Traveling wave solutions to modified Burgers and diffusionless Fisher PDE's

We investigate traveling wave (TW) solutions to modified versionsof the Burgers and Fisher PDE’s. Both equations are nonlinear parabolicPDE’s having square-root dynamics in their advection and reaction terms.Under certain assumptions, exact forms are constructed for the TW solutions.

For physical systems, u(x, t) is either a particle number or density and, as a consequence, must be non-negative, i.e., u(x, t) ≥ 0. (1.5) In fact, for the general physical meaningful solutions to Eqs. (1.1) and (1.2), the condition is required [2,5,11] where M is a fixed positive constant. We assume, without proof, that this requirement holds.
In Section 2, we use the method of variable scaling [8,9] to rewrite Eqs. (1.1) and (1.2) in terms of dimensionless variables. Section 3 provides a brief summary of the important general properties of TW solutions. In the next two Sections 4 and 5, we respectively, derive the TW solutions to dimensionless forms of Eqs. (1.1) and (1.2), and discuss several of their mathematical properties. In the final Section 6, we give a summary of the obtained results and consider several possible extensions of this work.
The following abbreviations will be used in the remainder of the paper: MBESRD -modified Burgers equation with square-root dynamics MFESRD -modified, diffusionless Fisher equation with square-root dynamics PDE -partial differential equation TW -traveling wave WF -wave front where (T, L, U ) are the respective time, space and dependent variable scales [6,7], and (t,x,ū) are the corresponding dimensionless new variables. Substitution into Eq. (1.1) and rewriting, gives If we replace U by the positive constant M appearing in Eq. (1.6) and then set the coefficients to one, then we obtain for which solving for T and L gives the following time and space scales Note that M is the scale for the u variable. Using the results of Eq. (2.3) and dropping the bars, gives the dimensionless form for the MBESRD In a similar manner, the MFESRD, Eq. (1.2), can be rewritten to the form with the rescaling variables and the dropping of bars on the new variables.
In the calculations to follow the dimensionless PDE's, Eqs. (2.5) and (2.6), are used. Further note that for both PDE's there are no free parameters, i.e., the original PDE's having, respectively, two and three dimensional parameters, have been rescaled such that the dimensionless equations do not contain any parameters.
Finally, it should be noted that an earlier publication by Soluyan and Khokhov [10] derived an equation similar to our Eqs. (2.5) and (2.6), using the approximate equations of relaxation gas dynamics; see their Eq. (20). However, these calculations are not directly relevant to our current paper and, as a consequence, no discussion will be given of their findings.
3. Properties of TW solutions. Consider a general parabolic PDE [2,4,8,9] We assume H(. . . ) has properties such that solutions exact, whether unique or not. A TW solution to Eq. (3.1) has the following properties [2,5,11] (i) For the purposes of this paper, we select the following values for f 1 and f 2 TW solutions generally exist in two forms. First, there does not exist a finite z 0 , such that Second, there exists a z 0 , such that f (z 0 ) = 0 and Note that at z = z 0 , the derivative may or may not exist [2,9]. After the preliminaries given in Sections 2 and 3, we are now at a point to examine whether the MBESRD and MFESRD have TW solutions. 4. TW's for MBESRD. The substitution u(x, t) = f (x − ct) into Eq. (2.5), with a rearrangement of the terms, gives We also assume the following conditions to hold Integrating Eq. (4.1) once gives where A is a constant. If the Lim z → +∞ is taken, then Likewise, taking the Lim z → −∞, gives  This equation is a separable first-order ordinary differential equation and it can be solved using elementary methods, as follows: where B is an integration constant. Making the transformation of dependent vari- The integral on the left-side can be easily calculated using the method of partial fractions. Carrying out this calculation and selecting v(0) = 1 2 , an arbitary choice, we finally obtain (4.10) (4.11) Inspection of the result in Eq. (4.11) immediately shows that Note that the original differential equation, Eq. (5.1), has two fixed-points or constant solutions,f Consequently, we have Since v(z) has negative slope, the non-negativity of v(z), Eq. (5.2) and the satisfaction of the boundary conditions, Eq. (5.5) gives from which it follows that v(z) is a piece-wise continuous function whose mathematical structure takes the form Thus, the TW solution, for c = 1, is Figure 1 provides a sketch of v(z) and f (z). The third case is for c > 1. Since Eq. (5.3) is a separable equation, it can be rewritten to the expression where, using the method of partial fractions [8], its solution is Using the mathematical properties of v(z) and f (z) = v(z) 2 , allows the general features of these functions to be determined. Figure 2 is a representation of these results. Note, in particular, that v(z) has a discontinuous derivative at z = z 0 , while f (z) has a continuous derivative with the value f (z 0 ) = 0. Remarkably Eq. (5.15) can be explicitly solved for v(z) and expressed in terms of the Lambert-W function [4,8,11]. This exact solution is (c > 1) Note that W 0 is the principal branch of the W -function.
Let us now calculate the magnitude of the jump discontinuities in dv/dz and df /dz for Eqs. (5.10) and (5.13). Note, however, that both v(z) and f (z) are continuous for all finite values of z.
If a function g(z) is discontinuous at z =z, then the magnitude of the jump discontinuity is defined as follows Since W 0 (0) = 0 [4,8,11], we obtain v(∞) = 1, (5.22) an expected result based on our prior discussion.
We can obtain more detailed information, on the z → −∞ limit, by using [4,8,11] the relation Since [8,11] W 0 (λe λ ) = λ, From this expression it follows that the magnitude of the jump discontinuities in v (z) and f (z) = 2v(z)v (z), are In summary, f (z) = v(z) 2 and f (z) = 2v(z)v (z) are continuous for all values of z, including z = z 0 . However, v(z) is continuous for all z, but its derivative, v (z), is discontinuous at z = z 0 . Mathematically, we have and 6. Discussion. We have investigated two nonlinear PDE's with regard to their TW solutions. One was a modified version of the standard Burgers equation, while the other corresponded to a modified Fisher equation, but for which the diffusion term is absent. An important feature of these two equations is the fact that the usual nonlinear advection term, uu x , is replaced by √ uu x . The major consequence of this change is that the TW stops at some value of z = z 0 , and for z > z 0 , the TW is zero. This is another confirmation of the, in general, "finite dynamics" of nonlinear differential equations containing fractional powers of the dependent variable and/or their derivatives [1,2,6,7,8]. A significant finding is that both of our PDE's can be solved to yield exact solutions which can be expressed in terms of either the elementary or Lambert-W functions.
A future project is to investigate the following Fisher type, nonlinear PDE This differs from one of the PDE's examined in this work by the inclusion of a diffusion term.