solutions are related through one or more invariance transformations, we say they are "equivalent" (for example, see Note 1.0 in the Table). If all the transformations in question are real, they do not alter the shape of the function on which they operate; we therefore say that such solutions are "isomorphic" (for example, Note 1.0). In the Table we list only real solutions (some of which are equivalent through complex transformations such as (7b)), and in the Not