2.1. Dressing method and linear bundles in pole gauge

The dressing method represents an indirect way to solve completely integrable equations, i.e. one constructs new solutions to a given NLEE from a known (seed) solution. In doing this, one essentially uses the existence and the form of Lax representation.

In order to see how the method works, let us consider an arbitrary NLEE from the integrable hierarchy related to (1.4) and denote by (u₀, v₀) a known solution subject to (1.3) and (2.1). Then, as it is shown in [23], the NLEEs has a Lax representation

where L₀ and A₀ are given by:

The coefficients Ak(0), k = 1,...,N can be uniquely expressed through (u₀,v₀) and their x-derivatives of order up to N − k. Here we shall not specify the form of the coefficients of the second Lax operator because only their asymptotic behavior as x → ±∞ is essential for our further considerations. We refer the reader who is interested in this issue to our previous paper [23] where they can find more detailed explanations.

Similarly to (1.4)–(1.6), the above Lax operators fulfill the following symmetry conditions:

where † stands for Hermitian conjugation and ε = ±1.

Let ψ₀ be an arbitrary fundamental solution to the auxiliary linear problem:

Due to (2.2), ψ₀ also fulfills the linear problem:

where is the dispersion law of the NLEE. Above, is the gauge transform diagonalizing S⁽⁰⁾, see [23] for more explanations. The dispersion law of a NLEE is an essential feature encoding the time dependence of its solutions and that way it labels the NLEE within the integrable hierarchy. The dispersion law of (1.2) is

Linear problems (2.7) and (2.8) will be referred to as bare (seed) linear problems and their fundamental solutions will be called bare (seed) fundamental solutions. We shall denote the set of all bare fundamental solutions by ℱ₀.

As discussed in [23], (2.5) and (2.6) are due to the action of the reduction group 𝕑₂ × 𝕑₂ on the set of bare fundamental solutions, see [13, 14] for more explanations. Indeed, in our case the following 𝕑₂ × 𝕑₂-action

leads to (2.5) and (2.6) respectively.

Now, let us apply the gauge (dressing) transform

where 𝒢 (x, t, λ) is a 3 × 3-matrix with unit determinant. Since the Lax operators are transformed through: the operators L₁ and A₁ commute as well. We shall require that the auxiliary linear problems remain covariant under the dressing transform, i.e. we have for L₁ and A₁ being of the same form as L₀ and A₀. Thus we can write where u₁ and v₁ are some new (unknown) functions solving the same NLEE and subject to (1.3) and (2.1). The coefficients Ak(1), k = 1,...,N depend on (u₁, v₁) and their x-derivatives of order up to N − k in the same way as Ak(0) depend on (u₀, v₀) and their x-derivatives. Therefore, the dressed Lax pair has the same asymptotic behavior as the bare one.

Finding (u₁, v₁) means that we have obtained a new solution to the same NLEE and this is the main idea underlying the dressing method. Symbolically, it could be presented as the following sequence of steps:

Though the covariance of the linear problems is a strong condition, it does not determine the dressing factor 𝒢 uniquely. Indeed, after comparing (2.7) and (2.8) with (2.14), we see that 𝒢 must solve the system of linear partial differential equations:

Equations (2.17) and (2.18) tell us nothing about the λ-dependence of 𝒢 . This is why we need to make a few assumptions about its behavior with respect to λ in order to obtain more specific results. Let us assume the dressing factor and its derivatives in x and t are defined in the neighbourhood of λ = 0. After setting λ = 0 in (2.17) and (2.18), we immediately see that

These relations imply that 𝒢 should depend non-trivially on λ otherwise it will be merely a constant. We shall pick up 𝒢|_λ=0 = 𝟙 since it does not lead to any loss of generality. In fact, this normalization corresponds to the normalization of the fundamental analytic solutions to the auxiliary linear problems at λ = 0, see [23].

Equation (2.17) allows one to find u₁ and v₁ in terms of the seed solution and the dressing factor. At this point, we require that 𝒢 as well as its derivatives in x and t are regular when |λ| → ∞. After dividing (2.17) by λ and setting |λ| → ∞, we derive the following interrelation:

Since S⁽⁰⁾ is determined by (u₀,v₀) and S⁽¹⁾ is determined by (u₁,v₁), the above relation allows us to construct another solution of our system starting from a known one.

The form of the dressed operators and the zero curvature condition imply L₁ and A₁ obey (2.5) and (2.6) too. Therefore, the set of the dressed fundamental solutions is subject to the 𝕑₂ ×𝕑₂-action

The 𝕑₂ ×𝕑₂ action on ℱ₀ and ℱ₁ implies that the dressing factor is not entirely arbitrary but obeys the symmetry relations:

A simple ansatz for dressing factor meeting all the requirements discussed so far is given by:

Generally speaking, the dressing procedure does not preserve the spectrum of scattering operator, see [6]. Indeed, let us assume for simplicity that L₀ has no discrete eigenvalues, i.e. the spectrum of the bare scattering operator coincides with the real line, see [23]. Denote the resolvent of L₀ by R₀(λ). According to (2.13), the dressing transform maps the bare resolvent onto

Equation (2.25) implies that the singularities of the dressing factor and its inverse “produce” singularities of the dressed resolvent operator R₁(λ). This way the dressing procedure adds new discrete eigenvalues to the bare scattering operator.

In order to find L₁ and A₁ through (2.19), we need to know B_j(x,t) = Res(𝒢 (x,t,λ); μ_j). The algorithm to find the residues of (2.24) consists in two steps. In the first step, one considers the identity 𝒢 𝒢 ⁻¹ = 𝟙 which gives rise to a set of algebraic relations for B_j. The form of those relations crucially depends on the location of μ_j— whether the poles of the dressing factor are arbitrary complex numbers with nonzero real and imaginary parts (generic case) or the poles are either imaginary or real numbers. If the poles are complex numbers in generic position or purely imaginary then we obtain soliton type solutions. For poles lying on the real line, we have degeneracy in the spectrum of the scattering operator. As a result, we obtain quasi-rational solutions. Due to all these differences, we shall consider the three cases in separate subsections.

Though the algebraic relations may be different, they always imply that the residues of 𝒢 are some singular (degenerate) matrices which could be decomposed as follows:

where X_j(x, t) and F_j(x, t) are rectangular matrices of certain rank and the superscript T stands for matrix transposition. After substituting (2.26) into the algebraic relations, we are able to express X_j through F_j.

The factors F_j are determined in the second step. For this to be done, we consider (2.17) and (2.18). As for the algebraic relations discussed above, the calculations here depend on the location of the poles of 𝒢 . However, (2.17) always leads to the following result:

allowing one to construct F_j through a bare fundamental solution defined in the neighbourhood of μ_j and some arbitrary x-independent matrices F_j,₀. The matrices F_j,₀ depend on t in a way governed by (2.18). Regardless of the location of μ_j, we derive exponential t-dependence. Thus in order to recover the time dependence in all formulas, we may use the rule below: where f(λ) is the dispersion law of the NLEE, see (2.9). We shall demonstrate in more detail that procedure in the subsections to follow.

2.2. Soliton type solutions I. Generic case

Let us start with the case when the poles of (2.24) are complex numbers in generic position, i.e. μj2∉ℝ for all j. From the symmetry condition (2.23), we immediately deduce that

Thus the dressing factor and its inverse have poles located at different points.

Let us consider the identity 𝒢 (x, t, λ)[𝒢 (x, t, λ)]⁻¹ = 𝟙. Since it holds identically in λ, the residues of 𝒢 𝒢 ⁻¹ should vanish. After evaluating the residue of 𝒢 𝒢 ⁻¹ at μi* we easily obtain the following algebraic relation:

Evaluation of the residues at ±μ_i and −μi* leads to equations that can easily be reduced to (2.28), thus giving us no new constraints.

It is seen from (2.28) that each B_i should be a degenerate matrix, hence it can be factored

where X_i and F_i are two rectangular matrices. After substituting (2.29) into (2.28), one obtains the linear system for the factors X_j. Solving it, allows one to express X_j through F_j. This is easier when the dressing factor has just a single pair of poles: μ and −μ. Assuming X and F are column-vectors, we get:

Let us return now to the general case. The factors F_j can be found from (2.17). For that purpose we rewrite (2.17) in the following way

and calculate the residues of its right-hand side at λ = μ_i. After taking into account (2.30), we obtain the linear differential equation which is immediately solved to give

Above, F_i,0 are “integration constant” matrices which depend on t however. Evaluation of the residues at −μ_i or ±μi* gives equations that are easily reduced to (2.32) after taking into account the symmetries of S1(0).

The t-dependence of F_i,₀ is determined from (2.18) after it is rewritten as follows:

Evaluation of the residues of both hand sides of (2.34) for λ = μ_i gives

Yet again, we do not need to consider the residues for λ=±μi* and λ = −μ_i due to the symmetries of the coefficients of A(λ).

After substituting (2.33) into (2.35), we derive a linear differential equation for F_i,₀, namely

where f(λ) is the dispersion law of the NLEE. Equation (2.36) is easily solved and allows us to state the rule: in order to recover the t-dependence in all formulas, one has to make the following substitution

Now let us apply the general results obtained above to some seed solution. An obvious choice for seed solution satisfying (1.3) and (2.1) is

One can prove that the bare scattering operator in this case has no discrete eigenvalues.

For a bare fundamental solution we shall choose

This fundamental solution is invariant with respect to (2.11) and (2.12) which makes it rather convenient for the calculations to follow.

From this point on, we shall focus on the simplest case when the dressing factor has a single pair of poles and X and F are column-vectors. After substituting (2.29) and (2.31) into (2.24) and (2.27), equation (2.19) can be written in components as follows:

where F^T = (F¹,F²,F³). It is not hard to check that (2.40) and (2.41) satisfy (1.3) for any F.

On the other hand, (2.33) leads to the following expression for F:

In order to evaluate X, we shall need the following quadratic expressions as well:

where ω = Re μ > 0, κ = Im μ > 0 and φ=argF01−argF03. Taking into account the structure of (2.39)–(2.42), it is natural to consider in detail the following three elementary cases, see [9].

1. (1)

   First, let us assume F02=0 and F01≠±F03. Then for F we have

   F(x)=(F01cos(μx)+iF03sin(μx)0F03cos(μx)+iF01sin(μx))(2.45)

   and (2.43), (2.44) now could be simplified to

   FTQεF*=Acosh(2κx+ξ0),(2.46)

   FTQεHF*=−Acos(2ωx+δ0)(2.47)

   where

   coshξ0=|F01|2+|F03|2A,   sinhξ0=−2|F01F03|cosφA,(2.48)

   A=(|F01|2+|F03|2)2−4|F01F03|cos2φ,(2.49)

   cosδ0=|F01|2−|F03|2A,   sinδ0=−2|F01F03|sinφA.(2.50)

   After substituting (2.45)–(2.50) into (2.40) and (2.41), we obtain the dressed solution to (1.2) at a fixed moment of time:

   u1(x)=0,(2.51)

   v1(x)=exp[4i arctanκcos(2ωx+δ0)ωcosh(2κx+ξ0)].(2.52)

   In order to recover the time evolution in (2.51) and (2.52), one needs to make the substitution:

   φ→φ,   ξ0→ξ0,   δ0→δ0,   A→Aexp(−4ωκt3)(2.53)

   that follows directly from (2.10) and (2.37). It is seen that (2.51) and (2.52) remain invariant under transformation (2.53), so the dressed solution is stationary and non-singular. The solution just derived coincides with that found in [9] (see formula (3.26)) for the generalized HF with Hermitian reduction.

2. (2)

   Let us assume F02≠0. Without any loss of generality we could just set F02=1. There exist two elementary options: F01=F03 and F01=−F03. Let us consider the former one. After recovering the time evolution, the vector F(x) is given by:

   F(x,t)=(F01eiμx+iμ2t3e−2iμ2t3F01eiμx+iμ2t3).

   The solution in this case reads:

   u1(x,t)=4ωκ[2ωeϑ(x,t)+ε(ω−iκ)e−ϑ(x,t)]e−i[ωx+(ω2−κ2)t+δ](ω−iκ)[2ωeϑ(x,t)+ε(ω+iκ)e−ϑ(x,t)]2,(2.54)

   v1(x,t)=1−8εωκ2(ω−iκ)[2ωeϑ(x,t)+ε(ω+iκ)e−ϑ(x,t)]2(2.55)

   where

   ϑ(x,t)=−κ(x+2ωt)+ln|F0,1|,   δ=(π2+argF0,1).

   Solution (2.54), (2.55) has no singularities and goes into the one obtained in [9] (see equation (3.27)) when ε = 1.

3. (3)

   Assume now F02=1 and F01=−F03. In this case F(x,t) is given by:

   F(x,t)=(F01e−iμx+iμ2t3e−2iμ2t3−F01e−iμx+iμ2t3).

   The corresponding dressed solution looks as follows:

   u1(x,t)=4ωκ[2ωeϑ˜(x,t)+ε(ω−iκ)e−ϑ˜(x,t)]e−i[ωx+(κ2−ω2)t+δ˜](ω−iκ)[2ωeϑ˜(x,t)+ε(ω+iκ)e−ϑ˜(x,t)]2,(2.56)

   v1(x,t)=1−8εωκ2(ω−iκ)[2ωeϑ˜(x,t)+ε(ω+iκ)e−ϑ˜(x,t)]2(2.57)

   where

   ϑ˜(x,t)=κ(x−2ωt)+ln|F0,1|,   δ˜=π2−argF0,1.

Formally, this solution could be derived from (2.54), (2.55) by applying the transform x → −x and F₀,₁ → F₀,₃.

All the solutions constructed explicitly in this subsection are connected with four distinct poles (quadruples) of 𝒢 and 𝒢⁻¹, i.e. four discrete eigenvalues of the dressed scattering operator. This is why we call such solutions “quadruplet” solutions.

2.3. Soliton type solutions II. Imaginary poles

Let us assume now that all the poles of 𝒢 are purely imaginary, i.e. we have μ_j = iκ_j for some real numbers κ_j ≠ 0. Thus we could write

for the dressing factor and for its inverse. Since 𝒢 and 𝒢 ⁻¹ have the same poles, the identity 𝒢 𝒢 ⁻¹ = 𝟙 gives rise to algebraic relations that do not follow from (2.28). It is easily checked that the following equations hold true: where

There is no need to perform similar calculations for λ = −iκ_i since all algebraic equations can be reduced to (2.60) and (2.61).

Relation (2.60) implies that B_i(x,t) is degenerate, so (2.29) holds again for some rectangular matrices X_i and F_i (we shall assume that these matrices are simply column vectors). Due to (2.60) the factors F_i satisfy the quadratic relation

After substituting (2.29) into (2.61), we see that there exist functions α_i such that:

These relations allow one to find all vectors X_i in terms of F_i and α_i. In the simplest case when the dressing factor has a single pair of poles ±iκ (2.63) is easily solvable to give:

After taking into account (2.29), (2.58), (2.59) and (2.64), equation (2.19) could be written down in components as follows:

It is easy to check that (2.65) and (2.66) are interrelated through |v|² +ε|u|² = 1 for any real-valued α and F obeying (2.62).

Like in the generic case (see the previous subsection) we can find F_i and α_i by analyzing (2.17) and (2.18). Considerations similar to those in the generic case lead to the following linear differential equations:

Equation (2.67) shows that (2.33) holds true for the vectors F_i while (2.68) implies that the scalar functions α_i can be expressed in terms of the seed fundamental solution as given by:

Above, α_i,₀ is an integration constant and K_ψ(λ) = [ψ(λ)]⁻¹HQ_ε[ψ^†(−λ^*)]⁻¹ Q_εH measures the “deviation” of the solution ψ from invariant solutions, i.e. if ψ is invariant with respect to (2.11) and (2.12), then K_ψ = 𝟙.

Finally, we need to recover the time dependence in all formulas. For this to be done we consider (2.18) as we did in the previous subsection. As a result, we derive a set of equations for F_i and α_i, namely:

Equation (2.70) coincides with (2.35). Thus after substituting (2.33) into (2.70), we derive (2.36) and (2.37) holds again. On the other hand, (2.71) is reduced to

If the bare fundamental solution is invariant with respect to both reductions, i.e. K_ψ0 = 𝟙, then (2.72) is simplified to

Let us consider the case when the seed solution S⁽⁰⁾ is picked up as in (2.38) and the corresponding fundamental solution is given by (2.39). As we discussed in the previous subsection, for (2.39) we have K_ψ0 = 𝟙. From this point on, we shall assume that 𝒢 has a single pair of imaginary poles.

In this case the 3-vector F reads:

Our further considerations depend on whether or not F02 is equal to 0.

1. (1)

   Let us first assume F02=0. Then we may set |F01|=|F03|=1 and (2.74) and (2.69) now give

   |F1(x)|2=cosh(2κx)−sinh(2κx)cos(φ),   2φ=argF01−argF03(2.75)

   α(x)=α0+2xsin(2φ)(2.76)

   where α₀ is t-independent in this case. After substituting (2.75) and (2.76) into (2.65) and (2.66), then taking into account (2.74), we get the following stationary solution to (1.2):

   u1(x)=0,(2.77)

   v1(x)=[cosh(2κx)−sinh(2κx)cos(2φ)+iκ(α0+2xsin(2φ))cosh(2κx)−sinh(2κx)cos(2φ)−iκ(α0+2xsin(2φ))]2.(2.78)

2. The dressed solution just derived does not “feel” the presence of ε, i.e. whether the reduction is Hermitian or pseudo-Hermitian, hence it formally coincides with the one obtained in [9]. The right-hand side of (2.78) could be rewritten as:

   v1(x)=exp[4iarctan(κ(α0+2xsin(2φ))cosh(2κx)−sinh(2κx)cos(2φ))].

   When we have F₀,₁ = ±F₀,₃, it simplifies to

   v1(x)=[e∓2κx+iκα0e∓2κx−iκα0]2.

   It is seen that v₁ is nontrivial only if α₀ ≠ 0.

3. (2)

   Assume now that F02≠0. Thus we may simply set F02=1. Our further analysis depends on the value of ε. Let us pick up ε = 1 first. Due to (2.62) and the invariance of (2.39) under the reductions, we have

   F0TQεHF0*=0⇒|F01|2−|F03|2=1.(2.79)

   A natural parametrization for F01 and F03 is the following one:

   F01=cosh(θ)ei(δ+φ),   F03=sinh(θ)ei(δ−φ),   θ>0.(2.80)

Then using (2.80), we have

In the end we derive the following dressed solution:

The result we obtained coincides with the one found in [9].

Let us consider now the case when ε = −1. Relation (2.79) is rewritten as

hinting at the following parametrization:

Then we have

and substituting into (2.65) and (2.66) we obtain:

It is not hard to see that Δ_m could vanish for particular values of the parameters and thus we obtain singular solutions.

Unlike the quadruplet solutions in the previous subsection, the soliton-like solutions we derived here are related to a pair of discrete eigenvalues (doublet) of the scattering operator. This is the reason we call such solutions “doublet” solutions.