# Decomposition of 2-Soliton Solutions for the Good Boussinesq Equations

- DOI
- https://doi.org/10.1080/14029251.2020.1819610How to use a DOI?
- Keywords
- N-Solitons, Wronskian Solutions, KdV, Boussinesq
- Abstract
We consider decompositions of two-soliton solutions for the good Boussinesq equation obtained by the Hirota method and the Wronskian technique. The explicit forms of the components are used to study the dynamics of 2-soliton solutions. An interpretation in the context of eigenvalue problems arising from KdV type equations and transport equations is considered. Numerical examples are included.

- Copyright
- © 2020 The Authors. Published by Atlantis Press and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

## 1. Introduction

The *N*-soliton solutions of nonlinear PDE describing shallow water waves (SWW) as the Kortewegde Vries (KdV), the KP hierarchy, Boussinesq and more recent Camassa-Holm (CH), Depasperies-Procesi (DP) and other PDE have been subject of numerous recent studies. The interest stems from the mathematical and the physical importance of those solutions. Explicit forms of the N-soliton solutions also can provide better modeling of SWW. There are well developed analytical methods for obtaining explicit solutions via recursion operators, Hirota bi-linear method or the Wronskian technique but the physical interpretation is obscured by the complexity of the formulas. A few papers on interpretations and discussion of possible decomposition are [2], [13], [21], [5].

The intend in this work is to study the properties of 2-soliton solutions for one of the simplest and oldest models, the Boussinesq equation. The general form of the Boussinesq equation is

*a*,

_{i}*i*= 1, 2, 3 are real numbers and

*a*

_{2}

*a*

_{3}≠ 0. In the case

*a*

_{3}> 0 it is known as the ‘good’ Boussinesq equation and is equivalent to

We consider the ‘good’ Boussinesq (gB) equation as defined in [6]

The equations has a well known Lax pair

PDE with a Lax pair is usually a definition of integrable system, [11], and can be solved by the inverses scattering transform (IST) introduced by Clifford, Greene, Kruskal, and Muira, [9]. By utilizing IST Ablowitz and Segur, [1] provide an almost linearization of the KdV

Many of the SWW equations posses a single traveling wave of the form sech* ^{r}*(

*ax*−

*bt*) for

*r*= 1, 2 and

*a*,

*b*independent of

*x*. The duality of a wave and a particle of the one soliton solution is often considered, [5]. The results presented in this work are also interpreted in a direction of that comparison for the case of two-soliton solutions. In order to clarify this we present a brief discussion on the duality of one soliton solution in the context of KdV and gB.

A one soliton solution to (1.2) for *p* > 0 is *k*(*x*, *t*) = *p*sech^{2} *p*(*x* − 4*p*^{2}*t* − *w*), where *w* is a real phase shift. On the other hand a one soliton solution to (1.1) is not defined for all *p* but for a range of *p* that can be represented by two of the solutions, *m*_{1}, *m*_{2} of the cubic equation *p* is the function *b*(*x*, *t*) = *p*sech^{2} *p*(*x* − 2*Mt*), [6]. The time coefficients in the phases of the two ‘sech’ functions are different and we can not compare *k* and *b* for any time *t* and any *x*. We will compare *k* and *b* at any fixed time *t* = *t*_{0} with a phase shift for KdV *w*(*t*_{0}) = (4*p*^{2} − *M*)*t*_{0}, and hence *v*(*x*, *t*_{0}) = *k*(*x* − *w/p*, *t*_{0}).

The two equations describe traveling waves that in turn represent an energy transport. The function *ψ*(*x*, *t*) = sech *p*(*x* − 4*p*^{2}*t* − *w*) is a solution of the time independent equation from the Lax pair for KdV *ψ _{xx}* = (

*p*

^{2}−

*k*)

*ψ*with potential

*k*=

*pψ*

^{2}. The function Φ(

*x*,

*t*) = sech

*p*(

*x*−

*Mt*) is a solution of the transport equation

*b*=

*p*Φ

^{2}. The equation for

*ψ*is an eigenstate equation. The function

*ϕ*=

_{t}*ϕ*+

_{xx}*bϕ*. Since

*k*(

*x*−

*w*(

*t*

_{0})/

*p*,

*t*

_{0}) =

*b*(

*x*,

*t*

_{0}) for any

*t*

_{0}we can consider the function

*ψ*defines the steady state of a ‘particle’, and

*k*=

*pψ*

^{2}can be considered as a weighted energy distribution and potential. The function

*ϕ*is a solution of the real time-dependent Schrodinger equation corresponding to the energy state

*ψ*(

*x*,

*t*

_{0}) so

In the next section we include brief overview of decomposing *N*-soliton solution into components with an emphasis on KdV type equations.

Section 3 contains decomposition of 2-soliton solutions of gB with components that satisfy similar differential equations as in the case of a single soliton. In sections 4, 5 we compare the resulting solutions to provide interpretation in the context of the discussion for one soliton, including numerical examples. The last section is a discussion of the results presented and further directions.

## 2. Decomposition for 2-Soliton Solutions

A general natural way method for decomposing a *N*-soliton solution *u _{N}* onto

*N*interacting components (interacting solitons)

*u*) and a corresponding hierarchy, [8], [4],

In this case for real *u _{N}* =

*u*(

_{N}*ξ*

_{1},...,

*ξ*),

_{N}The operators Φ and *K′ _{m}* are a Lax pair

*N*-soliton solution decomposes as follows

*N*-soliton decomposition for Burgers, KdV, mKdV, sine-Gordon, cubic Schrodinger and other nonlinear PDE’s. For the Boussinesq equation a recursion operator is presented in [20] but no decomposition of N-soliton solution discussed.

The Lax pair (Φ, *K′ _{m}*) is often not the ‘standard’ Lax pair related to the linear problem

*ψ*and

_{k}*w*as well as the eigenvalues

_{k}*λ*and

_{k}*c*are interrelated. For the KdV for example

_{k}Our interest is in nonlinear PDE modeling SWW. These equations usually are either directly derived from or related to the Fluid Eulers’ Equation. The simplest model equations are KdV and Boussinesq equations. They model unidirectional solitons and posses explicit and relatively simple *N*-soliton solutions. Models of higher order of approximation to the Euler’s equations and multi-soliton solutions are Camassa-Holm (CH), [23], Depasperis-Procesi (DP), [17], Novikov (N), [17], and Sawada-Kotera (SK), [22]. Recursion operators for some of those equation are known but with no simple analytical descriptions.

One of the goals in this work is to investigate decomposition of 2-soliton solution onto components related to lower order linear PDE. We consider known solution and the lower order equation from a Lax pair. For KdV this is the second order eigenproblem but for the Boussinesq *L* is a third order and we consider the second order wave equation corresponding to *A _{m}*.

The Hirota method, with Wronskian solutions, Darboux and Backlund transformations are more often used to model 2-soliton solutions and their interactions in the setting of SWW PDE. For some of them fully explicit 2-soliton solutions are not known. For example, in [23] a 2-soliton solution for Negative order KdV is transformed to express the 2-soliton solution for CH. The 2-soliton solutions for the other shallow water waves present the same difficulty due to the complexity of the functions describing them. All of the equations posses a single soliton solution in sech(*ax* − *bt*) form but the construction of the 2-soliton solutions from two seed solutions is more involved. A common element in the most of the constructions is the Wronskian determinant

*f*,

*g*defined as

*p*

_{1},

*p*

_{2}.

The most popular model for SWW is still the Boussinesq eqaution due to its simplicity but even in this case there are difficulties in considering 2-soliton solutions. In [17] a merging of two solitons, fusion, was reported. This is likely due to the fact that the Lax pair for gB has a third order spatial equation. Explicit *N*-soliton solutions for KdV and gB with elastic interaction, i.e. after the interaction only phase shifts occur, can be obtained by Hirota’s method, [10] and Wronskian determinants, [12], [6].

For the ‘good’ Boussinesq equation *f* and *g* are of the form

*m*

_{1},

*m*

_{2}are two solutions to the equation

*ε*= −1 corresponds to waves traveling in the same direction while

*ε*= 1 corresponds to waves traveling in opposite directions.

Next by using the Wronskian technique and Hirota’s log substitution one can get 2-soliton solutions of KdV and gB equations in the form,

We introduce the notations *c _{i}*(

*x*,

*t*) = cosh(

*p*

_{i}*x*−

*T*(

_{i}*t*)),

*s*(

_{i}*x*,

*t*) = sinh(

*p*

_{i}*x*−

*T*(

_{i}*t*)), and

*i*= 1, 2. The

*x*derivatives of

*c*,

_{i}*s*are

_{i}*c*

_{i}_{,}

*=*

_{x}*p*

_{i}*s*,

_{i}*s*

_{i}_{,}

*=*

_{x}*p*

_{i}*c*, and |

_{i}*t*| < 1. We also define the KdV type Wronskian determinants

_{i}In the case of unidirectional solitons *p*_{2} > *p*_{1} > 0 and hence *K* > 0. The next result is an extension of results from [2] for characterization of *k* = 2(ln*K*)* _{xx}*.

### Lemma 2.1.

*If* *and* *then*

*and ψ*

_{i}*are solutions of the eigenvalue problem ψ*= (

_{xx}*λ*

^{2}−

*k*)

*ψ for the eigenvalues p*

_{1}

*and p*

_{2}

*correspondingly.*

### Proof.

By the Hirota’s substitution with

Next, one can differentiate *x* and get that

Using the expressions for *K _{x}* and

*K*we get the identity

_{xx}*ψ*

_{1}follows. Similarly,

*ψ*

_{2}is an eigenfunction corresponding to

*p*

_{2}.

For *T _{i}* and we consider the potential function

*k*as a weighted energy distribution of particles in the two states

*ψ*with no interactions. There is no immediate PDE that can be associated to

_{i}*k*for any choice of

*T*and we will refer to

_{i}*k*as KdV type potential. Ant advantage of the KdV type potentials is that they are well defined for solitons traveling in opposite directions.

## 3. 2-Soliton solutions for the ‘good’ Boussinesq Equation

In this section we derive decomposition of the 2-soliton solution for gB. The Wronskian determinant *f* and *g* were defined in (2.1), generates a 2-soliton solution 2*b* = (ln*V*)* _{xx}* for gB. The functions

*f*and

*g*are solutions of the following ODEs

*P*

_{1}(

*λ*) = (

*λ*−

*m*

_{1})(

*λ*−

*m*

_{2}) and

*P*

_{2}(

*λ*) = (

*λ*−

*n*

_{1})(

*λ*−

*n*

_{2}). Let

*D*=

*n*

_{1}+

*n*

_{2}−

*m*

_{1}−

*m*

_{2}, and

*L*=

*m*

_{1}

*m*

_{2}−

*n*

_{1}

*n*

_{2}.

### Theorem 3.1.

*For functions f and g, defined in* (2.1)*, a two soliton solution b* = 2(ln*V*)_{xx}*for the gB has the following representation*

*The functions* *and* *are two different solutions to the wave equation ϕ _{t}* =

*ϕ*+

_{xx}*vϕ and*

*The functions* Φ_{i}*= e*^{−θi}*ϕ _{i}*

*are solutions of the transport-eigenvalue problem*Φ

*+ 2*

_{t}*R*Φ

*= Φ*

_{x}*+ (−*

_{xx}*λ*

^{2}

*+ v*) Φ

*with R = M and λ = p*

_{1}

*for*Φ

_{1}

*, and*

*and λ = p*

_{2}

*for*Φ

_{2}

*and*

### Proof.

To compute (ln*V*)* _{x}* we use a column-wise differentiation of determinants

Denoting *b* = 2*ν*_{1,}* _{x}* + 2

*ν*

_{2,}

*. Differentiating*

_{x}*ν*

_{1}row-wise with respect to

*x*we get

The last identity can be verified by using the Sylvester’s method to expand *W*(*f*, *f _{x}*,

*g*) along the second and third rows and first and second columns.

Since *f* and *f _{x}* satisfy (3.1), multiplying the first row by

*m*

_{1}

*m*

_{2}and the second by −(

*m*

_{1}+

*m*

_{2}) and adding them to the last we get that

Since, *g _{xx}* − (

*m*

_{1}+

*m*

_{2})

*g*+

_{x}*m*

_{1}

*m*

_{2}

*g*= (

*n*

_{1}+

*n*

_{2}−

*m*

_{1}−

*m*

_{2})

*g*+ (

_{x}*m*

_{1}

*m*

_{2}−

*n*

_{1}

*n*

_{2})

*g*=

*Dg*+

_{x}*Lg*then we obtain the the representation for 2

*ν*

_{1,}

*. For the second term the determinant*

_{x}*f*− (

_{xx}*n*

_{1}+

*n*

_{2})

*f*+

_{x}*n*

_{1}

*n*

_{2}

*f*= −(

*n*

_{1}+

*n*

_{2}−

*m*

_{1}−

*m*

_{2})

*f*− (

_{x}*m*

_{1}

*m*

_{2}−

*n*

_{1}

*n*

_{2})

*f*= −(

*Df*+

_{x}*Lf*) and thus (3.2) follows.

Next we show that the functions *V _{x}* =

*fg*−

_{xx}*f*

_{xx}*g*it follows that

Since *f _{t}* = −

*f*and

_{xx}*g*= −

_{t}*g*we have that

_{xx}Finally, from the relation

*ϕ*

_{1}. Similarly we can obtain the result for

*ϕ*

_{2}and (3.3).

By using the relation *ϕ _{i}* =

*e*

^{θi}Φ

*and substituting in the wave equation we see that Φ*

_{i}*are solutions to the corresponding transport-eigenvalue problems and*

_{i}*b*solves (3.4).

The decomposition for *b* holds in the case of solitons traveling in the same or opposite directions. The function *g* has a zero in the case of opposite directions but

The two components of *k* from lemma 2.1 could be considered as two energy states of the eigenvalue problem with potential *k*, the steady state, *ψ*_{2} (has no zeros) and the excited state, *ψ*_{1} (has one simple zero). For gB we obtained two representations, (3.3) with components being solutions either to the wave equation with a potential *b* but with damping factors *e*^{−2θi} or in case of (3.4) the transport-eigenvalue problem, with potential *b*. Next we compare the potential functions *k* and *b* and the corresponding components for the 2-soliton solutions corresponding to gB and KdV type potentials. For *t* → ±∞ both Φ* _{i}* and

*ψ*, up to a constant shift approach

_{i}*ψ*

_{2}, while the other transition into an excited state,

*ψ*

_{1}.

Let *P*_{1}* _{i}* =

*P*

_{1}(

*n*) and

_{i}*P*

_{2}

*=*

_{i}*P*

_{2}(

*m*),

_{i}*i*= 1, 2. Then we have the following

### Theorem 3.2.

*Let* *and in case of solitons*, *ε* = 1, *and in case of chasing solitons traveling in opposite directions, ε* = −1, *then*

### Proof.

First we notice that in the Hirota substitution the solution for gB can be generated by using the following determinant

*V*. Indeed, since

*θ*

_{1}and

*θ*

_{2}are linear in

*x*we have that

From the identity

Since *D* = *n*_{1} + *n*_{2} − *m*_{1} − *m*_{2} and *L* = *m*_{1}*m*_{2} − *n*_{1}*n*_{2} the identities 2*Dp*_{1} = −*P*_{22} − *P*_{21} and 2(*DM* + *L*) = *P*_{22} − *P*_{21} hold. The second term of *b*, in (3.1), in both cases is

To evaluate the first term of (3.1) we consider separately the cases for *ε*. For *ε* = −1 we have *x* derivative *f* the first term of *b* in (3.1) is

Summing up we get that the two-soliton solution for gB for *ε* = −1 is

*ε*= 1 and

*x*derivative

*f*we have that the first term of

*b*in (3.1) is

*ε*= 1 is

From the choice of *u*, *v* we get that

By solving for the first term in the parenthesis

Similarly by expressing the terms depending on *u* and taking into and account that *ε* = −1 and *P*_{12} and *P*_{22} are negative we obtain

By adding up the last two identities we get the formula in case of chasing solitons. The derivation for solitons traveling in opposite directions is similar.

Next we show that the 2-soliton solution for gB obtained in theorem 3.1 is always positive.

### Lemma 3.1.

*For any x and t the* 2-*soliton solution b is positive.*

### Proof.

In the case of colliding solitons the coefficients *b*_{1} and *b*_{2} are always positive so we need to consider only the case of chasing solitons. In this case *n*_{1} < *m*_{1} < *m*_{2} < *n*_{2} and hence *P*_{11}, *P*_{12} are positive while *P*_{21}, *P*_{22} are negative. Since

Furthermore since *p*_{2} > |*p*_{1}| and *σ* > 0 we need to show that

Thus *n*_{1} = cos(*β*) > *m*_{1} and *n*_{2} is either

By direct evaluation we get that

*U′*(

*ξ*) = 4

*ξ*+ 3

*m*

_{3}is negative for any

*x*<

*m*

_{2}. Indeed,

*U*(

*ξ*) > 0 for any

*ξ*<

*m*

_{2}. Both of the choices for

*n*

_{3},

*m*

_{2}the statement in the lemma is established.

In the next section we obtain necessary condition on the distribution of m’s and n’s such that the determinant *B* is nonzero. We also the properties of

## 4. Analysis of the parameters of the 2-Soliton Solution for gB

In order to have non-singular solutions it is necessary to require that the Wronskian determinant *B*(*x*, *t*) > 0 for any real *x* and *t*. Initially, we consider separately the cases for *ε*, starting with chasing solitons i.e. *ε* = −1,

*t*

_{1},

*t*

_{2}. Since

*t*

_{1}and

*t*

_{2}are hyperbolic tangents in terms of

*x*and

*t*it is clear that the values of the pair (

*t*

_{1},

*t*

_{2}) are in the square with vertices

*t*

_{1},

*t*

_{2}plane. The bi-linear function

*Tc*attains its maximum and minimum at the elements of

*E*. Considering the values at the vertices we obtain sufficient conditions for the distributions of the

*n*’s and

*m*’s. Direct computations lead to

*c*

_{1},

*c*

_{2}are always positive we get that

*B*is positive for any

*x*,

*t*.

In the case of solitons traveling in opposite directions *ε* = 1 and

Again, direct computations lead to

*c*

_{1},

*c*

_{2}are always positive we get that

*B*is positive for any

*x*,

*t*.

The results from the previous section suggest that *b* is closely related to the KdV type potentials *k*, they both are expressed in terms of *c*_{1}, *c*_{2}. Similarly to the case of one soliton from the introduction we consider the spatial shifts *u*, *v* defined in theorem 3.2 in the two seed solitons,

*s*= sinh

_{v}*v*,

*s*= sinh

_{u}*u*and

*c*,

_{c}*c*are the hyperbolic cosines. We associate a bi-linear function in the variables

_{u}*t*

_{1},

*t*

_{2}

The values of *T* at the vertices of *E* are

Since *p*_{2} > *p*_{1} > 0 and hence *T* is positive for any choice of the parameters *u* and *v*. In the case of solitons traveling in opposite directions, (4.2), *p*_{1} < 0 and a necessary condition all values to be positive is *p*_{2} > |*p*_{1}|. In what follows we assume that indeed *p*_{2} > |*p*_{1}|.

Next we show that *B* and *K* are equivalent in the sense that there exist positive real constants *a*_{1}, *a*_{2} such that *a*_{1}*K* ⩽ *B* ⩽ *a*_{2}*K* for any *x* and *t*. In such a case we write *B* ≈ *K*.

### Lemma 4.1.

*Let*

*then*

*or B*≈

*K, where θ*= min(

_{m}*η*

_{1},

*η*

_{2})

*and*Θ

*= max(*

_{M}*η*

_{1},

*η*

_{2}).

### Proof.

In the case of chasing solitons the positive bi-linear functions *Tc* and *T* attain their extreme values at the elements of

Direct substitution of *u*, *v* show that the ratio *η*_{1} and *η*_{2} as possible values at *E* and this determins the choice for *θ _{m}* and Θ

*. In the case of solitons traveling in opposite direction we notice that*

_{M}*To*(

*e*,

_{i}*e*) =

_{j}*Tc*(

*e*,

_{i}*e*),

_{j}*i*,

*j*= 1, 2 and hence we arrive at the same estimate.

The explicit expressions obtained so far suggest that the main difference between *b* and *k*^{*}
is in the ‘interaction’

### Lemma 4.2.

*For the choices of the parameters as in* *theorem* 3.2*, I and J are bell-shaped,* *,* *, with a single maximum tending to zero as t* → ±∞*, and*

### Proof.

From lemma 3.1 it follows that 2(ln(*B*))* _{xx}* =

*b*> 0 and hence its anti-derivative

*B*(−∞)

_{x}*B*(∞) < 0 it follows that

_{x}*B*has exactly one zero on the

_{x}*x-*axis or

*I*has a single maximum. For

*J*we have that

*s*

_{2}(

*x*,

*t*) = sinh(

*p*

_{2}

*x*− 2

*p*

_{2}

*Nt*) = 0 or

*x*(

*t*) = 2

*Nt*. Thus

*t*→ ±∞. Since

*B*≈

*K*and both are positive, it follows that

*I*also decreases to 0 when

*t*→ ±∞. The asymptotic for

*x*is clear from the estimates

*I*,

*T*is either

*Tc*or

*To*from lemma 4.1 and in both cases

*T*is positive function uniformly bounded from below.

For the choice of *u*, *v* as above we observe that

Since,

In the next section we consider numerical estimates and examples.

## 5. Numerical Examples

In this section we derive numerical estimates for dynamics of *b* and *k*^{*}
in terms of *m _{i}*,

*n*. We start by discussing the distribution of the roots of

_{i}The parameters *m _{i}*,

*n*depend on

_{i}*α*and

*β*and the

*η*’s are nonlinear functions in

*α*,

*β*. When

*α*=

*β*the function

*K*

^{*}has always a zero and thus in the numerical examples we consider

*α*<

*β*. If we further consider

*β*−

*α*> 0.1 then

*α*,

*β*) are plotted in Fig. 1.

The deviation is bigger for smaller values of both parameters and approaches ∞ when *α* approaches *β*.

For the numerical estimates we pick values for *β* and *α*.

*The case of chasing solitons*: We pick

*p*

_{1}= 0.3522 and

*p*

_{2}= 0.6124. The shifts are

*u*= 0.1562,

*v*= 0.2636, and

*The case of solitons traveling in opposite directions*: We pick

*p*

_{1}= −0.1130 and

*p*

_{2}= 0.4330. The shifts are

*u*= −0.1577,

*v*= −0.5169, and

The two examples illustrate that the interaction therm

## 6. Discussion and Conclusions

In the previous sections we presented decomposition of the KdV type potentials, which include the 2-soliton solutions of KdV, as *γ _{i}* are constants depending only on

*p*

_{1},

*p*

_{2}. The two functions

*ψ*are eigenfunctions of

_{i}*ψ*= (

_{xx}*λ*

^{2}−

*k*

^{*})

*ψ*for eigenvalues

*are solutions of the transport-eigenvalue equations Φ*

_{i}*+ 2*

_{t}*R*Φ

*= Φ*

_{x}*+ (−*

_{xx}*λ*

^{2}+

*v*)Φ and

*h*

_{1}and

*h*

_{2}are variable coefficients. In theorem 3.2 we established that

*ϕ*≈

_{i}*ψ*for

_{i}*i*= 1, 2. The 2-soliton solution for gB can be considered as an energy transport process of two energy waves with influence on each other via the eigenfunctions

*ψ*. For

_{i}*t*→ ±∞ the components Φ

*and*

_{i}*ψ*for

_{i}*i*= 1, 2 correspondingly are bell-shaped and differ from the single soliton case primarily by a phase shift and the interaction terms tend to 0. From the signs of

*b*

_{1},

*b*

_{2}it also can be observed that during the interaction the two waves exchange energy, this might be concluded from the signs of the interaction terms and the fact that Φ

_{1}has negative values. The dynamics is different for the solitons traveling in the same or opposite directions. In the case of chasing solitons, the two pairs of components are almost identical and the interaction results in loss of energy at the pick of

*b*

_{1}and

*b*

_{2}, as also can be seen from Fig. 3.

Another observation that can be made is that *ψ*_{1} away from the interaction is up to a phase shift a single soliton while during the interaction it looks like transition from a steady state to excited state. This also might be in agreement with the fact that after interaction the smaller and slower soliton, *ψ*_{1}, exhibits a time delay while the faster and taller, *ψ*_{2} exhibits positive time shift. The horizontal velocity of a particle is approximated by *kdv* and the slowing of the horizontal velocityof *ψ*_{1} can be explained by the more chaotic movement during the interaction while the acceleration in horizontal direction of *ψ*_{2} might be explained with the fact that the vertical displacement is lesser during interaction while the horizontal is about the same.

An interesting observation is that the results obtained in this work can be extended for gB type potentials

*a*,

_{i,j}*b*),

_{i,j}*ε*,

_{j}*i*,

*j*= 1, 2. Straight substitution and computations lead to similar decomposition as in theorem 3.1. The main difference is that the corresponding modes

## Acknowledgement

An anonymous reviewer is thanked for critically reading the manuscript and suggesting important improvements.

## References

### Cite this article

TY - JOUR AU - Vesselin Vatchev PY - 2020 DA - 2020/09 TI - Decomposition of 2-Soliton Solutions for the Good Boussinesq Equations JO - Journal of Nonlinear Mathematical Physics SP - 647 EP - 663 VL - 27 IS - 4 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1819610 DO - https://doi.org/10.1080/14029251.2020.1819610 ID - Vatchev2020 ER -