Regularized degenerate multi-solitons

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Abstract

We report complex ......-symmetric multi-soliton solutions to the Korteweg de-Vries equation that asymptotically contain one-soliton solutions, with each of them possessing the same amount of finite real energy. We demonstrate how these solutions originate from degenerate energy solutions of the Schrödinger equation. Technically this is achieved by the application of Darboux-Crum transformations involving Jordan states with suitable regularizing shifts. Alternatively they may be constructed from a limiting process within the context Hirota's direct method or on a nonlinear superposition obtained from multiple Bäcklund transformations. The proposed procedure is completely generic and also applicable to other types of nonlinear integrable systems.

Published for SISSA by Springer Received: June 13, 2016

Revised: August 4, 2016 Accepted: August 26, 2016 Published: September 1, 2016

JHEP09(2016)008

Regularized degenerate multi-solitons

Francisco Correaa,b and Andreas Fringc

aInstituto de Ciencias F sicas y Matem aticas, Universidad Austral de Chile,

Casilla 567, Valdivia, Chile

bInstitut fur Theoretische Physik and Riemann Center for Geometry and Physics,

Leibniz Universitat Hannover,

Appelstrae 2, 30167 Hannover, Germany

cDepartment of Mathematics, City University London,

Northampton Square, London EC1V 0HB, U.K.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We report complex PT -symmetric multi-soliton solutions to the Korteweg

de-Vries equation that asymptotically contain one-soliton solutions, with each of them possessing the same amount of nite real energy. We demonstrate how these solutions originate from degenerate energy solutions of the Schrodinger equation. Technically this is achieved by the application of Darboux-Crum transformations involving Jordan states with suitable regularizing shifts. Alternatively they may be constructed from a limiting process within the context Hirotas direct method or on a nonlinear superposition obtained from multiple Backlund transformations. The proposed procedure is completely generic and also applicable to other types of nonlinear integrable systems.

Keywords: Integrable Field Theories, Integrable Hierarchies, Space-Time Symmetries

ArXiv ePrint: 1605.06371

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP09(2016)008

Web End =10.1007/JHEP09(2016)008

Contents

1 Introduction 1

2 Degenerate complex multi-soliton solutions from DC transformations 32.1 Darboux-Crum transformations, generalities 32.2 Degenerate complex KdV multi-soliton solutions 52.3 Degenerate two-solitons 62.4 Degenerate three-solitons 8

3 Degenerate complex multi-soliton solutions from Hirotas direct method 10

4 Degenerate complex multi-soliton solutions from superposition 12

5 Conclusions 13

1 Introduction

Soliton solutions to nonlinear integrable wave equations play an important role in nonlinear optics [1]. The rst successful experiments to detect them have been carried out more than forty years ago [2]. A particularly important and structurally rich class of solutions are multi-soliton solution which asymptotically behave as individual one-soliton waves. This feature allows to view N-soliton solutions as the scattering of N single one-solitons with di erent energies.

In analogy to von Neumanns avoided level crossing mechanism in quantum mechanics [3], it is in general not possible to construct multi-soliton solutions possessing asymptotically several one-solitons at the same energy. The simple direct limit that equates two energies in the expressions for the multi-solitons diverges in general. Some attempts have been made in the past to overcome this problem. One may for instance construct slightly modi ed multi-soliton solutions that allow for the execution of a limiting process towards the same energy of some of the multi-particle constituents [4, 5]. However, even though the solutions found are mathematically permissible, they always possess undesired singularities at certain points in space-time and have in nite amounts of energy. These features make them non-physical objects.

Inspired by the success of PT -symmetric quantum mechanics [6{8], many experi

ments have been carried out in optical settings, exploiting the formal analogy between the Schrodinger and the Helmholtz equation. In particular, the existence of complex soliton solutions in such a framework has recently been experimentally [9{11] veri ed and it was shown [12] that such type of solutions may posses real energies and lead to regular solutions despite being complex. Here we will employ a similar idea and demonstrate that they can be used to overcome the above mentioned in nite energy problem related to degenerate multi-soliton solutions. Starting from a quantum mechanical setting we

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JHEP09(2016)008

show that the degeneracy is naturally implemented by so-called Jordan states [13] when Darboux-Crum (DC) transforming [14{17] degenerate states of the Schrodinger equation. Finiteness in the energy is achieved by carefully selected complex PT -symmetric shifts in

the dispersion terms.

Subsequently we show how such type of solutions are also obtainable from other standard techniques of integrable systems. For Hirotas direct method [18] this can be achieved by reparameterizing known solutions such that they will become suitable for a direct limiting process that leads to degeneracy together with a tting complexi cation that achieves the regularization. For the other prominent scheme, the Backlund transformations we also demonstrate how the limit can be carried out on a superposition of three solutions in a convergent manner.

Here we consider in detail one of the prototype nonlinear wave equations, the Kortewegde Vries (KdV) equation [19], for the complex eld u(x; t)

ut + 6uux + uxxx = 0; (1.1)

depending on time t and space x. When taking the complex eld to be of the form u(x; t) = p(x; t) + iq(x; t) with p(x; t), q(x; t) 2 R and subsequently separating it into its

real and imaginary part one may view it as set of coupled equations for the real elds p(x; t) and q(x; t). Those equation reduce to some well studied systems, the Hirota-Satsuma [20] and Ito equations [21] in the limits (pq)x ! pqx and qxxx ! 0, respectively. The KdV

equation is known to arise from standard functional variation from the Hamiltonian density

H(u; ux) = u3 +

In general, for PT -symmetric models the energy

E = [integraldisplay]

1

1 H

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1

2u2x: (1.2)

H[u(x; t)]duux ; (1.3)

remains real despite the fact that the Hamiltonian density is complex [22]. The PT -

symmetry is realized as PT : x ! x, t ! t, i ! i, u ! u, leaving (1.1) invariant. As

we will demonstrate below it is essential to have complex contributions to u in order to render the energy nite.

Our manuscript is organized as follows: in section 2 we discuss the general mechanism that allows to implement degeneracies into Darboux-Crum transformations. We show that degenerate states in the Schrodinger equation need to be replaced by Jordan states in order to obtain nonvanishing and nite, up to singularities, solutions. Subsequently we elaborate in detail on the novel features of degenerate two and three soliton solutions and explain how the regularizing shifts need to be implemented. In section 3 and 4 we explain how Hirotas direct method and nonlinear superpositions obtained from four Backlund transformations need to be altered in order to allow for the construction of degenerate complex multi-soliton solutions with nite energy. We state our conclusions in section 5.

{ 2 {

[u(x; t)]dx =

I

2 Degenerate complex multi-soliton solutions from DC transformations

2.1 Darboux-Crum transformations, generalities

The Darboux-Crum transformations [14{17] are well-known to generate covariantly an entire hierarchy of Schrodinger equations to the same eigenvalue E = 2 in a recurrence

procedure. It allows to solve the hierarchy of equations

@2x (n) + V (n) (n) = 2 (n) ; n = 0; 1; 2; : : : (2.1)

with potentials

V (n)( 1; : : : ; n) = V (n1) 2 [parenleftBig]

ln (n1)

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xx = V 2 ln [W ( 1; : : : ; n)]xx ; (2.2)

by the wave functions

(n) ( 1; : : : ; n) = D (n1)

n

(n1)

[parenrightBig]=n

Yk=1D (k1) k (0) =

W

1; : : : ; n; (0)

[parenrightBig]

W ( 1; : : : ; n) ; (2.3)

for all i [negationslash]= j, i; j = 1; 2; 3; : : : with V (0) = V and (0) = .

Here we will be dealing the degenerate when i = j for some i and j. Let us explain this in detail: following [17] we recall here that in case of degeneracy one has to replace the eigenstates of the Schrodinger equation by so-called Jordan states (k) de ned as solutions of the iterated Schrodinger equation

^

Hk+1 (k) =

@2x + V E( )

k+1 (k) = 0; (2.4)

with potential V and eigenvalue E( ) depending on the spectral parameter . Thus for k = 0 the corresponding Jordan state simply becomes the eigenfunction of the Schrodinger equation, that is (0) = or (0) = with denoting the second fundamental solution to the same eigenvalue E( ) obtainable via Liouvilles formula (x) = (x)

[integraltext]

x [ (s)]2 ds

from the rst solution . The general solution to (2.4) is easily seen to be

(k) =

k

Xl=0dl (l) ; cl; dl 2 R; (2.5)

with ~(k) := @k =@Ek and (k) := @k =@Ek. Some identities that will be useful below immediately arise from this. Di erentiating the Schrodinger equation with respect to

E yields

H

^ h~(1)

[bracketrightBig]= ; and ^H h (1)

[bracketrightBig]= ; (2.6)

which can be employed to derive

Wx

; ~(1)

[parenrightBig]= 2 ; and Wx [parenleftBig] ; (1)

[parenrightBig]= 2 : (2.7)

Here W ([notdef]) denotes the Wronskians W (f; g) = fgx gfx.

{ 3 {

Xl=0cl~(l) +

k

Let us see how these states emerge naturally in degenerate DC-transformations. With E( ) = 2, the rst iterative step in this procedure is simply to note that the equation

@2x (1) + V (1) (1) = 2 (1) ; (2.8)

with same eigenvalue as in (2.4) for k = 0, but new potential1

V (1) = V 2 (ln )xx (2.9)

is solved by(1) =

(D ( 1) = W ( ; 1) 1 for [negationslash]= 1D ( ) = 1 for = 1; (2.10)

where D () := x ( x= ) is the Darboux operator. The hierarchy of Schrodinger

equations is then obtained by repeated application of these transformations. It is clear that a subsequent iteration of the degenerate solution in (2.10) will simply produce again the potential V and hence nothing novel. However, using the second fundamental solution (1) = (1) (x)

[integraltext]

x h (1) (s)

i2 ds to the level one equation yields something novel. In this case the new potential becomes

V (2) = V (1) 2 [parenleftBig]

ln (1)

xx = V (1) 2 [bracketleftbigg] ln

[parenleftbigg]

1

(x)

[integraldisplay]

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x [ (s)]2 ds

[parenrightbigg][bracketrightbigg]xx

; (2.11)

= V 2

ln[parenleftbigg][integraldisplay]x[ (s)]2 ds[parenrightbigg][bracketrightbigg]xx= V 2

ln[parenleftbigg][integraldisplay]x Ws

; ~(1)

[parenrightBig] ds

[parenrightbigg][bracketrightbigg]xx

;

; ~(1)

[parenrightBig][bracketrightBig][bracketrightBig]xx ;

where we used identity (2.7) through which the Jordan states enter the iteration procedure. The corresponding wave function to this potential is(2) = D(1)

[D ( )] : (2.12)

Proceeding in this way, the solutions to the hierachy of equations (2.1) with potentials (2.2)

and wavefunctions (2.3) have to be replaced by

V (n)( ) = V (n1) 2 [parenleftBig]

ln (n1)

= V 2

hln

hW

xx = V 2 ln

hW

; ~(1) ; ~(2) ; : : : ; ~(n1)

[parenrightBig][bracketrightBig]xx ; (2.13)

(n) =

n

Yk=1D(k1) (D ( )) =

W

; ~(1) ; ~(2) ; : : : ; ~(n1) ;

[parenrightBig]

W

; ~(1) ; ~(2) ; : : : ; ~(n1)

[parenrightBig]:

Evidently we may also chose to have a partial degeneracy keeping some of the is di erent from each other, in which case we simply have to replace consecutive i by Jordan states.

For instance, taking 1 [negationslash]= 2 and 3 = 4 = 5 = we obtain the potential

V (5)( 1; 2; ; ; ) = V 2 ln [bracketleftBig]

W

1; 2; ~(1) ; ~(2) ; ~(3)

[parenrightBig][bracketrightBig]xx ; (2.14)

with either = 1 or = 2. Notice from (2.11) the sequence of Jordan states always has to accompanied by a ~(0) = . Let us now see how this procedure can be employed in nding degenerate multi-soliton solutions by means of inverse scattering.

1Here and in what follows we always understand (ln f)x as a short hand notation for fx/f.

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2.2 Degenerate complex KdV multi-soliton solutions

The di erent methods in integrable systems take various equivalent forms of the KdV equation as their starting point. The Darboux-Crum transformation exploits the fact that the central operator equation underlying all integrable systems, the Lax equation Lt = [M; L], may be written as a compatibility equation between the two linear equations

L = , and t = M , with = (x; t; ); 2 R: (2.15)

For the KdV equation (1.1) the operators are well-known to take on the form

L = @2x u; and M = 4@3x 6u@x 3ux: (2.16)

Thus L becomes a Sturm-Liouville operator, such that the rst equation in (2.15) may be viewed as the Schrodinger equation (2.1) with L H being interpreted as a Hamiltonian

operator. Considering now the free theory with u = 0 and taking the wave function in the form (kx + !t), the second equation in (2.15) is solved by assuming the nonlinear dispersion relation 4k3 + ! = 0. For = 2=4 the two linear independent solutions

to (2.15) are simply

, (x; t) = cosh

12( x 3t + )[bracketrightbigg]; , (x; t) = sinh

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12( x 3t + )[bracketrightbigg]: (2.17)

We allowed here for a constant 2 C in the argument and normalized the Wronskian as

W ( ; ) = x x = =2. Suitably normalized, i.e. dropping overall factors, the rst

Jordan states resulting from (2.17) are computed to

~(1), = 2@ ,

@ = x 3 2t [parenrightbig]

, ; (2.18)

~(2), = x 3 2t

2 , 2

x + 3 2t[parenrightbig], ; (2.19)

(1), = 2@,

@ = x 3 2t

[parenrightbig]

, ; (2.20)

(2), = x 3 2t

x + 3 2t[parenrightbig] , : (2.21)

Using these explicit expressions the crucial identities (2.7) in the above argument

Wx

, ; ~(1),

[parenrightBig]= 2, ; and Wx , ; (1),

[parenrightBig]= 2, ; (2.22)

are easily con rmed. We also verify

H

^ h~(1),

[bracketrightBig]= , ;^H h (1),

[bracketrightBig]= , ;^

H2 h~(2),

[bracketrightBig]= 2 3 , ; ^

H2 h (2),

[bracketrightBig]= 2 3, ;(2.23)

which yield the de ning relations for the Jordan states upon a subsequent application of the energy shifted Hamiltonian ^

H as de ned in (2.4).

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2 , 2

2.3 Degenerate two-solitons

To compute the degenerated two-soliton solution we use the above expressions to evaluate the Wronskian W ( , ; ~(1), ) involving one Jordan state. As indicated in (2.5) we may take the constants cl, dl di erent from zero, which we exploit here to generate suitable regularizing shifts. First we compute

W

h , ; ~(1),

[bracketrightBig]= W

, ;

x 3 2t[parenrightbig],

[bracketrightbig]=

x 3 2t[parenrightbig]

W [ , ; , ] + , ,

= 12

x 3t +

[parenrightbig][bracketrightbig]; (2.24)

where we used the identity (2.18) and the property of the Wronskian W (f; gh) = W (f; g)h+ fghx. We note that one of the dispersion terms already includes a shift . Next we demand that also the dispersion term x 3 3t is shifted by a constant , which is uniquely

obtained from

W

h , ; ~(1), + ,

[bracketrightBig]= 12

x 3 3t + + sinh

x 3 3t + sinh

x 3t +

[parenrightbig][bracketrightbig]: (2.25)

The degenerate two-soliton solution u = 2(ln W )xx resulting from (2.2) and (2.25) reads

u, ; , (x; t) = 2 2[bracketleftbig][parenleftBigg] x

3 3t +

[parenrightbig]

sinh x 3t +

[parenrightbig]

2 cosh x 3t +

[parenrightbig]

2

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[ x 3 3t + + sinh ( x 3t + )]2

:

(2.26)

This solution becomes singular when the Wronskian vanishes, which is always the case for some speci c x and t when ; 2 R. However, for the PT -symmetric choice = i^

,

= i^

, ^

; ^

2 R this solution becomes regularized for a large range of choices for ^

and ^

.

From

W = 12

cos ^ sinh

x 3t[parenrightbig]+ x 3 3t[bracketrightbig]+ i

^ + sin ^ cosh( x 3t)

[bracketrightbig]; (2.27)

we observe that whenever ^

= sin ^

> 1 the imaginary part of W can not vanish and

therefore u, ; , will be regular in that regime of the shift parameters. Furthermore, we observe that u, ; , involves two di erent dispersion term x 3t + and x 3 3t + ,

each with a separate shift. In the numerator the latter becomes negligible in the asymptotic regimes where the degenerate two-soliton behaves as two single solitons traveling at the same speed with one slightly decreasing and the other with slightly increasing amplitude due to the time-dependent pre-factor. In the intermediate regime, when the linear term x 3 3t term in the numerator contributes, it produces a scattering between the two

one-solitons with the same energy. We depict this behaviour in gure 1. In addition to the regularization, this entire qualitative behaviour is due to the fact that our solutions are complex. For a more detailed analytical discussion of the asymptotic behaviour we refer the reader to [26].

We observe that the larger and smaller amplitudes have exchanged their relative position in the two asymptotic regimes with their mutual distance kept constant. This is of course di erent from the standard nondegenerate case where the solitons continuously approach each other before the scattering event and separate afterwards. Again this is

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Figure 1. Degenerated KdV two-soliton compound solution with = = 2, = i3=5 and

= i=5 at di erent times.

Figure 2. Degenerated KdV two-soliton compound solution with = = 2, = i=5 at xed

moment in time t = 1 and varying shift parameter = r + i3=5.

achieved through the complexi cation of our solution. Here the scattering is governed by some internal breatherlike structure as in con ned to a certain region.

As demonstrated in gure 2 this internal structure can be manipulated by varying the shift .

For a xed instance in time we can employ to increase or decrease the distance between the single soliton amplitudes and even nd a value such that the distance becomes zero. However, this value is in the intermediate regime and as time evolves the two solitons will separate again to some nite distance in the asymptotic regime.

Our interpretation is supported by the computation of the energies resulting from (1.3) with Hamiltonian density (1.2) for the solution u, ; , . Numerically we nd the nite real energies

E, ; , =

[integraldisplay]

1 [u, ; , ; (u, ; , )x]dx = 2

5

5 = 2E; ; (2.28)

i.e. precisely twice the energy of the one-soliton u; , reported for instance in [12].

In order to compare with various other methods it is useful to note that the degenerate

Wronskians may be obtained in several alternative ways. We conclude this subsection

{ 7 {

by reporting how the expression for the Wronskian (2.25) can be derived by mean of a limiting process directly from the two-soliton solution. This is seen from by starting from the de ning relation for the Jordan state ~(1)

W

h , ; ~(1),

[bracketrightBig]= 2 lim ! @@ W [ , ; , ] (2.29)

= 2 lim

!

lim

h!0

W

, ; , +h , h[bracketrightbigg]

(2.30)

= 2 lim

h!0

W

, ; , +h , h[bracketrightbigg]

(2.31)

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= 2 lim

h!0

1hW [ , ; , +h] (2.32)

= 2 lim

!

1

W [ , ; , ] ; (2.33)

where in the last step we chose h = . The shift can now be implemented by determining

from the limit of the expression

W [ + , ; , ] = W [ , ; , ] cosh2 [parenleftbigg]

2

[parenrightbigg]

W [, ; , ] sinh2

2[parenrightbigg](2.34)

+12 sinh ( ) [W [, ; , ] W [ , ; , ]] :

It it is obvious that for the limit (2.33) of the shifted expression to be nite we require ( ) with constant of proportionality chosen in such a way that it yields 1=2 in

the limit. Hence we obtain

W

h , ; ~(1), + ,

[bracketrightBig]= 2 lim ! 1 W

+ + , ; + ,

[bracketrightbigg]; (2.35)

= 2 lim

!

W

+ + , ; + ,

[bracketrightbigg]: (2.36)

These identities will be useful below when we relate this approach to Hirotas direct method.

2.4 Degenerate three-solitons

To nd the degenerate three-soliton solution we may once again compute the Wronskian, albeit now involving two Jordan states. As discussed in the previous section, the expression for W ( , ; ~(1), ; ~(2), ) will inevitably lead to solutions with in nite energy. Thus we will again exploit (2.5) with nonvanishing constants cl, dl to generate the regularizing PT -

symmetric shifts. Demanding regularized shifts, the coe cients in the generically expanded Jordan states are uniquely xed as

W

, ; ~(1), + , ; ~(2), + 2 (1), +2 4 , [bracketrightbigg]

=

1 +

(3);

2+ cosh

(1);

[parenrightBig][bracketrightbigg]sinh (1);

2

[parenrightBigg]

(9) ; cosh (1);

2

!; (2.37)

{ 8 {

Figure 3. Degenerated KdV three-soliton compound solution with = = = 2, = i3=5,

= i3=10 and = i=10.

where we abbreviated the di erent dispersion terms as

( ); := x 3t + ; (2.38)

Notice that we have now three di erent shifted dispersion terms (1); , (3); and (9) ; , where

the rst governs the asymptotic behaviour and the remaining ones the additional structure in the intermediate regime. The solution u, ,; , , = 2(ln W )xx is depicted in gure 3.

We observe that asymptotically we have three single one-solitons moving at the same speed. They exchange their positions in the intermediate region near the origin, when the linear terms in (2.37) contribute.

For the general three-soliton solution we have also additional options available, namely to produce the degeneracy only in two of the one-solitons while keeping the remaining one at a di erent velocity. A suitable choice that produces the desired shifts is

W

h , ; ~(1), + , ; ,

[bracketrightBig]=

2 + 28 sinh

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(1);

[parenrightBig] 2 28 (3) ; [bracketrightbigg]cosh (1);

2

[parenrightBigg]

2 cosh2[parenleftBigg][parenleftBigg]

(1);

2

!: (2.39)

We depict the corresponding KdV solution u, ,; , , = 2(ln W )xx in gure 4.

We clearly observe that asymptotically we have a degenerated two-soliton and a onesoliton solution with the faster two-soliton overtaking the slower one-soliton.

Let us nish this section by reporting an alternative form of the degenerate three-soliton solution suitable for a comparison with other methods. We nd

W

, ; ~(1), + , ; ~(2), + 2 (1), +2 4 , [bracketrightbigg]

= 16 lim

, !

1( )( )( )

!sinh

(1); 2

W

+f( , , ), ; +f( , , ), ; +f( , , ),

[bracketrightbig](2.40)

= 8 lim

, !

W

+f( , , ), ; +f( , , ), ; +f( , , ),

[bracketrightbig]; (2.41)

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Figure 4. Degenerated KdV two-soliton compound solution scattering with a one-soliton with

= = 2, = 1:7, = i3=5, = i3=10 and = i=10.

where we introduced the shift function

f(x; y; z) := 49 [bracketleftbigg]

x2 + yz

(x + y)(x + z) 2

x(y2 + z2)

(x + y)(x + z)(y + z)

[bracketrightbigg]

x2 yz(x + y)(x + z): (2.42)

It will be important below to note that the sum of all shifts adds up to zero, f( ; ; ) + f( ; ; ) + f( ; ; ) = 0.

Once again our interpretation is supported by the computation of the corresponding energies. Numerically we nd

E, ,; , , =

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+ 43

[integraldisplay]

1 [u, ,; , , ; (u, ,; , , )x]dx = 3

5

5 = 3E; ; (2.43)

E, ,; , , =

[integraldisplay]

5

1 [u, ,; , , ; (u, ,; , , )x]dx = 2

5

5

5 = 2E; + E; ; (2.44)

which are again nite and real energies irrespective of whether the shifts are taken to be complex or real.

3 Degenerate complex multi-soliton solutions from Hirotas direct method

Hirotas direct method [18] takes a di erent equivalent form for nonlinear wave equations as starting point. The system at hand, the KdV equation (1.1), can be converted into Hirotas bilinear form

D4x + DxDt

[parenrightbig]

[notdef] = 0; (3.1) by means of the variable transformation u = 2(ln )xx. The required combination of Hirota derivatives in terms of ordinary derivatives are

D4x [notdef] = 2xxxx 4xxxx + 6xxxx; (3.2)

DxDt [notdef] = 2xt 2xt: (3.3)

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The -function can be identi ed with the Wronskian in the previous section, up to the ambiguity of an overall factor exp [c1x + c2 + f(t)] with arbitrary constants c1, c2 and function f(t). Remarkably equation (3.1) can be solved with a perturbative Ansatz =

P1k=0 "kk in an exact manner, meaning that this series terminates at N-th order in " for the corresponding N-soliton solution. Order by order one needs to solve the following set of linear equations

D4x + DxDt

[parenrightbig][parenleftBigg][parenleftBigg]1

[notdef] 1 + 1 [notdef] 1 [parenrightbig]

= 2(1)xt + 2(1)xxxx = 0; (3.4)

D4x + DxDt

[parenrightbig][parenleftBigg][parenleftBigg]1

[notdef] 2 + 1 [notdef] 1 + 2 [notdef] 1 [parenrightbig]

= 0; (3.5)

D4x + DxDt

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= 0: (3.6)

Let us rst see how the Hirota equations are solved using Wronskians involving Jordan states. We start with the two-soliton solution and take 1 = Wx

; ~(1)

[parenrightbig][parenleftBigg][parenleftBigg]1

[notdef] 3 + 1 [notdef] 2 + 2 [notdef] 1 + 3 [notdef] 1 [parenrightbig]

. Using

; ~(1)

[bracketrightbig]= c 2, the rst order Hirota equation (3.4) reads

(1)xt + (1)xxxx = (Wx)t + (Wx)xxx = c( 2)t + c( 2)xxx = 2c t + 2 x

[parenrightbig]

identity (2.7) in the form Wx

= 0: (3.7)

This equation is solved using the above mentioned nonlinear dispersion relation, i.e. by taking (x; t) = (x 2t).

Next we show how one may carry out the limit to our degenerate solutions directly on the Hirota multi-soliton solutions. The two-soliton -function is known to be of the form

, (x; t) = 1 + c1e + c2e + c1c2{( ; )e + (3.8)

with := x 3t and {( ; ) := ( )2=( + )2. Usually the constants c1 and c2 are

set to one. Evidently carrying out the limit ! in this variant will simply produce a

one-soliton solution. However, when making use of the freedom to multiply the -function with an overall factor we de ne

, ; , (x; t) = 8( )

e

+

2 +W

+ + , ; + ,

[bracketrightbigg]; (3.9)

which produces the series expansion form (3.8) of the -function with coe cients

c1 =

+

e+

+ : (3.10)

+ ; and c2 = +

e

In this form the limit is easily performed

, ; , (x; t) = lim

!

, ; , (x; t) = 1 2( x 3 3t + )e + e2 +2: (3.11)

Since the factor in (3.9) has the form of the general ambiguity, the expression 2 (ln , ; , )xx produces the same two-soliton solution (2.26) as previously obtained.

Similarly, using the identity

, ,; , , (x; t) =

64 exp

+ + +32[parenrightBig]

( )( )( )

W

+f( , , ), ; +f( , , ), ; +f( , , ),

(3.12)

{ 11 {

we obtain the series expansion form (3.8) of the -function for the 3-soliton solution

, ,; , , (x; t) = 1 + c1e + c2e + c3e + c1c2{( ; )e + + c1c3{( ; )e +

+c2c3{( ; )e + + c1c2c3{( ; ){( ; ){( ; )e + + (3.13)

with coe cients

c1 = c( ; ; ); c2 = c( ; ; ); c3 = c( ; ; ); (3.14)

where

c(x; y; z) = (x + y)(x + z)

(x y)(x z)

e+f(x,y,z): (3.15)

Clearly without the information from the previous section it is not obvious at this stage how to determine the coe cients ci in general, especially the regularizing shifts.

4 Degenerate complex multi-soliton solutions from superposition

It is well-known that the combination of four Backlund transformations combined in a Bianchi-Lamb [23, 24] commutative fashion gives rise to a \nonlinear superposition principle", e.g. [12]. Introducing the quantity u = wx, it takes on the form

w12 = w0 + 2 1 2 w1 w2

; (4.1)

for the KdV equation where w0, w1, w2 and w12 correspond to di erent solutions. Relating w1 and w2 to the standard one-soliton solution and setting w0 to the trivial solution w0 = 0, the general formula (4.1) becomes

w,; , = 2 2 w; w;

1

2 ( x 3t + )

, 1 = 2=2 and 2 = 2=2, see [12]. Remarkably in this form the limit lim ! w,; , can be performed directly

lim

!

8

>

<

>

:

: (4.3)

The corresponding solution the KdV equation will still be singular, but when implementing the same shifts as in (2.35) we compute

lim ! w+ + , + ; , x= lim !

(1);

[parenrightBig]

w+ + , + ; ,

{ 12 {

JHEP09(2016)008

; (4.2)

with w; (x; t) = tanh

[bracketleftbig]

w,; , =

0 for [negationslash]= ^

2

1+cosh

(1);

[parenrightBig]

(3)0; +sinh

for = ^

x= u, ; , ; (4.4)

and thus recover precisely the solution (2.26). The relation to the treatment in section 2 involving DC-transformations is achieved by considering (2.2) for n = 0 with V (0) = 0.

Then we read o the identi cation w; = 2 (ln , )x, which is con rmed by the explicit expression (2.17).

Similarly we may carry out the limit on higher soliton solutions. For instance, iterating (4.1) once more we obtain the three-soliton solution

w, ,; , , = w; + 2 2 w, ; , w,; ,

; (4.5)

which yields the non-trivial limit

lim

, !

w,,; , , =

2

[bracketleftbigg][bracketleftbigg]

1 +

(3)0;

2+ cosh

(1);

[parenrightBig][bracketrightbigg] sinh

(1); 2

[parenrightbigg]

(9)0; cosh

(1); 2[parenrightbigg][bracketrightbigg]x

1 +

(3)0;

JHEP09(2016)008

2+ cosh

(1);

[parenrightBig][bracketrightbigg] sinh

(1); 2[parenrightbigg] (9)0; cosh

(1); 2[parenrightbigg]:

(4.6)

When implementing the appropriate shifts and di erentiating once more this produces precisely the same three-soliton solution as previously constructed in section 2.4.

5 Conclusions

We have constructed a novel type of compound soliton solution composed of a xed number degenerate one-soliton constituents with the same energy. Asymptotically, that is for large and small time, the individual one-solitons travel at the same velocity with almost constant amplitudes. In the intermediate regime they scatter and exchange their relative position. Thus the entire collection of one solitons may be viewed as a single compound object with an internal structure only visible in a certain regime of time. As we have shown, one may construct solutions in which these compounds scatter with other (degenerate) multi-solitons at di erent velocities.

Technically these compound structures arose from carefully designed limiting processes of multi-soliton solutions. We have demonstrated how these limits can be performed within the context of standard techniques of integrable systems, employing Darboux-Crum transformations involving Jordan states, Hirotas direct method with specially selected coe -cients and on the nonlinear superposition obtained from Backlund transformations. While the limits led to mathematically admissible nonlinear wave solutions, they always possess singularities such that their energy becomes in nite. In order to convert them into physical objects it was crucial to implement in addition some complex regularizing shifts.

When comparing the di erent methods, the DC-transformations require the most substantial modi cation by the introduction of Jordan states. This approach is very systematic and the modi ed transformations always constitute degenerate soliton solutions. To carry out the limit within the context of Hirotas direct method requires some guesswork in regards to the appropriate choice of coe cients, which we overcame here by relying on the information from the DC-transformations. The nonlinear superposition of three solutions appears to be the most conductive form for taking the limit directly. The disadvantage in this approach is that expressions for higher multi-soliton solutions are rather cumbersome when expressed iteratively. So far in all approaches the regularizing shift were introduced in a somewhat ad hoc fashion.

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There are various open issues left to be resolved and not reported here. Evidently the suggested procedure is entirely generic and not limited to the KdV equations or the particular type of solutions and boundary conditions considered here [25]. It would be interesting to apply them to other types of integrable systems as that might help to unravel some further universal features. For instance, one expects that the regularizing shifts can be cast into a more universal form that might be valid for any arbitrary number of degeneracies when exploiting further their ambiguities. Furthermore it is desirable to complete the argument on why the energies of these complex solutions are real. This follows immediately when they and the corresponding Hamiltonians are PT -symmetric.

As demonstrated in [12], this can be achieved with suitable real shifts in time or space, but in addition one also requires the model to be integrable. We report on these issues in more detail elsewhere [26].

Acknowledgments

FC would like to thank the Alexander von Humboldt Foundation (grant number CHL 1153844 STP) for nancial support and City University London for kind hospitality.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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