2 Λ $^{′}$( r ) = 1 $_{2}$(8 π ) 1+1 $^{/α}$r ( r $^{2}$)1 $^{/α}$A 1 $^{/α}$e 2Λ( r ) [ 1 − e − 2Λ( r ) − 2 re − 2Λ( r $^{)}$Φ ′ ( r ) ] 1 /α − 1 2 r ( e 2Λ( r ) − 1 ) . (12) ℓ$_{1}$ ( r ) = ∫ r r 0 e Λ 1 ( r $^{$^{′}$)}$dr ′ q$_{1}$ ( r ) = Λ ′ $_{1}$( r ) + 1 K 1 2 r ( e 2Λ 1 ( r ) − 1 ) ( K − 1) (7) q$_{2}$ ( r ) = Λ ′ $_{2}$( r ) + 1 K 1 2 r ( e 2Λ 2 ( r ) − 1 ) ( K − 1) (8) Λ ′ $_{2}$( r ) = 1 K [ − [Φ $^{′}$( r ) + η $^{′}$( r )] − 1 2 r ( e 2Λ 2 ( r ) − 1 ) ( K − 1) ] (9) Φ( r ) = φ$_{0}$ + 1 + Ad 2(1 − d ) ln | r − r$_{0}$ | − 1 2 ln r − 32 π $^{2}$B d (1 − d ) r 4 0 ln | r − r$_{0}$ | − 32 π $^{2}$B d (1 − d ) [ 1 4 ( r − r$_{0}$ ) 4 + 4 r 3 $_{0}$( r − r$_{0}$ ) +3 r 2 $_{0}$( r − r$_{0}$ ) 2 + 4 3 r$_{0}$ ( r − r$_{0}$ ) 3 ] . (13) Λ ′ $_{1}$( r ) + ϵ $^{′}$( r ) = 1 K [ − [Φ $^{′}$( r ) + η $^{′}$( r )] − 1 2 r ( e 2Λ 1 ( r )+2 ϵ ( r ) − 1 ) ( K − 1) ] , (10) A = 1 d $_{2}$(64 π $^{2}$Br4 0 − d ) . (14) Φ( r ) = φ$_{0}$ − 1 2 ln r − 32 π $^{2}$B d (1 − d ) [ 1 4 ( r − r$_{0}$ ) 4 +4 r 3 $_{0}$( r − r$_{0}$ ) + 3 r 2 $_{0}$( r − r$_{0}$ ) 2 + 4 3 r$_{0}$ ( r − r$_{0}$ ) 3 ] , (15) ℓ$_{2}$ ( r ) = ∫ r r 0 e Λ 2 ( r $^{$^{′}$)}$dr ′ (11) by Chaplygin and generalized Chaplygin gas [7], modified Chaplygin gas [2], and even van der Waals quintessence [8]. For the case of a generalized Chaplygin gas, the EoS is p = − A/ρ $^{α}$, 0 < α ≤ 1, and where A is a constant [7]. After substituting and solving for Λ $^{′}$( r ), we get [4] For an “arbitrary” Λ( r ), this integral is not likely to be finite, given the asymptotic behavior of Λ( r ). For cer- tain choices of Φ( r ), however, ℓ ( r ) does exist, thereby producing a traversable wormhole. Examples are the ex- act solutions for Λ( r ) obtained by choosing the following redshift functions: Φ( r ) ≡ constant [6] and Φ( r ) = ln k r [3, 10]. Next, let us suppose that Λ$_{1}$( r ) is one of the above exact solutions [corresponding to some Φ( r )], so that is finite. Given that Φ $^{′}$( r ) is finite by assumption, we deduce from Eq. (5) that According to Ref. [7], there is an exact solution for Φ ′ ≡ 0, making ℓ ( r ) finite. In analogous manner, we now let Λ$_{1}$( r ) be an exact solution of Eq. (12). Then Λ$_{2}$( r )+ ϵ ( r ) is a new solution, corresponding to some Φ( r ) + η ( r ). (Assume that ϵ ( r ) and η ( r ) satisfy the same conditions as before.) The resulting proper distance ℓ$_{2}$ ( r ) is again finite. is also finite. To generate a new solution, we let ϵ = ϵ ( r ) be a function with a continuous derivative satisfying the conditions ϵ ( r$_{0}$ ) = ϵ $^{′}$( r$_{0}$ ) = 0 and ϵ ( r ) → 0 as r → ∞ . Define Λ$_{2}$( r ) = Λ$_{1}$( r ) + ϵ ( r ). Then by Eq. (7) An example that illustrates the problem of generalizing an exact solution even better comes from the modified Chaplygin gas model, whose EoS is p = Aρ − B/ρ $^{α}$, 0 < α ≤ 1, and where A and B are constants. This time we begin with Λ( r ) and determine Φ( r ). According to Ref. [2], one may choose b ( r ) = r$_{0}$ + d ( r − r$_{0}$ ), where d must be less than unity to meet the flare-out condition. The shape function b ( r ) determines Λ( r ), as well as Φ( r ) [2]: is finite. Now consider the equation for some finite differentiable function η = η ( r ). Writing Eq. (9) in the form This Φ( r ) is not finite unless A has the (corrected) value we see that, by comparison to Eq. (5), η $^{′}$( r ) must sat- isfy the conditions η $^{′}$( r$_{0}$ ) = 0 and η ( r ) → 0 as r → ∞ , thereby retaining the asymptotic flatness. The function η $^{′}$( r ) is defined only implicitly in Eq. (10), but since we are strictly interested in the existence of new solu- tions, it is sufficient to note that the formal relationship η $^{′}$( r ) = − Φ $^{′}$( r ) − Kq$_{2}$ ( r ) implies that η $^{′}$( r ) exists and is sectionally continuous, so that η = η ( r ) exists, as well. Remark: Since A is part of the EoS, it would be more realistic to state the condition as follows: determine d (0 < d < 1), if it exists, so that Eq. (14) is satisfied. For such a choice of d , we find that We conclude that Λ$_{2}$( r ) is a solution to Eq. (5) cor- responding to some finite redshift function Φ( r ) + η ( r ). Moreover, is finite since ϵ ( r$_{0}$ ) = 0. The result is the existence of an entire family of solutions, one for each ϵ = ϵ ( r ). Similar arguments can be used to obtain new classes of solutions from exact solutions of wormholes supported which has a well defined and continuous derivative. If we now substitute Eqs. (2) and (3) in the EoS and solve for