Infinite Number of Conservation Laws

For some evolution equations such as the Sasa–Satsuma equation 3.4, the potential matrix Q admits more than one symmetry; thus the discrete scattering data also admits more than oneinvol

Notice that the only information used to construct the N -soliton solutions in the hierarchy

(3.20) is the discrete scattering data{ζk, ¯ζk , vk0,¯vk0, 1≤ k ≤ N} and the dispersion relation

of the hierarchy (3.20) as reflected in the exponents of Eqs (3.89)–(3.90) For a particularevolution equation in this hierarchy (3.20), the only caution the reader should heed isthe symmetry reduction of this equation from the hierarchy This symmetry reduction

corresponds to symmetry properties of the potential matrix Q in Eq (3.42), and it induces

the corresponding involution properties in the discrete scattering data{ζk , ¯ζk, vk0,¯vk0, 1≤ k ≤

N} For instance, the symmetry reduction (3.35) leads to the symmetry (3.75) of the potential

matrix Q and involution properties (3.80) and (3.82) of the discrete scattering data For some evolution equations (such as the Sasa–Satsuma equation (3.4)), the potential matrix Q

admits more than one symmetry; thus the discrete scattering data also admits more than oneinvolution These involution properties of the discrete scattering data must all be respected

in order to obtain the correct N -soliton solutions To illustrate, explicit N -soliton solutions

for the Manakov equations (3.1)–(3.2), the focusing-defocusing NLS system (3.29)–(3.30),and the Sasa–Satsuma equation (3.4) will be given later in this chapter (see Secs 3.10–3.12)

3.6 Infinite Number of Conservation Laws

The hierarchy (3.20) also admits an infinite number of conservation laws These tion laws can be derived analogously to what we did for the NLS equation in Sec 2.3 Theonly main difference is that conservation laws here will be generated by a coupled Riccatisystem rather than a single Riccati equation

conserva-Let us consider a solution Y = (y1, y2, y3)T to the third-order scattering operator (3.41)

Defining

µ(1)= y1/y3, µ(2)= y2/y3, (3.95)then it is easy to find from (3.41) that

(ln y3)x= iζ + ˆuµ(1)+ ˆvµ(2)

Using the temporal equation (3.9) of this hierarchy, an analogous equation for (ln y3)t

can also be derived Cross-differentiating these two equations with respect to t and x, respectively, we see that if we expand µ(1)and µ(2)into the power series,

n would be the density of a local conservation law, and an infinite number

of conserved quantities are

To determine the expansion coefficients µ(1,2)n , we use the scattering equation (3.41).

Simple calculations show that µ(1,2)satisfy the following coupled Riccati system:

the Lax pair to derive the (ln y3)tequation, then insert the expansions (3.97) The coefficients

of (ln y3)tat various orders of ζ −nwould then be the fluxes of local conservation laws.

For a particular evolution equation in the hierarchy (3.20), if the potential Q admits

symmetry reductions, then these symmetries should also be inserted into the above generalconservation laws For instance, the Manakov system (3.1)–(3.2) admits the symmetry re-ductionsˆu = −u∗,ˆv = −v∗ Utilizing these symmetries, the first three conserved quantities

of the Manakov system are then

I1=

 ∞

3.7 Closure of Eigenstates in the Higher-Order Scattering Operator

In this section, we establish the closure of eigenstates in the third-order scattering operator(3.41) This proof is a simple extension of the one for the Zakharov–Shabat system inSec 2.5 and will be only sketched below The proof of closure for the more generalscattering operator (3.8) could be similarly given

First, we define functions

R+(x, y, ζ ) = χ+(x, ζ ) [θ (y − x)H1− θ(x − y)H2] (χ+)−1(y, ζ ), (3.112)

R−(x, y, ζ ) = χ−(x, ζ ) [θ (x − y)H1− θ(y − x)H2] (χ−)−1(y, ζ ), (3.113)

where

χ+= (φ1, φ2, ψ3) , χ−= (ψ1, ψ2, φ3), (3.114)

and θ (x) is the standard step function given in Eq (2.187) of the previous chapter Functions

R±are meromorphic for ζ∈ C±, respectively, and are bounded as ζ → ∞ From similar

relations as (2.188)–(2.189) but with H1,2replaced by (3.54), we see that

det χ+(x, ζ ) = e −iζ x ˆs33(ζ ), det χ−(x, ζ ) = e −iζx s

33(ζ ). (3.115)

Thus R±has pole singularities at the zeros ofˆs33and s33, respectively Then we define two

complex contours, γ+and γ−, with γ+starting from ζ = −∞+i0+, passing over all zeros of

ˆs33(ζ ) inC+, and ending at ζ = ∞+i0+, and with γ−starting from ζ = −∞+i0−, passing

under all zeros of s33(ζ ) inC−, and ending at ζ = ∞+i0− Using the large-ζ asymptotics

(3.55)–(3.56) and (3.61)–(3.62) of Jost solutions and then bringing the contours to the realaxis, we get

Res&

where ζj∈ C+and ¯ζj∈ C−are the zeros ofˆs33(ζ ) and s33(ζ ) Following similar calculations

as in Sec 2.5, we also have



φ+ 3

Substituting the above two equations into (3.116)–(3.120), the final closure relation is then

obtained Assuming the zeros (ζj, ¯ζj) are all simple, then the closure relation reads

Note that det(s)= ˆs33, thus s−1(ζ ) has pole singularity at the zeros ζj ofˆs33, and s−

j in the

above equation is the residue of s−1(ζ ) at ζ

j The closure relation (3.126) shows that forthe third-order scattering operator (3.41), its discrete and continuous eigenstates also form

a complete set For multiple zeros of (ζj , ¯ζ j), the closure relation is similar, except that theresidue terms in (3.126) need to be calculated differently

3.8 Squared Eigenfunctions of the Higher-Order Scattering Operator

In this section, we derive the squared eigenfunctions for the third-order scattering operator(3.41) and its integrable hierarchy Most of the derivations are analogous to those for theZakharov–Shabat system (2.2) in Sec 2.6, so our derivation will be brief in general Somenew features do arise though Such features will be elaborated in detail We will first assume

the potential functions (u, v, ˆu, ˆv) in (3.42) to be independent, and hence derive the generic

squared eigenfunctions for the third-order scattering operator (3.41) Afterwards, we will

discuss the case when (u, v, ˆu, ˆv) are dependent on each other We will describe how to

obtain the corresponding squared eigenfunctions from the generic ones through symmetry

reduction, and we explain how squared eigenfunctions can become sums of products of Jost

functions due to the potential reduction

3.8.1 Squared Eigenfunctions for Generic Potentials

We first consider the generic case when the potential functions (u, v, ˆu, ˆv) are independent from each other We also assume as before that s33 and ˆs33 have no zeros (the general

case of s33andˆs33 having zeros will be added later) Following our procedure described

in Sec 2.6, we first calculate variations of the scattering data in terms of variation of thepotential Repeating the same calculation as in Sec 2.6, we also get the variation relations(2.221)–(2.224), i.e.,

ϕ j = s33φ j − s 3j φ3, ϕ j = ˆs33φ j − ˆs j3φ3, j= 1,2 (3.134)

Now we simplify these expressions of ϕ jandϕ jand show that they are analytic in the samehalf planes of ψ3and ψ3, respectively For this purpose, we first rewrite ϕjandϕ jin terms

From the first formula of B in (3.139), we readily obtain the first two rows of B as the

inverse of the 2×2 block in the upper left corner of S−1, followed by a zero column From

the second formula of B in (3.139), we readily obtain the third row of B as the third row

of S divided by s33 The matrix Bcan be similarly determined from the two formulae in

(3.140) Thus matrices B and  Bare found to be

Inserting Eqs (3.138) and (3.141) into (3.134), the expressions for ϕjandϕ jthen reduce to

Recalling the analytic properties of Jost solutions and scattering matrices summarized in

(3.63)–(3.65), we see that (ϕ1, ϕ2) are indeed analytic inC−as ψ3, and (ϕ1,ϕ2) are indeedanalytic inC+as ψ3; thus products of Jost solutions and adjoint Jost solutions in the variation

formulae (3.132)–(3.133) for δρj and δ ˆρj are analytic, as we desired Furthermore, these

products are only between Jost solutions  and their inverse −1whose boundary conditions

are both set at x= +∞

The variation relations (3.132)–(3.133) can be rewritten into the following more venient form:

2} are the adjoint squared eigenfunctions Notice that in the variation

relations (3.144)–(3.145), the potential variation takes the form of (δu, δv, −δ ˆu, −δ ˆv) T rather than (δu, δv, δ ˆu, δ ˆv) T

The same goes to the expansion of the potential variation(3.156) below This form of the potential variation corresponds to the form of the integrablehierarchy (3.20) and is necessary so that the corresponding squared eigenfunctions and their

adjoints are also eigenfunctions of the recursion operator LR and its adjoint operator LA R

(see the next section) For the Zakharov–Shabat system (2.2), ˆu = −u∗ Thus the above

form of the potential variation is consistent with that in variation relations (2.230)–(2.231)and (2.254) for the Zakharov–Shabat system

Next, we calculate variation of the potential in terms of variations of the scatteringdata Defining new Jost solutions

we find that the Riemann–Hilbert problem (3.66) becomes

F+(ζ ) = F−(ζ )  G (ζ ), ζ ∈ R, (3.150)where

Taking variation to this Riemann–Hilbert problem and repeating the same calculations as

in Sec 2.6, we get (Yang and Kaup (2009))

When the potential expansion (3.156) is inserted into the variation relations (3.144)–

(3.145), we get the inner products between squared eigenfunctions and their adjoints as

where I4is the unit matrix of rank 4

In the general case where s33andˆs33have zeros, the above closure relation needs toinclude the discrete-spectrum contribution As before, this discrete-spectrum contribution

is nothing but the residues of integrand functions in the above closure relation Notice from

the above expressions of squared eigenfunctions that Z−

2 are analytic in the upper half plane Also

notice that 1/s332 and 1/ˆs2

33are meromorphic in the lower and upper half planes, respectively

Thus the residues of integrand functions in (3.160) can be easily obtained These residueterms are similar to those in Eq (2.264) of Sec 2.6 Specifically, these terms are

us consider the reduction (3.27) which gives the Manakov equations In this case, due to

the involution properties of zeros (ζj , ¯ζ j) and vectors{¯vj0, v j0}, the discrete spectral data in

Eqs (3.73)–(3.74) for each pair of zeros contains only six real parameters, i.e., ζ j and two

complex parameters in vj0(since the third complex parameter in vj0can be normalized tounity) This means that a Manakov single soliton contains only six real parameters As aresult, the linearized Manakov equations around their single soliton only have six discrete(localized) solutions which are induced by variation of the soliton to its six free parameters.

Since discrete squared eigenfunctions also satisfy the linearized Manakov equations (seeSecs 2.7 and 3.9), their number then can only be six, not eight This means that the eight

terms for each pair of zeros (ζj , ¯ζj) in (3.161) are not linearly independent This is indeed

the case, and it is an important fact of the third-order scattering problem (3.41)

then

χ+= H1+ H2= (H1+ S−1H2) (3.165)Utilizing this equation and (3.138), we get

k (x, ζj)= −ˆsk3(ζj +

0(x, ζj), k= 1,2, (3.173)where

3.8.2 Squared Eigenfunctions under Potential Reductions

Next, we consider the case when the potential functions (u, v, ˆu, ˆv) are related to each other.

Here we consider two representative cases One is that

as in the Manakov system (3.1)–(3.2) This case corresponds to the general reduction (3.25)

(i.e., (3.75)), with B1= 1 and B2= I2 In this case, there are two independent complex

variables (u, v) in the potential matrix Q The other case is that

as in the Sasa–Satsuma equation (3.4) This case corresponds to two simultaneous

reduc-tions: one is (3.25) with B1= 1 and B2= I2, and the other one is v = u∗ In this case, there

is only one independent complex variable u in the potential matrix Q.

In the first case (3.177), since complex conjugates (u∗, v∗) should be treated as being

different from (u, v), squared eigenfunctions for this case are exactly the same as the generic

ones obtained above The only difference is that due to the symmetry reduction (3.177),Jost solutions and scattering elements satisfy the involution properties (see (3.77)–(3.80))

to vectors of length two These are distinctive features which we will elucidate below

Before we derive squared eigenfunctions under the potential reduction (3.178), weneed to detail the involution properties of Jost solutions and scattering elements under thisreduction This reduction (3.178) corresponds to two simultaneous symmetry reductions,

invo-σ−1∗(−ζ∗)σ and σ−1∗(−ζ∗)σ also satisfy the scattering equation (3.41) Then in view

of the boundary conditions of Jost solutions  and  at x= ±∞, we see that these matrix

solutions must be equal to (ζ ) and (ζ ), respectively Thus,  and  satisfy the additional

σ φ1(ζ )= φ2T(−ζ ), σ φ2(ζ )= φ1T(−ζ), σφ3(ζ )= φ3T(−ζ) (3.186)Now we derive the squared eigenfunctions under the potential reduction (3.178)

First, we re-examine the formulae (3.144)–(3.145) for variations of the scattering data due

to variation of the potential Under the reduction (3.178),

−δ ˆu = δv = δu∗, −δ ˆv = δu. (3.187)Thus formulae (3.144)–(3.145) reduce to

−and +are the adjoint squared eigenfunctions under the symmetry reduction (3.178).

These adjoint squared eigenfunctions become sums of products of Jost solutions The reason

is clearly due to the potential reduction (3.187), which leads to the combination of terms inthe generic formulae (3.144)–(3.145)

Next, we re-examine the formula (3.156) for variation of the potential due to variations

in the scattering data In view of the involution relations (3.185), we see that the terms onthe right-hand side of (3.156) can also be combined so that (3.156) becomes

are the squared eigenfunctions under the symmetry reduction (3.178) The last two rows in

Eq (3.191) are redundant due to the involution relations and can be dropped These squared

eigenfunctions are also sums of products of Jost functions The reason here is due to the

involution relations (3.185) between scattering coefficients, which lead to the combination

of terms in the generic potential expansion (3.156) Of course, the involution relations(3.185) are induced by the potential reduction (3.178) in the first place The inner productsand closure relations of these squared eigenfunctions can be trivially obtained by insertingthe potential expansion (3.192) and scattering-data variation formulae (3.188)–(3.189) intoeach other as we did before for the generic case, and so they will not be presented

3.9 Squared Eigenfunctions, the Linearization Operator, and the Recursion Operator

As we have seen for the NLS equation in the previous chapter, squared eigenfunctions areintimately related to the linearization operator and the recursion operator of the integrableequation Specifically, squared eigenfunctions are the eigenfunctions of the recursion oper-ator, and time-dependent squared eigenfunctions satisfy the linearized integrable equation

These relations hold for the hierarchy (3.20) associated with the higher-order scatteringequation (3.8) too To demonstrate, we will show these relations for the hierarchy associ-ated with the third-order scattering equation (3.41) in this section In our discussions, we

will first consider the generic case where the potential functions (u, v, ˆu, ˆv) are independent

of each other The nongeneric case with potential reductions will be considered at the end

In order for these squared eigenfunctions to be related to this recursion operator, the

antiderivative ∂−1in the definition (3.18) of this operator must be taken as ∂−1= x

−∞dy.

In that case, these squared eigenfunctions are also eigenfunctions of this recursion

opera-tor LR To show this, we first notice that Jost solutions  satisfy the scattering equation

(3.41), i.e.,

while adjoint Jost solutions −1satisfy the adjoint scattering equation of (3.41), i.e.,

−(−1)x= −iζ−1 + −1Q (3.196)Then using the definitions (3.157) of squared eigenfunctions and the above two equations for

Jost solutions, together with these Jost solutions’ boundary conditions of (x) → e −iζxas

x→ −∞, we can easily verify that these squared eigenfunctions are indeed eigenfunctions

of the recursion operator with the eigenrelations

LR Z±

±

j are eigenfunctions of the

adjoint recursion operator LA Ras well

Next, we consider the relation between squared eigenfunctions and the linearizationoperator of the hierarchy (3.20) This hierarchy is

whereL is the linearization operator It is important to emphasize that this linearized

equation is formulated for the perturbations (δq,−δr) rather than (δq,δr) This is necessary

so that the form of perturbations (δq,−δr) is consistent with the form of potential variations

(δu, δv, −δ ˆu,−δ ˆv) T in the expansion (3.156) of the previous section It is also necessary sothat the form of the linearized equation (3.199) is consistent with the form of the hierarchy(3.198) These consistencies must be followed in order to establish the relations betweenthe linearization operator, the recursion operator, and squared eigenfunctions below

Now we follow the general procedure of Sec 2.7 to make the connection betweensquared eigenfunctions and the linearization operatorL above For this purpose, we sub-stitute the potential-variation expansion (3.156) into the above linearized equation (3.199)and get

Here the time variable has been restored in the squared eigenfunctions and the

scattering-data variations since the potentials (q, r) are now varying with time (see (3.199)) For

simplicity, we have also assumed that the potentials do not contain discrete eigenvalues

Recalling the time evolution equations (3.86)–(3.88) of the scattering data in this hierarchy

(3.20), we see that for j= 1,2,

δρ j(ξ , t) = δρ j (ξ , 0)e i (2ξ ) n t, δ ˆρj(ξ , t) = δ ˆρj (ξ , 0)e −i(2ξ) n t (3.201)Then defining time-dependent squared eigenfunctions

Because the initial scattering-data variations can be arbitrary, in order for the above equation

to hold, we must have

LZ j −(t) (x, t, ξ ) = LZ j +(t) (x, t, ξ ) = 0, j = 1,2, (3.204)

for all x, t, ξ ∈ R In other words, the time-dependent squared eigenfunctions Z j ±(t)satisfythe linearized equation of the hierarchy (3.20) Note that these time-dependent squaredeigenfunctions are nothing but the original squared eigenfunctions (3.157) with Jost solu-

tions  replaced by the time-dependent Jost solutions

These time-dependent Jost solutions satisfy both of the Lax pair (3.41) and (3.43) Similarly,

by introducing time-dependent adjoint squared eigenfunctions

LA −(t)

j (x, t, ξ )= LA +(t)

j (x, t, ξ ) = 0, j = 1,2, (3.207)

for all x, t, ξ∈ R, where LAis the adjoint operator ofL under the inner product (3.148)

Now we know that these time-dependent squared eigenfunctions satisfy the linearized

equation (3.199) In addition, they are also eigenfunctions of the recursion operator LRinview of Eqs (3.197) and (3.202), and thus we have

(LLR− LRL)Zj ±(t) = 0, j = 1,2. (3.208)Then since these squared eigenfunctions form a complete set, we see that

i.e., the recursion operator and the linearization operator of the hierarchy (3.20) commute

Similarly, their adjoint operators also commute

Next, we discuss the case where the potentials admit reductions If the reduction isonly (3.25), i.e.,

where B1and B2are constant Hermitian matrices, then since q and its complex conjugate

are treated as independent variables, all the above relations still hold without change underthis reduction An example is the Manakov equations (3.1)–(3.2) where all the results abovestill apply But if there are further reductions in the potentials beyond (3.210), then certainmodifications will need to be made in order for the above relations to hold Let us take theSasa–Satsuma equation (3.4) as an example This equation has two simultaneous potential

reductions One is rT = −q†, which is of the type (3.210), and the other one is v = u∗

(see Sec 3.2) For this equation, the squared eigenfunctions become vectors of length two(rather than vectors of length four), as was shown in the previous section Its linearization

operator, for the variables (δu, δu∗)T, is also a 2× 2 matrix operator rather than the generic

4× 4 matrix operator In accordance with these length reductions, the recursion operatorfor the Sasa–Satsuma equation should also be reduced from the generic 4× 4 operator LR

in Eq (3.18) to a 2× 2 operator (this 2 × 2 recursion operator was given by Sergyeyev andDemskoi (2007)) After these reductions, analogous relations as obtained earlier in thissection would then carry over to the Sasa–Satsuma equation as well

3.10 Solutions in the Manakov System

In the remaining three sections, we consider three special integrable equations in the erarchy (3.20) of the third-order scattering operator (3.41) and illustrate their soliton andmultisoliton dynamics These equations are chosen partially due to their physical relevance(or potential physical relevance) But more importantly, they are chosen because their soli-ton and multisoliton solutions exhibit interesting features which are quite different fromthose in the NLS equation of the previous chapter

hi-In this section, we consider the Manakov system (3.1)–(3.2), i.e.,

iu t + uxx + (|u|2+ |v|2

iv t + vxx + (|u|2+ |v|2

which arises frequently in nonlinear optics and nonlinear water waves (see Sec 1.3) The

N-soliton solutions in this system can be readily derived from the general formulae (3.91)–

(3.93) after proper involution properties are included From Secs 3.2 and 3.3, we see thatthis system possesses the following involution properties:

¯ζ k = ζ∗

k, ¯vk = v†

Also n= 2 in the temporal equation (3.43) of the Lax pair Inserting these relations into

the general formulae (3.91)–(3.93), the N -soliton solution in the Manakov system (3.211)–

(3.212) would be obtained Without loss of generality, we let

Single-Soliton Solutions

To get single-soliton solutions, we set N= 1 in the above formulae Letting

ζ1= ξ + iη, $|α1|2+ |β1|2= e −2ηx0, (3.218)and introducing the unit polarization vector

where c†c= 1, then the single-soliton solution in the Manakov system can be rewritten as

(u, v) T (x, t) = c · 2η sech [2η (x + 4ξt − x0)] exp



−2iξx − 4i
ξ2− η2

t

 (3.220)

Without the polarization vector c, this Manakov soliton would just be the NLS soliton

(2.128), which has a sech profile and moves at velocity−4ξ The role of this polarization

vector is to control the relative power distribution between the two components in thissoliton However, the total power of this Manakov soliton, defined as the sum of its twocomponents’ individual powers, is

P =

 ∞

−∞(|u|2+ |v|2)dx = 4η, (3.221)

which depends only on the discrete eigenvalue ζ1, but not on the polarization vector c This

is an important feature of the Manakov system, and it has direct implications for solitoncollisions below

Collisions of Manakov Solitons

When N = 2 and ξ1
= ξ2, where ξk = Re(ζk), solutions (3.216) describe the collision of

two Manakov solitons In this collision process, while the discrete eigenvalue ζk of each

soliton does not change, its polarization vector ckdoes Thus after collision, while the power

of each Manakov soliton remains the same as before, its polarization can rotate, causing aredistribution of power among its two components This polarization rotation after collision

is a distinctive feature of Manakov solitons which has no counterpart in the NLS equation

To illustrate, we take

ζ1= 0.1 + 0.7i, ζ2= −0.1 + 0.7i, α1= α2= 1, β1= 0.25, β2= 0, (3.222)and the corresponding solution (3.216) is plotted in Fig 3.1 As we can see, the left soliton

before collision has only a u-component and no v-component But after collision when

xt

|v|

xt

Figure 3.1 Polarization rotations in the collision of two Manakov solitons The

solution parameters are given in Eq (3.222).

this soliton has re-emerged on the right side, its v-component appears In other words, its

polarization has rotated due to the collision The polarization of the other soliton has rotated

as well since the relative power distribution between its two components has also changedafter collision (see Fig 3.1) This phenomenon of polarization rotation provides anotherscenario of soliton collisions which is different from the elastic collision of NLS solitons

However, the power of each Manakov soliton remains the same before and after collision

In other words, the power of each Manakov soliton still passes through completely

This polarization rotation of Manakov solitons can be explicitly analyzed Repeatingthe same asymptotic analysis as for NLS-soliton collisions in Sec 2.2, we can find that

when ξ1
= ξ2, the two-Manakov-soliton solution from (3.216) decomposes into two single

Manakov solitons as t → ±∞ Let us assume that ξ1> ξ2, i.e., at t= −∞, soliton-1 is onthe right side of soliton-2 and moves slower Then polarizations of the two single Manakovsolitons before and after collision are related by the following formulae (Manakov (1973),Tsuchida (2004)):

(c−†

1 c−

2) c− 1

These formulae show that polarizations of Manakov solitons always rotate after collision,

except two special cases where the original polarization vectors c−

3.11 Solutions in a Coupled Focusing-Defocusing

differ-exhibits self-focusing nonlinearity in its u-component but self-defocusing nonlinearity in its v-component As a consequence, solution dynamics in this FDNLS system exhibits

significant differences from that in the Manakov system

The N -soliton solutions in this FDNLS system can be derived from the general

for-mulae (3.91)–(3.93) after proper involution properties are incorporated For this system,

while zeros (ζk, ¯ζk ) are still related by ¯ζk = ζ∗

k (see Eq (3.80)), the involution property for

eigenvectors (vk,¯vk) now becomes

where M is an N × N matrix given by

then this single-soliton solution can be rewritten as

(u, v) T (x, t) = c · 2η sech [2η (x + 4ξt − x0)] exp



−2iξx − 4i
ξ2− η2

t

, (3.235)which is a familiar sech-shaped solitary wave similar to the Manakov case This polarization

vector c also controls the relative power distribution between the two components in this soliton, but it is not a unit vector anymore (i.e., c†c
= 1) As before, we also define thepower of this FDNLS soliton as the sum of its two components’ individual powers Then

P =

 ∞

−∞(|u|2+ |v|2)dx = 4η |α1|2+ |β1|2

|α1|2− |β1|2 (3.236)

It is important to notice that this power depends not only on the soliton’s discrete eigenvalue

ζ1, but also on its polarization vector c This power dependence on the polarization vector

is a new feature of this FDNLS system, and it leads to new scenarios of soliton collisionsdifferent from those in the Manakov system This will be discussed next

Collisions of FDNLS Solitons

When two FDNLS solitons collide with each other, the discrete eigenvalues ζkassociated

with each soliton will not change after collision, but polarization vectors ck of the twosolitons will change as in the Manakov case The new feature here is that, since eachsoliton’s power depends on its polarization, this power will then change after collision Inother words, the two FDNLS solitons will transfer power from one to the other due to thecollision (Kanna et al (2006)) This contrasts the Manakov case where such power transferdoes not take place In addition to this power transfer between the two solitons, relativedistributions of each soliton’s power among its two components will change as well Thisfeature is analogous to the polarization rotation in collisions of Manakov solitons

To illustrate this power-transfer collision in the FDNLS system, we take parametervalues

ζ1= 0.1 + 0.7i, ζ2= −0.1 + 0.7i, α1= 1, α2= 0.8, β1= 0.25, β2= 0.1 (3.237)The corresponding solution (3.230) is illustrated in Fig 3.2 This solution describes thecollision between two FDNLS solitons As we can see from this figure, before the collision,the right soliton (soliton-1) has much higher power than the left one (soliton-2) But whensoliton-1 emerges out of collision (now on the left side), its power drops significantly Onthe other hand, soliton-2, which originally has little power, acquires a lot of power fromsoliton-1 and hence emerges out of collision as a much stronger soliton Thus power transferhas taken place from soliton-1 to soliton-2 during collision This collision visually lookslike the two solitons are reflected off by each other—a phenomenon which can be calledsoliton reflection This is a new scenario of soliton collisions which has no counterpart inthe NLS and Manakov systems

3.12 Solutions in the Sasa–Satsuma Equation

In this section, we consider solutions in the Sasa–Satsuma equation (3.4), i.e.,

u t + uxxx + 6|u|2u x + 3u(|u|2

This equation is interesting since its single-soliton solutions have more complicated profilesand can be double humped (Sasa and Satsuma (1991)) This contrasts the NLS equation,the Manakov equations, the FDNLS equations, and many others where single solitons areall single humped This double-hump structure of single solitons is induced by the doublesymmetries (3.181)–(3.182) of the potential matrix in this equation, which in turn imposefurther constraints on the discrete eigenvalues so that they must appear as quadruples ratherthan pairs in the complex plane

xt

|v|

xt

Figure 3.2 Soliton reflection in the FDNLS system (3.226)–(3.227) The solution

parameters are given in Eq (3.237).

To construct N -soliton solutions in the Sasa–Satsuma equation, we first detail its

involution properties on eigenvalues and eigenvectors Since the potential matrix in thisequation satisfies the double symmetries (3.181)–(3.182), involution properties (3.179)–

(3.180) and (3.183)–(3.184) on Jost solutions and the scattering matrix then are both satisfied

From the involution relation (3.184), we get

k) This differs from the previous systems where

discrete eigenvalues appear in pairs (ζk, ζ∗

k) under the single potential symmetry (3.75)

Eigenvectors in the kernels of Riemann–Hilbert solutions P± at the zeros (ζk , ζ∗

k,

−ζk,−ζ∗

k) satisfy additional constraints too From the double involution properties (3.179)

and (3.183) of Jost solutions as well as the definitions (3.53) and (3.60) of P±, we see that

P±satisfy double involutions relations

Incorporating the above involution properties of eigenvalues and eigenvectors into thegeneral formulae (3.91)–(3.93), multisoliton solutions in the Sasa–Satsuma equation will

be obtained Assuming that there are N quadruples of eigenvalues {(ζk , ζ∗

Next, we set N= 1 in the above formula and examine its single-soliton solutions

First, we consider the special cases where α1β1= 0 Solutions for the two cases of α1= 0

and β1= 0 are equivalent, and thus we take β1= 0 below After simple algebra, thesingle-soliton solution from formula (3.245) becomes

u (x, t) = ψ(x − vt − x0) e −2iξx−iλt+iσ0, (3.250)where

This solution is a moving solitary wave Its intensity profile is

|ψ(x)|2= 16η2e 4ηx + 2|γ |2+ |γ |2e −4ηx



e 4ηx + 2 + |γ |2e −4ηx2, (3.253)

which depends only on η and |γ | The parameter η controls the width of this soliton, while

|γ | controls its shape An interesting property of this soliton is that its shape can be single or

double humped depending on the parameter|γ | Indeed, it can be easily checked that this

soliton is single humped when|γ | > 0.5, but becomes double humped when |γ | < 0.5 To illustrate, we take η = 1 and |γ | = 0.1 The corresponding shape function |ψ(x)| is plotted

in Fig 3.3(a) It is seen that this soliton has two intensity peaks This is quite unusual inintegrable systems where single-soliton solutions are often single humped

The solitary waves (3.250) presented above are only part of the single-soliton

solu-tions (3.245) (with N = 1) When α1β1
= 0, these single-soliton solutions are no longersolitary waves Instead, they become spatially localized and temporally periodic boundstates (Mihalache et al (1993a)) To illustrate, we take

The corresponding single-soliton solution (3.245) is displayed in Fig 3.3(b) This solution

is a moving breather The reason for this breather is that this single soliton has four

dis-crete eigenvalues (ζ1, ζ∗

1,−ζ1,−ζ∗

1), which double those in single solitons of many otherintegrable systems (such as the NLS equation and the Manakov system) Thus this breathermay be viewed as the counterpart of a two-soliton breather state of the NLS equation in

0 0.5 1 1.5

Figure 3.3. Single-soliton solutions in the Sasa–Satsuma equation (3.238) :

(a) shape function of a double-hump solitary wave (3.250) with |γ | = 0.1 and η = 1; (b) a

bound state with ζ1= 0.5 + i and α1= β1= 1.

Fig 2.1(b) However, these single-soliton breathers in the Sasa–Satsuma equation are bust and will persist if the initial conditions are perturbed This contrasts the NLS breatherswhich generally will break up under perturbations (because the two constituent solitons inthose breathers will generically acquire different velocities under perturbations) Note thatthis Sasa–Satsuma breather solution in Fig 3.3(b) resembles the complexiton solutions in

ro-a coupled KdV system; see Hu et ro-al (2006)

Eq (1.50) for optical pulses in fibers for instance) In a perturbed system, solitons maynot propagate stationarily anymore Their shapes may be distorted over time In addition,energy radiation can be excited which can affect the soliton’s evolution in nontrivial ways.

In order to describe soliton evolution under perturbations, a soliton perturbation theory isrequired

Perturbation theories for single solitons have been developed for many integrableequations Examples include the KdV equation (Karpman and Maslov (1977), Kaup andNewell (1978a), Kodama (1985), Herman (1990), Grimshaw and Mitsudera (1993), Yanand Tang (1996)), the NLS equation (Kaup (1976a), Keener and Mclaughlin (1977), Kaupand Newell (1978a), Karpman and Maslov (1978), Kaup (1990), Hasegawa and Kodama(1995)), the sine-Gordon equation (Fogel et al (1977)), the Benjamin–Ono equation (Kaup

et al (1999)), the derivative NLS equation (and the related modified NLS equation) esnovich and Doktorov (1999), Chen and Yang (2002)), the Manakov equations (Lakoba andKaup (1997), Shchesnovich and Doktorov (1997)), the massive Thirring model (Kaup andLakoba (1996)), the fifth-order KdV equation (Yang (2001a)), the complex modified KdVequation (Yang (2003), Hoseini and Marchant (2009)), and many others (see also the reviewarticle by Kivshar and Malomed (1989) and the references therein) In addition to thesesingle-soliton perturbation theories, perturbation theories for soliton collisions have alsobeen developed for the KdV equation (Kodama (1987)), the NLS equation (Kano (1989)),

(Shch-the Benjamin–Ono equation (Matsuno (1995)), (Shch-the Manakov equations (Yang (1999)), andthe complex modified KdV equation (Hoseini and Marchant (2006)) All these perturba-tion theories can be separated into several categories One category is the perturbationtheory based on the inverse scattering transform (Kaup (1976a), Karpman and Maslov(1977, 1978), Kaup and Newell (1978a)), which has a recent development based on theRiemann–Hilbert problem (Shchesnovich and Doktorov (1997, 1999)) In this method, onecalculates the shift of the scattering data of the soliton due to the perturbation, and thenuses inverse scattering to reconstruct the perturbed solution Another category is the directsoliton perturbation theory (Fogel et al (1977), Keener and Mclaughlin (1977), Kaup (1990),Herman (1990), Grimshaw and Mitsudera (1993), Matsuno (1995), Yan and Tang (1996),Kaup and Lakoba (1996), Lakoba and Kaup (1997), Kaup et al (1999), Yang (1999, 2001a,2003), Chen and Yang (2002), Hoseini and Marchant (2009)) In this method, one solves thelinearized wave equation around a soliton by expanding its solution into a set of completeeigenfunctions of the linearization operator Suppression of secular growth in the linearizedsolution gives the evolution equations of soliton parameters, and the radiation is given by thelinearized solution through integrals of these complete eigenfunctions This method doesnot rely explicitly on the inverse scattering transform and is often easier to apply But itsconnection to the integrable theory is still visible since these eigenfunctions of the linearizedequation are simply the squared eigenfunctions of the underlying scattering operator (see theprevious two chapters) Both of these two methods can give not only the evolution of solitonparameters, but also the perturbation-induced radiation The third category is the normalform method (Kodama (1985, 1987), Kano (1989), Hoseini and Marchant (2006)) In thismethod, one transforms the perturbed equation into a normal form, and then uses the con-served quantities of the normal form to derive evolution equations of soliton parameters.

Among these different categories of perturbation theories, the direct soliton perturbationtheory is notable for its simplicity and versatility This perturbation theory and its applica-tions will be described in this chapter

4.1 Direct Soliton Perturbation Theory for the NLS Equation

We consider the perturbed NLS equation

where F is the perturbation term, and ε  1 When ε = 0, this equation has a soliton

solution (2.128) which can be rewritten as

This soliton has four free parameters: amplitude r, velocity v, initial position x0, and initial

phase σ0 We now consider how this soliton evolves under small perturbations (i.e., 0
=

ε 1) In the presence of perturbations, the four parameters of the soliton will slowlychange with time, the shape of the soliton will be modified, and energy radiation will beemitted These effects will be calculated below by a direct soliton perturbation theory

The essence of the direct soliton perturbation theory is a multiscale perturbation ysis Here the solution contains two time scales The four soliton parameters evolve on the

anal-slow time scale T = εt, while the other aspects of the solution (such as energy radiation) evolve on the fast time scale t According to a standard multiscale perturbation analysis,

we expand the solution u(x, t) into the following perturbation series:

the O(1) terms all cancel out because the soliton (4.2) is a solution of the NLS equation At

and the partial time derivative ∂t in (4.9) is with respect to the fast time t The operator

Lis closely related to the linearization operatorL in (2.268) for the NLS equation, and itwill also be called the linearization operator below The initial condition for the first-orderequation (4.9) is

since we are considering the evolution of an initial soliton under perturbations tion (4.9) can be solved by the eigenfunction expansion method, where one expands the

Equa-inhomogeneous term W as well as the solution A1into the complete set of operator L’s

eigenfunctions This method is analogous to the Fourier transform method for solving linearPDEs with constant coefficients In this method, both eigenfunctions and adjoint eigenfunc-

tions of L are needed Below, we first derive L’s eigenfunctions and adjoint eigenfunctions,

then we solve Eq (4.9) and finish the soliton perturbation theory

4.1.1 Eigenfunctions and Adjoint Eigenfunctions of the Linearization

Operator

In this subsection, we show that eigenfunctions and adjoint eigenfunctions of the

lineariza-tion operator L are simply squared eigenfunclineariza-tions and adjoint squared eigenfunclineariza-tions of

the NLS equation (2.1) for the soliton potential (4.2) Explicit expressions of these functions will also be derived

eigen-In order to obtain L’s eigenfunctions, we notice that L is related to the linearization

operatorL of the NLS equation around the soliton solution u = (θ)e ir2t(see (2.268) with

where Z ±(t) are given by (2.256) and (2.271) For the soliton potential u = (θ)e ir2t, the

corresponding analytical functions P±are simply  and −1given by (2.72), (2.73), and

(2.109), with N = 1, ζ1= ir/2, v10= (1,1)T , and with x replaced by θ In addition, the scattering matrix S is diagonal From this information, we can obtain all Jost solutions as well as the scattering matrix In particular, we get J−E = (φ1, φ2) as

φ2(θ , t, ζ )= e iζ θ

2iζ + r



 r sechrθ e ir2t 2iζ − r tanh rθ

Z +(t) (θ , t, k)= e −ir

2k2t (ik− 1)2diag(1, e −2ir2t

)Z2(θ , k), (4.19)where

λ (k) = r2

ThusZ1(θ , k) andZ2(θ , k) are L’s continuous eigenfunctions Note from (4.18)–(4.19) that these continuous eigenfunctions of L are simply squared eigenfunctions Z −(t) and Z +(t)

evaluated at time t= 0, multiplied by a constant which does not affect eigenrelations

Like-wise, L’s discrete eigenfunctions are also proportional to the discrete squared eigenfunctions

Z−(ζ∗

1), ˙Z−(ζ∗

1), Z+(ζ1), and ˙Z+(ζ1) (as seen in the closure relation (2.264)) with ζ1= ir/2 and time set to t= 0 These discrete eigenfunctions can be more easily derived directlyfrom the continuous eigenrelations (4.22) Evaluating (4.22) and their derivative equations

(with respect to k) at k = ∓i, we obtain

LZ1(θ , −i) = LZ2(θ , i)= 0, (4.24)

L ˙Z1(θ , −i) = −2ir2Z1(θ , −i), (4.25)

L ˙Z2(θ , i) = −2ir2Z2(θ , i). (4.26)Here the dot above Zj represents its derivative with respect to k Thus Z1(θ , −i) and

Z2(θ , i) are L’s discrete eigenfunctions with zero eigenvalue, while ˙Z1(θ , −i) and ˙Z2(θ , i)

are L’s generalized discrete eigenfunctions at the zero eigenvalue Since the set of squared eigenfunctions is complete for arbitrary time t, while L’s eigenfunctions are proportional

to these squared eigenfunctions with time set to t = 0, L’s eigenfunctions hence form a

complete set as well

Regarding L’s discrete eigenfunctions, it is more convenient to use the following

equivalent but simpler functions:

ZD,1(θ ) = θ



 11





which are linearly related to the above discrete eigenfunctionsZ1(θ , −i), Z2(θ , i), ˙Z1(θ , −i),

and ˙Z2(θ , i); see Yang (2000) Here (θ ) is given in (4.3) It is easy to verify that

LZG,1(θ )= ZD (θ ), LZG,2(θ ) = 2rZD,2(θ ). (4.30)Thus ZD (θ ) and ZD,2(θ ) are L’s discrete eigenfunctions with zero eigenvalue, while

ZG (θ ) andZG (θ ) are L’s generalized discrete eigenfunctions at the zero eigenvalue.

These four new discrete eigenfunctions have clear physical meanings They are the called Goldstein modes which are generated by the four free parameters in the NLS soliton(4.2): initial phase, initial position, amplitude, and velocity Indeed, we know that the

so-unperturbed NLS equation (4.1) (with x replaced by θ ) has the following soliton solution:

u (θ , t) = (θ − vt + θ0) exp

.1

t= 0, we will get the eigenrelations (4.29)–(4.30)

To solve the first-order equation (4.9), we also need L’s adjoint eigenfunctions, which are eigenfunctions of the adjoint operator L A To define this adjoint operator, the innerproduct is taken as

f,g =

 ∞

−∞f

which is the same as that used in the previous two chapters (see (2.233) and (3.148)) Under

this inner product, L A is equal to the transpose of L Analogously to eigenfunctions of L, joint eigenfunctions ϒ1,2(θ , k) are related to time-dependent adjoint squared eigenfunctions

ad-±(t) (θ , t, ζ ) as

−(t) (θ , t, ζ )= −e −ir

2k2t (ik+ 1)2diag(e −2ir2t , 1)ϒ1(θ , k), (4.33)

+(t) (θ , t, ζ )= − e ir

2k2t (ik− 1)2diag(1, e 2ir2t )ϒ2(θ , k), (4.34)

Here λ(k) is given in (4.23) Discrete adjoint eigenfunctions can be obtained from discrete

adjoint squared eigenfunctions in the closure relation (2.264) with time set to zero; or they

can be obtained from continuous adjoint eigenrelations (4.37) above by setting k = ∓i Their more convenient forms can be obtained by noticing that operator σ3Lis self-adjoint under

the inner product (4.32), i.e., (σ3L)A = σ3L , where σ3= diag(1,−1) Thus L A = σ3Lσ−1

3

As a result, adjoint discrete eigenfunctions are simply

ϒ D (θ ) = σ3ZD (θ ), ϒ D (θ ) = σ3ZD (θ ), (4.38)

ϒ G,1(θ ) = σ3ZG (θ ), ϒ G (θ ) = σ3ZG,2(θ ), (4.39)and their eigenrelations are

L A ϒ D,1(θ ) = L A ϒ D (θ )= 0, (4.40)

L A ϒ G (θ ) = ϒD (θ ), L A ϒ G (θ ) = 2rϒD (θ ). (4.41)

Inner products between L’s continuous eigenfunctions and adjoint continuous

eigen-functions can be obtained from the inner products of continuous squared eigeneigen-functions

(2.261)–(2.263) Here for the soliton potential, s11(ζ ) = s22−1(ζ ) = (ζ − ir/2)/(ζ + ir/2).

Then by setting ξ = rk/2, Eqs (2.261)–(2.263) give the following inner products:

Z1(θ , k), ϒ1(θ , k) = Z2(θ , k), ϒ2(θ , k) = 2π

r (k2+ 1)2δ (k − k), (4.42)

and the other two inner products are zero Here δ(·) is the Dirac delta function These inner

product relations can also be obtained directly For instance, to derive the first inner product

in (4.42), notice from the eigenrelations (4.22) and (4.37) that

Here the second step is obtained through integration by parts Inserting the large-θ

asymp-totics ofZ1and ϒ1into the right-hand side of the above equation and simplifying, wefind that

Regarding inner products between L’s discrete eigenfunctions and adjoint discrete

eigenfunctions, they can be calculated directly It is easy to verify that the only nonzeroinner products are

ZD (θ ), ϒG (θ ) = ZG,1(θ ), ϒD,1(θ )  = −r, (4.47)

ZD,2(θ ), ϒG (θ ) = ZG (θ ), ϒD,2(θ ) = 2 (4.48)

Other inner products between L’s discrete eigenfunctions and adjoint continuous

eigenfunc-tions are all zero

4.1.2 Solution for the Perturbed Soliton

After the eigenfunctions and adjoint eigenfunctions of the linearization operator L have

been obtained, we can now solve the first-order equation (4.9) and derive the solution forthe soliton under perturbations

To solve Eq (4.9), we first expand the forcing term W into the complete set of

the coefficients of the same eigenfunction of L, we obtain the following relations:

ih 1t + h3= c1, ih 2t + 2rh4= c2, (4.55)

ig 1t + λ(k)g1= α1(k), ig 2t − λ(k)g2= α2(k). (4.57)

As in (4.9), the partial time derivatives in these equations are with respect to the fast time t.

Due to the initial condition A1|t=0= 0, initial conditions for h j and g jin the above equations

are all zero Notice that the coefficients cj (1≤ j ≤ 4) depend on the slow time T only.

Thus solutions to Eq (4.56) are h3= −ic3t and h4= −ic4t, which grow linearly with time

Such linearly growing terms are called secular terms in the literature These terms would

make the perturbation series (4.6) invalid over the long time scale t = O(ε−1) and thus

must be suppressed Suppression of such secular terms in h3and h4requires c3= c4= 0.

Similarly, suppression of secular terms in h1and h2requires c1= c2= 0 Thus, we get

−∞Re(F0)· sechrθ (1 − rθ tanh rθ)dθ. (4.63)

Evolution equations for the position ν and phase σ then can be found from these equations

which are bounded for all times Substituting solutions (4.59) and (4.66) into (4.54) and

(4.6), we finally obtain the perturbed soliton solution (up to O(ε)) as

where φ and θ are given in (4.4), the perturbation function F is given in (4.53), and

evolution equations for r, v, ν, and σ are given by Eqs (4.60)–(4.61) and (4.64)–(4.65).

One can see from this expression that when t → ∞, the radiation terms, which involve

e ±ir2(k2+1)t in the integrals of Eq (4.67), disperse and decay at the rate of t −1/2(see also

Whitham (1974)) Since group velocities of these radiation modes in the NLS equation

are proportional to the wavenumber k, after a short time, the high-wavenumber modes

quickly escape to the far field; hence the remaining radiation modes near the soliton have

wavenumbers k approximately zero For these k≈ 0 modes, their oscillation frequency is

approximately r2(see (4.67)) Thus these radiation modes cause the shape of the perturbed

soliton to oscillate at the frequency r2 After a long time, when all the energy radiationhas dispersed and the shape oscillation of the perturbed soliton stopped, the solution willasymptotically approach a stationary state,

It is noted that in the above soliton perturbation theory, most of the effort was spent

on calculating squared eigenfunctions and the first-order radiation field of the perturbedsoliton If one is interested only in deriving the dynamical equations (4.60)–(4.65) forthe soliton’s parameters, then he can do so quite easily by utilizing the soliton’s Goldsteinmodes (4.27)–(4.28) and their adjoint modes (4.38)–(4.39), and requiring the forcing term

Wto be orthogonal to those adjoint Goldstein modes These requirements lead to the sameequations (4.58) and hence the same dynamical equations (4.60)–(4.65); thus expressions ofsquared eigenfunctions are not needed at all Indeed, formulae for squared eigenfunctionsare necessary only when energy radiation needs to be calculated The fundamental reason

for this is that dynamical equations for the soliton’s parameters, at time scale T = εt,

are decoupled from the first-order radiation field of the perturbed soliton, and thus can

be calculated separately This phenomenon occurs for all perturbed integrable equations

Hence dynamical equations for the soliton’s parameters, at time scale T = εt, can always

be derived readily without the knowledge of squared eigenfunctions In fact, even for

perturbed nonintegrable equations, derivation of evolution equations for the solitary wave’s parameters (at time scale T = εt) is an easy matter by utilizing the solitary wave’s Goldstein

modes (Kodama and Ablowitz (1981), Yang and Kaup (2000)) In some situations, however,

interesting dynamics of the perturbed soliton occurs at the longer time scale T2= ε2t Atthis time scale, evolution of the soliton’s parameters will be coupled with the first-orderradiation field, and hence explicit calculations of squared eigenfunctions and the first-order

radiation field then become necessary Such a soliton perturbation theory will be presented

in Sec 4.4

4.1.3 Evolution of a Perturbed Soliton in the NLS Equation

In the integrable NLS equation

For the purpose of demonstration, we consider the following initial condition:

u (x, 0) = (1 + ) sechx,   1, (4.70)which is a slightly amplified NLS soliton We want to determine the final state of thesolution, and also characterize how this initial condition relaxes to the final state Thisproblem has been treated before by other techniques (see Anderson (1983), Kath and Smyth(1995), Kuznetsov et al (1995)), but our treatment below gives simpler and more accurateresults

First we turn this problem into a soliton perturbation problem For this purpose, wedefine a scaled variable

which satisfies the perturbed NLS equation

iU t + Uxx + 2|U|2U = −4|U|2U (4.72)

Here the O(2) term has been neglected The initial condition for U is

which is soliton solution of the unperturbed NLS equation (4.72) with initial amplitude

r = 1 and initial position x0= 0 Thus this problem has become a soliton perturbation

problem analyzed in the previous section Here F0= −4r2sech3θ Substituting this F0into

dynamical equations (4.60)–(4.61) for r and v, we find that

dr/dT = dv/dT = 0, (4.74)