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charge-free solutions 16 were found to be an excellent ap- proximation for ideal dielectric materials. 4 The condition ∇ · P = 0 implies that the polarization vector variation is restricted to a plane perpendicular to s (the trajectory is constrained to P$_{s}$ = P$_{s}$ ( ±∞ ) plane). In fact, the polarization trajectories representing the 

T180 and T90 walls calculated under ∇· P = 0 constraint were found to be the straight lines connecting the bound- ary values. 1,2,4,16 This greatly simplifies the algebra and the solution of the variational problem can be found an- alytically. Therefore, we decided to impose the condition of a direct, straight polarization trajectory for the varia- tional problem of all domain wall species of BaTiO$_{3}$. This condition, further referred as straight polarization path (SPP) approximation, implies that both P$_{s}$ and P$_{t}$ po- larization components are constant across the wall. We shall come back to the meaning and possible drawbacks of this approximation in Section VI. 

Polarization and strain in the mechanically compat- ible and electrically neutral SPP walls can be cast in the form: P = [ P$_{r}$ ( s ) , P$_{s}$ ( ±∞ ) , P$_{t}$ ( ±∞ )] and e = [ e$_{rr}$ ( ±∞ ) , e$_{ss}$ ( s ) , e$_{tt}$ ( ±∞ ) , 2 e$_{st}$ ( s ) , 2 e$_{rt}$ ( ) , 2 e$_{rs}$ ( s )], re- spectively. The s -dependent strain components are cal- culated from the mechanical equilibrium condition 

Boundary conditions for stress and the fact that all quan- tities vary only along direction s imply 

for ij ∈ { ss , st , rs } with the integration constant C to be determined from boundary conditions. 

The Euler-Lagrange equation for polarization reduces to 

where the strain components from Eqn. (14) were substi- tuted into f . Thus, the elastic field was eliminated. It turns out that the resulting Euler-Lagrange equation for P$_{r}$ is possible to rewrite in the form 

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where p ( s ) stands for P$_{r}$ ( s ), and where the coefficients g, a$_{1}$, a$_{11}$ and a$_{111}$ , different for each domain type, depend only on the material tensors. The boundary values are − P$_{r}$ ( ∞ ) = P$_{r}$ ( −∞ ) = p$_{∞}$ . Solution of the Euler-Lagrange equation (16) is well 

known. 1,16–19 We shall follow the procedure of Ref. 20. Integrating Eqn. (16) one can obtain the equation: 

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where (see Ref. 4) 

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The function f$_{EL}$ ( p ) is a double-well ”Euler-Lagrange” potential with two minima ± p$_{∞}$ , where 

The differential Eqn. (16) has the analytical solution 

where 

and 

Quantity A determines deviation of the profile (20) from the tanh profile, which occurs for the 4 $^{th}$-order potential (i.e., a$_{111}$ = 0 , A = 1). 

The domain wall thickness (Fig. 3) is defined as 

with U being the energy barrier between the domain states: 

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