The general one-dimensional solution of Burgers equation is developed using a series of transformations and the Green's function method. The solution gives the distribution of the reduced water content in a semi-infinite and in a finite domain for arbitrary initial moisture distributions. The boundary condition at the soil surface can be either a time-dependent flux or constant water content while the bottom boundary condition for the finite case is a constant water content value. Explicit results for a uniform, discrete, hydrostatic and steady state initial moisture profiles are derived. An analytical solution that spans the preponding to the postponding period of rainfall is also obtained. Expressions for the time to ponding and infiltration equations are presented for the various cases of initial moisture profiles and a shallow water table. These results allowed the quantification of the error in the various approximations to the time to ponding and the cumulative infiltration in the postponding period.
 Analytical solutions of the nonlinear Richards' equation are relatively few. Most closed-form solutions are derived from the linearized version of Richards' equation, which is based on the assumption of constant soil water diffusivity and a linear dependency of the hydraulic conductivity on the moisture content. The latter assumptions also imply that the dependence of the hydraulic conductivity on the pressure head is exponential. The resulting linear form of the governing equation allowed the derivation of a number of solutions for one-, two- and three dimensional soil water flows subject to various initial and boundary conditions, and root uptake forcing functions [e.g., Warrick, 1975; Philip, 1986; Basha, 2000].
 However, nonlinear exact solutions of Richards' equation are quite few and they are all restricted to one-dimensional flows. There are mainly two classes of closed-form nonlinear solutions. The first class is based on the assumption of constant soil water diffusivity and a quadratic dependence of the hydraulic conductivity on the moisture content. The resulting equation is known as the Burgers equation for soil water flow [Philip, 1974; Clothier et al., 1981]. The second class of exact nonlinear solutions is based on the Fujita  functions for the soil hydraulic properties. The diffusivity is expressed by an inverse quadratic function of the water content and the hydraulic conductivity function is given in a rational form. The governing nonlinear partial differential equation is then reduced after a series of transformation into the weakly nonlinear Burgers' equation [e.g., Sander et al., 1988]. All these transformations apply to one-dimensional flows only. However, the transformations used in the first class can handle flux and concentration boundary conditions and yield solutions that are explicit in depth. The Fujita-based solutions of the second class are based on three successive transformations that apply only to flux boundary conditions and yield solutions in parametric form and implicit in depth. The limiting cases of the Fujita-based solutions yield the Burgers' solutions or the linear solution depending on the limiting value of the parameter of the hydraulic conductivity function.
 In the first class, Philip  presented solutions of J. H. Knight for infiltration into a semi-infinite medium subject to a constant flux or to constant water content at the soil surface. The constant flux solution was presented again with minor corrections by Clothier et al.  and was compared with experimental data. Philip  used the constant water content solution of Burgers equation to derive expressions for the infiltration rate and the cumulative infiltration at the soil surface. Broadbridge and White  presented an expression for the time to ponding for a linearly varying rainfall using Burgers' equation. Broadbridge and Rogers  presented a one-dimensional drainage solution of Burgers' equation for an initially saturated vertical column in a semi-infinite medium. Hills and Warrick  published a solution of Burgers' equation for a finite region whereby the upper boundary condition is a time-dependent flux and the bottom condition is a constant water content. Finally, Warrick and Parkin  derived the same one-dimensional drainage solution of Broadbridge and Rogers  from the limiting cases of the Fujita-based solution and provided graphical comparisons and simplifying relationships for small and large times.
 In the present work, the class of analytical solutions of Burgers' equation is further explored. This class can be considered as a natural extension of the linear theory into the nonlinear one, since it is also based on a constant diffusivity but with a quadratic hydraulic conductivity function rather than a linear one. It therefore enables us to study the effect of the nonlinearity in the hydraulic conductivity on the moisture movement. The general solution of Burgers' equation in a bounded profile and in a semi-infinite medium is presented for the two types of boundary conditions at the soil surface: a time-dependent flux and a constant water content condition. The bottom condition is a constant water content to simulate the presence of a water table. The derived solutions extend the previous exact solutions of Burgers' equation to include the effect of an arbitrary initial profile and time-dependent surface flux in the presence of a shallow water table. It further allows the modeling of nonlinear infiltration from the preponding to the postponding stage.
Section 2 presents Burgers' equation with the various initial and boundary conditions and section 3 derives the general solution for the cases of a prescribed flux and water content condition at the soil surface. Section 4 presents various particular analytical solutions while section 5 discusses the results pertaining to the time to ponding and postponding infiltration. Section 6 concludes the study.
2.1. Governing Equation
 The nonlinear differential equation describing water flow in the unsaturated zone can be expressed by
where θ = volumetric moisture content, k = unsaturated hydraulic conductivity [L/T], ψ = the pressure head [L], z = the vertical coordinate (positive downward), and t = the time. The parameter D is the soil water diffusivity defined as
In the present work, the diffusivity D is taken as constant, and the hydraulic conductivity is assumed a quadratic function of the moisture water content [Philip, 1974; Clothier et al., 1981]
where, ks = k(θs), k(θr) = 0 and dk/dθ = 0 at θ = θr. The hydraulic conductivity can also be expressed in the general form k = a(θ − b)2 [Hills and Warrick, 1993], which is equivalent to (3) if b = θr and a = ks/(θs − θr)2. The parameters ks, θs and θr can then be considered as fitting parameters. For a constant diffusivity D, the pressure head dependence of the moisture characteristic can be derived from (2) using ψ = ψe = 0 at θ = θs
The hydraulic conductivity then becomes
Equations (4) and (5) are of a simple form because of the assumptions leading to equation (3). If k(θr) = kr ≠ 0, equations (4) and (5) would have involved tangent functions in an implicit fashion. Equations (3) and (5) show that the hydraulic characteristics of the soil are nonlinear functions; however the functional form of the nonlinearity is such that the diffusivity (2) becomes linear.
 Introducing the following dimensionless variables
Hence the transformation of the prescribed flux boundary condition (11) results in the simpler first type (Dirichlet) boundary condition (14).
2.2.2. Surface Condition: Constant Moisture
 For the constant moisture content boundary condition, , the corresponding transformed equation is from (8)
The transformation of a constant moisture content, , a first-type (Dirichlet) boundary condition in linear theory, yields the relatively more difficult third-type (Fourier) boundary condition (15) in Burgers equation.
2.2.3. Bottom Condition
 For the semi-infinite system, the far-field condition is taken as equal to the initial condition value at infinity, while for a finite depth domain, a constant water content is prescribed at depth L yielding
2.3. Initial Conditions
 The general initial condition can be expressed as
For a general form of , the initial condition becomes after transformation (8)
The integration constant was arbitrarily set to zero since it does not have any effect on the solution for θ.
2.3.1. Step Profile
 For a piecewise constant moisture profile over a defined depth D0
The transformed initial condition becomes
2.3.2. Hydrostatic Profile
 For a hydrostatic condition, the moisture profile can be integrated directly from Darcy's law (11) using = 0 and the condition at Z = L to get
In terms of v, the initial hydrostatic condition becomes
2.3.3. Steady State Profile
 The steady state initial moisture profile due to a steady flux i in a finite depth soil can also be obtained by integrating Darcy's law (11) using the condition at Z = L
For a semi-infinite medium, the steady moisture profile is uniform,, as expected from q = k and the definition of the hydraulic conductivity function (3). Hence the transformed initial condition is . For a dry uniform profile, and vi = 1.
3. General Solution
 The general solution of (9) is expressed in an integral form in terms of the Green's function. The form of the Green's function is dependent on the type of the boundary conditions at the surface and at the lower boundary. For the two types of boundary conditions at the soil surface and for the two soil domains, there are four Green's functions. The Green's function is obviously simpler in form for a semi-infinite domain than for a bounded domain with a prescribed water content at depth L, which involves infinite series terms and roots of transcendental functions. The complexity of the Green's functions is the result of the transformed third-type boundary conditions.
3.1. Prescribed Flux
3.1.1. Semi-Infinite Domain
 The general solution for the prescribed flux boundary condition at the land surface is [Basha, 1999]
where the first integral pertains to the initial condition and the second integral relates to the boundary condition at the surface. The Green's function for a semi-infinite domain with a flux (type I) boundary condition at Z = 0 is [Basha, 1999]
3.1.2. Finite Domain
 The general solution v for a finite domain can also be obtained from equation (25) whereby the upper limit of the first integral is L rather than infinity. However, the resulting series for the derivative term ∂v/∂z is not computationally suitable because of its slow convergence. An alternate form of the solution can be obtained by defining v = v1 + v2 where v1 satisfies the boundary conditions and v2 satisfies the transformed governing equation and initial condition. The solution v then becomes
where vh is the hydrostatic distribution given by (22) and v0 is the boundary condition at Z = 0 (14). The Green's function GLf for a finite domain with a constant moisture content (type III) boundary condition at Z = L is [Beck et al., 1992]
and βm is the solution of the transcendental equation
3.2. Prescribed Moisture Content
3.2.1. Semi-Infinite Domain
 The general solution is expressed in terms of the Green's function and the initial profile only
The integral pertaining to the surface boundary condition cancels out since the corresponding boundary conditions (15) is homogeneous (i.e., zero). The Green's function for the semi-infinite domain is [Basha, 1999]
3.2.2. Finite Domain
 The general solution is still given by (30) whereby the upper limit of the integral is L rather than infinity. The Green's function for a finite domain is derived using the separation of variable techniques. The solution is in a series form in terms of eigenvalues that are the infinite positive roots of a transcendental equation. However, for , the eigenvalues simplify to λm = mπ/L. The Green's function is
The restriction is a valid assumption for infiltration from a saturated soil surface toward a water table .
4. Results and Application
 The following solutions model the one-dimensional nonlinear infiltration process under a prescribed surface flux or water content, and for various initial moisture content distributions. The analytical solutions derived below are all expressed in terms of v from which the dimensionless moisture content can be derived using (8). The pressure profile can then be determined from .
4.1. Rainfall Infiltration
 The following solutions pertain to the one-dimensional infiltration into semi-infinite and finite media under a flux boundary condition at the land surface. They can also be useful in modeling isothermal evaporation by setting negative values to the surface flux. However, the numerical evaluation of the solutions requires a special treatment since the argument of the error function is complex in such a case.
4.1.1. Semi-Infinite Medium
22.214.171.124. Constant Flux Rate With Uniform Initial Conditions
 For a constant flux rate 0 and uniform initial conditions , the solution is obtained from the Green's function solution (25)–(26) using equations (14) and (18) The solution is the sum of two similar expressions: one accounting for the constant flux vqc and the other for the uniform profile vθu. Hence, where v = vθu + vqc where
The function f(Z, T, ϑ) is defined by
where erfc is the complementary error function. The derivative of v = vθu + vqc is
126.96.36.199. Constant Flux Rate With Steady Initial Conditions
 For a steady state initial moisture profile (23), the solution is obtained from (27)–(28) using (14) and (24)
Figure 1 presents the moisture propagation at various times in a finite depth soil of length L = 4 for a constant infiltration with a normalized flux 0 = 0.75 and an initial hydrostatic condition i = 0. The initial and final steady moisture profiles are also shown. One notices the formation of the traveling wave and its subsequent dissipation at the water table before its full development. This is due to the presence of a shallow water table and the effect of the capillary fringe. Note also that in the top part of the soil, the final steady state solution can be well approximated by the semi-infinite solution .
4.2. Ponded Infiltration
 The following solutions model the one-dimensional infiltration process into a semi-infinite and a finite media under moisture content conditions at the land surface.
4.2.1. Semi-Infinite Medium
188.8.131.52. Constant Moisture Content with Uniform Initial Distribution
 For a uniform initial condition, , the solution is obtained from (30) and (31)
where the function f is as defined in (35).The derivative is
For dry initial conditions,, the moisture distribution simplifies to the solution of J. H. Knight [Philip, 1974, equation 24].
184.108.40.206. Constant Moisture Content with Nonuniform Initial Profile
 The solution for ponded infiltration with initial condition that of the distribution at time of ponding can be obtained from (30) and any of the flux-based solutions previously derived. For example, for a constant infiltration 0 in a semi-infinite medium with dry initial condition , the solution at time of ponding is given by the sum of (33) and (34) with T = Tp. The solution for the postponding period T > Tp is then
Equations (33), (34) and (41) constitute an exact nonlinear solution that span the preponding and the postponding periods of rainfall. Equation (41) will be later used to obtain a better estimate of the infiltration rate and cumulative infiltration in the postponding stage than the traditional assumption of instantaneous ponding.
4.2.2. Shallow Water Table
220.127.116.11. Constant Moisture Content With Uniform Initial Distribution
 For ponded infiltration toward a shallow water table with a uniform initial distribution, , the solution is obtained using (32)
18.104.22.168. Constant Moisture Content With Steady State Initial Profile
 For ponded infiltration in the presence of a shallow water table with a steady initial distribution (23), the solution is given by
For a hydrostatic initial distribution, i = 0, the solution (43) can be further simplified as equation (44) becomes .
4.3. Drainage and Redistribution
 The above rainfall infiltration solutions can also model the process of drainage and redistribution noting however that capillary hysteresis reduces somewhat the relevance of these results. Equations (33) and (34) with 0 = 0 and gives the drainage solution of an initially saturated vertical column in a semi-infinite medium as derived by Broadbridge and Rogers [1990, equation 61] and Warrick and Parkin [1995, equation 11]. Drainage from a uniform profile with a nonzero flux 0 at the land surface is also given by equations (33) and (34) while the moisture redistribution from an initial step profile (19) is given by equations (37) and (34). An explicit solution for drainage from a linear moisture profile can be similarly derived. Redistribution solutions with evaporation at the land surface can also be obtained from the above flux-based solutions by setting the flux rate 0 to a negative value.
 Two important derivative results of the above solutions concern the estimation of the time to ponding for a given rainfall event and the infiltration rate in the postponding stage. The time to ponding expressions are obtained from the rainfall infiltration solutions by determining the time at which the moisture content at the soil surface is . The infiltration equations are obtained from the ponded infiltration solutions by deriving the equation of the flux at the soil surface using (13). The cumulative infiltration is the integral of the infiltration rate and is given simply by = log v. In the ensuing analysis, it is assumed that the moisture content at the bottom boundary is , which is a valid assumption for field applications. For laboratory settings, the derivation of the following results for can be similarly obtained from the analytical solutions.
5.1. Time to Ponding Expressions
 The time to ponding is defined as the time at which water first appears at the soil surface and runoff is thereby initiated. The time to ponding is mathematically defined as the time at which the water pressure potential becomes equal to zero. One must therefore solve for T given that the reduced moisture content at the surface Z = 0 is equal to 1. The following subsections present time to ponding expressions for various conditions.
5.1.1. Semi-Infinite Medium
 For a constant rainfall intensity in a semi-infinite medium, the solution is given by (33) and (34). Hence the time to ponding Tp is the solution of
For zero initial conditions , equation (45) yields the time to ponding explicitly in terms of the inverse error function
For nonzero initial conditions, the solution is obtained by solving (45) for Tp numerically using the Bisection or the Newton-Raphson method. Using Broadbridge and White's  finding, one can also obtain an approximate solution of (45) for a nonzero initial condition from
5.1.2. Finite Medium
 The time to ponding for a finite domain can be obtained from the corresponding analytical solutions. For example, the time to ponding expression for a steady initial distribution is
Figure 2 presents the time to ponding for various initial conditions in a finite and semi-infinite medium. The initial distribution is either a uniform profile or a steady state profile (i = 3−1). For a finite depth domain, the depth to water table is L = 0.5. Also shown are the time to ponding variation for a linear soil with dry initial conditions [Basha, 1999, equation 32] and the approximate expression (47). Figure 2 clearly shows that the time to ponding in a linear system is significantly higher than for the nonlinear one. The difference between the linear soil and a quadratic soil for = 2 is around 20% and it increases to 50% for = 1.2. This is due to the strongly diffusive character of a linear system that can transmit a larger volume of water. One also notices that for high flux rates ≥ 2, there is no significant difference between the time to ponding for a semi-infinite system and a finite one as shown in Figure 2 for and L = 0.5. However, for low flux rates, the effect of the water table depth becomes apparent. For = 1.2, there is a 32% difference in Tp between a semi-infinite system and a finite system with L = 1 and . For = 2, this difference decreases to less than 2%. However, for shallower water table with L = 0.5 and , the difference in Tp for = 2 is around 40% and much higher for lower . However, these error decrease with increasing initial water content value. For and L = 0.5, the difference in Tp for = 2 decreases from 40% to around 12% as shown in Figure 2. The expression (47) for nonzero initial conditions is also a roughly good approximation for Tp. The error decreases with increasing and decreasing . For = 1.5, the error is around 11% for and around 4% for .
 The effect of a water table on the time to ponding can be related to the value of the water content at the surface. Figure 2 presents the time to ponding curve for an initial steady state profile with = 1/3 and for an equivalent uniform profile. The equivalent uniform value for is set equal to the surface value of the initial steady distribution as obtained from (23). The uniform profile approximation approaches the exact curve for high flux rates ≥ 3, implying that the time to ponding is primarily affected by the surface value of the initial moisture distribution. However for lower flux rates, the moisture distribution in the deeper layer has a significant effect. For a wetter initial profile in the deeper layers, the time to ponding is shorter. Hence the shape of the initial distribution has a significant effect on the time to ponding, especially for low flux rates .
5.2. Infiltration Equations
 The infiltration rate and the cumulative infiltration equation are of practical importance in quantifying the infiltration component in rainfall-runoff models. They are also of use in providing a theoretical framework for estimating the hydraulic parameters of the soil. The following subsections present infiltration equations for various initial conditions.
5.2.1. Instantaneous Ponding
 For a semi-infinite medium with a uniform initial profile, the infiltration rate at the surface is obtained from (13) and (39)
The dimensionless cumulative infiltration can also be obtained in closed form using (39)
A series approximation for for small and intermediate T values can also be derived from (49)
The approximate cumulative infiltration is therefore
where ϕm for a steady initial distribution is given by
Equations (53) and (54) with s = 0 gives the infiltration equation for a hydrostatic initial distribution. A similar expression can also be derived for the case of a uniform initial profile. The cumulative infiltration at the soil surface for a steady state initial profile is
Figure 3 presents the infiltration rate for various initial conditions. These include the infiltration rate for a uniform profile and in a semi-infinite medium (49), uniform profile in a finite medium with L = 0.5 and L = 1.0, and for a steady state profile with = 1/3 and L = 0.5, equations (53) and (54). Also shown is the infiltration curve for a linear soil [Basha, 1999, equation 47], the approximation (51) for a semi-infinite soil, and a two-term approximation of (54). The infiltration rate for the linear soil is higher than for the nonlinear one due to the strongly diffusive character of the linear soil. However, the maximum percent difference does not exceed 10%. The initial condition has also a strong effect on the infiltration curve. The reduction in infiltration capacity for nonzero initial conditions is roughly proportional to for small T as can be deduced from (51). The expression (51) is a very good approximation of the exact equation (49) for T ≤ 1 and the accuracy increases for increasing values of . The error at T = 1 is around 3% for and it decreases to around 1% for . In case of a uniform initial profile, the semi-infinite solution is also a good approximation of the finite depth one at small times, especially for increasing values of the water table depth L. The difference is less than 1% for T ≤ 0.25 with L = 1 and for T ≤ 0.05 with L = 0.5. Figure 3 shows also the infiltration rate for a steady state initial profile along with the two-term series approximation. As expected, the infiltration rate is smaller than in the previous cases since the soil is wetter. The two-term series approximation produces results with excellent accuracy for T ≥ 0.04. The cumulative infiltration equation can also be approximated using the first two-term of the series and the results are of similar accuracy as the infiltration rate.
5.2.2. Postponding Infiltration
Equation (41) is an exact analytic solution of infiltration in the postponding period of rainfall. It allows us to estimate the error in the various approximations to the cumulative infiltration in the postponding stage. These approximations are mainly of two kinds. The first approach extends the time to ponding expression for a constant rate rainfall to an infiltration equation for the postponding period [Smith and Parlange, 1989]. In the time to ponding expression such as (45), the term f = 0 is defined as the infiltration flux for the postponding period and the product F = 0Tp is defined as the cumulative infiltration for the postponding period. The resulting equation becomes an expression relating f with F using Tp = F/f that is then used as the postponding infiltration equation. The second approach uses the instantaneous ponding solutions to estimate the ponding time and the infiltration in the postponding period. The latter approach is known as the time compression approximation (TCA) and consists of time shifting the infiltration curve by an appropriate value [Salvucci and Entekhabi, 1994]. In TCA, the time shift value is ΔT = Tp − Tc where Tc is the time at which the instantaneous ponding curve intersects the rainfall intensity and Tp is the time at which the cumulative infiltration 0Tp is equal to the cumulative infiltration at Tc as computed from (50). In the modified time compression approximation (MTCA) [Parlange et al., 2000], the time shift value is defined as ΔT* = Tp* − Tc* where Tp* is the exact value of the time to ponding (45) and Tc* is the time at which the cumulative infiltration in (50) is equal to 0Tp*.
 For the data 0 = 1.5 and used in Figure 4, the instantaneous ponding curve intersect 0 = 1.5 at Tc = 0.163. From (50), Q(Tc) = 0.393 and Tp = Q(Tc)/0 = 0.262. The time shift value to be used in TCA is therefore ΔT = Tp − Tc = 0.099. For MTCA, the exact time to ponding as computed from (45) is Tp* = 0.332 and the cumulative infiltration at time of ponding is 0Tp* = 0.498 that corresponds to Tc* = 0.237 using (50). Hence the time shift is ΔT* = Tp* − Tc* = 0.095. From the above numerical results, one notices that the difference between the two time shift values is less than 5%. The cumulative infiltration and infiltration rate after ponding are given by the instantaneous ponding results (49) and (50) but with T replaced by T − ΔT or by T − ΔT*. Although the equivalent time to ponding Tp underestimates the exact value Tp* by around 21%, both approximations calculate the cumulative infiltration to a very good accuracy. The maximum error in the cumulative infiltration in TCA is around 1.5% while the error in MTCA is half of that value. Both maximum errors occur around the time to ponding. The maximum error in the flux rates in both approximations is around 9%, noting further that the infiltration curve in MTCA is discontinuous [Parlange et al., 2000].
Figure 4 presents the variation of the exact infiltration rate as a function of the dimensionless time for a semi-infinite domain with a uniform initial condition . The exact dimensionless infiltration rate is constant and equal to the dimensionless rainfall rate 0 = 1.5 from time zero until ponding at which time the infiltration rate decreases according to the postponding solution (41). Also shown in Figure 4 is the infiltration capacity relationship as obtained from the instantaneous ponding case (IPA) (49), the constant flux case (CFA) (45), and the time shift approximations TCA and MTCA. One can notice the extent of the error in the various approximations noting that the area under the curve gives the cumulative infiltration. The flux-based solution (CFA) was proposed as a possible infiltration equation for the postponding stage. Figure 4 clearly shows that it overestimates the exact cumulative infiltration significantly. It intersects the exact curve only at the time of ponding and it deviates further as time increases. On the other hand, the instantaneous ponding curve underestimates the exact cumulative infiltration while the TCA and the MTCA approaches are very good approximations.
 In the present work, analytical solutions to Burgers' equation for various initial and boundary conditions were presented. Explicit results for a uniform, discrete, hydrostatic and steady state initial moisture profiles were derived. Analytical solutions were also obtained for infiltration that spans the preponding to the postponding period of rainfall. Solutions for other initial and boundary conditions can also be similarly derived from the general Green's function solution. These include the more realistic case of a time-varying rainfall distribution at the soil surface. The analytical solutions were further used to assess the effect of the initial conditions and the presence of the water table on the estimation of the time to ponding and the postponding infiltration. These results allowed us to quantify the error in Tp using the constant rainfall assumption and to estimate the error in the various approximations to the cumulative infiltration in the postponding stage. The above analytical solutions are also useful to numerical modelers for benchmarking especially because the nonlinear characteristics of soil water flow are maintained. Specifically, they provide a way of checking pressure-based numerical models since the governing equation is then fully nonlinear as opposed to the moisture content formulation whereby the equation is mildly nonlinear due to the constant diffusivity term. Although the constant diffusivity solutions have some limitations, they can still be useful in predicting integral properties such as the cumulative infiltration, and in providing generic insights into the nonlinear characteristics of infiltration, in contrast with the linear theory of infiltration in which these nonlinear characteristics are subdued.
 The thorough and constructive review of an anonymous reviewer is gratefully acknowledged. This work was partially supported by the University Research Board at the American University of Beirut.