Solvability Conditions

The solvability condition for (6.2.29b) (elimination of secular terms in P2) implies... 

The amplitude a, which remains undetermined at this stage is fixed by the solvability conditions imposed on the equations for the higher order terms of the perturbation expansion (5). We arrive in this way at the following set of bifurcation equations for the normalized amplitude (3 = ta. 

The first of these equations admits a solution of the type given in equation (7). In order to specify the amplitude of this solution, which remains undetermined in this stage, one has to solve equations (14) for j 2. As the operator °( ) admits a nontrivial null solution, one has to verify that these latter equations satisfy suitable solvability conditions. If this is the case for all j, then the imperfection constitutes a smooth perturbation whose effects can be handled by the expansion (13). In particular, bifurcation will subsist and the bifurcation points K will be identical to those determined in the absence of the imperfection. 

Now, in many instances it may happen that the solvability conditions are violated for at least one j. The simple expansion given in equation (13), frequently referred to as the outer expansion, then fails and must be substituted by an inner expansion, in which a nonanalytic dependence on g is allowed. This is achieved by the development... 

As stressed in Section II, the coefficients y, M, are to be determined from suitable solvability conditions. One finds that to order e2 the solvability conditions yield y, = 0. Thus, the amplitudes p, p2 cannot be determined to this order. To order e3 one obtains a nontrivial result, in the form of two coupled cubic equations for p, and p2. Among the possible solutions of these equations one obtains rotating wave solutions. Setting p2 = 0 one finds a clockwise wave ... 

The search for a possible oscillatory instability is slightly different from the procedure used in the stationary case. The solvability condition of the linearized set of equations determines both the neutral curve and the frequency along this curve. When searching for such a solution we scanned approximately the same parameter space as used for Figs. 5-7. Since the frequency tends to zero when the oscillatory neutral curve gets close to the stationary one, we concentrated on the frequency range 0 < co < 2 and checked in some cases for higher frequencies. 

For V2 < V3 a simple shear flow in a perpendicular alignment causes less dissipation than in a parallel alignment. The next step is to study the stability of these alignments in the linear regime. Following the standard procedure (as described above) we find a solvability condition of the linearized equations which does not depend on the shear rate 7 ... 

The solvability condition of (85) and (86) defines the neutral curve 9o(q) and its minimum directly gives the critical values 6C and qc (within the approximations of this section) ... 

This condition is the solvability condition for the second equation in (104) our argument above shows that it is a necessary condition in order that this equation be solvable and it can be proven [24] that this condition is also sufficient to that end. Equation (106) is automatically satisfied since... 

The solvability condition for this equation is also obtained by multiplying both sides by (27rA 5T) "/ e 2/5hU and integrating over v. Integrating by parts at the left hand-side and using (105), we arrive at... 

As the solvability condition at the third order, one obtains the evolution equation for the complex amplitude of the unstable, spatially-periodic mode, A X, T). The solvability condition at the fourth order yields the evolution equation for the real amplitude B X, T) of the zero mode associated with the conservation of mass. Together, the two equations form the system of coupled equations (7) with... 

Here, the matrix Bq is adjoint to Bq. The solvability conditions can then be written in the form... 

Next, applying the solvability conditions (3.108) to (3.104) with j = 3, we obtain a coupled set of Landau equations... 

This solution describes periodic waves traveling in opposite directions along the front. The as yet undetermined complex amplitudes Ri and Si of these waves are bounded functions of the slow variables and Hi can grow linearly in time on the slow time scales. Solvability conditions which are described below will provide equations for these amplitudes. 

Since the long time homogeneous problem (3.22)-(3.28) with j = 1 has nontrivial solutions, it is necessary that solvability conditions be satisfied in... 

Thus for j = 2, the three solvability conditions (3.47) imply that... 

Equations (3.61)-(3.62) are inhomogeneous versions of (3.54). Therefore, in order that solutions R2 and S2 exist, solvability conditions must be satisfied. Since the equations for R2 and S2 decouple we treat them separately. We first consider (3.61). The solvability condition is that... 

Singular perturbation analysis does not provide a fully analytical result for the very important case of KPP kinetics. It is not possible to go beyond the first order in 5, because the exact unperturbed solution is not known and the integrals in the solvability condition diverge. Proceeding as in Sect. 4.2.1 for (6.50) with KPP kinetics, we obtain to leading order the following equation for the action functional (e = 0) ... 

We are spared solving this equation by virtue of a solvability condition. Integrating the left hand side of (7.35) with respect to the momenta, we find... 

Applying the solvability condition to the O (e ) problem with v = v+ and v = v- shows that A and B depend only on the slow time Ta. When u = wq the solvability condition yields dU/dn = tq with ro as... 

We can uniquely define fi,(0) (i = 1,..., k) and d, (i = A + 1,..., m) from this system. Thus, IIiX (t) will be completely determined, and for the unspecified j3(t) the initial condition will be defined. This function is completely determined in the next step of the construction of the asymptotics during the solution of the equation for X2(0- The solvability condition for this equation provides a linear differential equation for fi(t)... 

---