Discrete Spectrum of 2 + 1-Dimensional Nonlinear Schrödinger Equation and Dynamics of Lumps

Copyright © 2016 Javier Villarroel et al.This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a natural integrable generalization of nonlinear Schrödinger equation to 2 + 1 dimensions. By studying the associated spectral operator we discover a rich discrete spectrum associated with regular rationally decaying solutions, the lumps, which display interesting nontrivial dynamics and scattering. Particular interest is placed in the dynamical evolution of the associated pulses. For all cases under study we find that the relevant dynamics corresponds to a central configuration of a certain N-body problem.


The Physical System
In this paper we consider the system of equations: where (, , ) is a complex function, depending on three real variables , , .We derive via the inverse scattering method several large classes of solutions that satisfy the boundary conditions ( 2 ≡  2 +  2 ),  (3) Due to the overall simplicity, elegance, and potential physical interest, the above system is a natural interesting nonlinear 2+1-dimensional integrable equation to study, even though it has not been related to any physical situation so far.To further motivate its study we note the following: (1) The more general problem corresponding to the boundary conditions lim →∞ || (, , ) = , where  > 0 is a given nonnull constant is also solvable and reducible to problem (2) as follows.Let û(, , ) be the unique solution to (1) under the boundary conditions (2).Then one proves easily that (, , ) ≡ û(, ,  2 ) also solves (1) and satisfies (4).
(2) System (3) arises as the compatibility of a Lax pair (see Section 2 below) and hence it is integrable.We show here that the operator has also a discrete spectrum, corresponding to potentials that solve (1) and (2): that is, it can be written as  = 1 + ũ, where ũ is a regular, weakly decaying function.
(6) There is an infinite number of conservation laws for this equation (see [5]), which includes, in particular, the functional  () ≡ ∫ (|| 2 (, ) − 1) d d.(7) We note that the "amplitude" || 2 (, ) − 1 of interesting solutions to (3) is naturally expected to be the physical observable but it is not definite positive; hence, the renormalized mass of the field  ≡  + −  − can take both signs.(7) An interesting parabolic real version of (1) (i.e., letting   →   ) is obtained as a particular reduction of the Self-Dual Yang Mills Equations; see [6].
Lump configurations (i.e., regular and rationally decaying solutions) are paradigmatic solutions of integrable equations in 2 + 1 dimensions and as such they have been extensively studied in the last years.Their spectral interpretation was first unlocked in relation to KPI [7] (see also [8,9]).A description of the KP equation, its physical origins, integrability, and soliton solutions can be found in [10].Subsequently, lumps have been found in other integrable equations like Davey-Stewartdon II (DSII) (see [11]) and the 2 + 1-Toda lattice (see [12][13][14]).
The dynamics of standard lumps on the plane is trivial: the motion is uniform; further, upon interaction, only a parallel shift on the asymptotic trajectory is to be found.Such uniform motion has been considered to rule the dynamics of localized pulses of integrable equations up until the late nineties.Remarkably it was found that KPI possess a new class of localized, real valued solutions that have a nontrivial asymptotic dynamics.Even though the simplest of these solutions has been known for a long time, the interesting scattering properties that they exhibit went unnoticed up until 1995 [15,16].The spectral interpretation along with a derivation of the general class of those solutions was unlocked in [17,18].Nonstandard lumps were found to be associated with a new discrete spectrum of the time dependent Schrödinger operator corresponding to meromorphic eigenfunctions with poles of higher multiplicity and to what we term nonstandard pole divisors.(A comprehensive account of the spectrum of both KPI and KPII is given in [19].)The extension of these ideas and solutions to DSII equation via spectral analysis of the Dirac operator on the plane has been considered in [20].We also note that KPI possesses, in addition, other localized, nondecaying solutions like line solitons [21,22].The solution of the Cauchy problem in such a background is considered in [14,23].
In this paper we show that the spectral operator that linearizes (3) possesses a rich discrete spectrum corresponding to smooth, rationally "decaying" lump configurations whose dynamical evolution and scattering are nontrivial.The physical behavior of these solutions is reminiscent to the aforementioned discrete spectrum and nonstandard lumps for KPI.We find that they are associated with higherorder pole meromorphic eigenfunctions of a similar discrete spectrum.Characterization of the discrete spectrum of this operator involves giving the pole multiplicity and an adequate pole divisor, which could be, in the spirit of the ideas of [18,20], associated with integer winding numbers.
Direct methods to study this class of solutions have also been developed; see [24,25] for KPI and [26] for DSII.See also [27] for related ideas.These direct algebraic methods, although they do not shed any light on the associated spectral problem, are quite powerful to undertake a general classification of the relevant class of solutions.In this regard note that in [28][29][30][31] Painleve's test for (3) was first considered and some special solutions, like line solitons, lumps, and dromion solutions, were found.
The organization of the paper is as follows.A convenient form of the linearizing Lax pair associated with (3) with boundary conditions (2) is introduced in Section 2, following the ideas of [32].In Section 3 we study the discrete spectrum of meromorphic eigenfunctions.Different relations between the Laurent coefficients (LC) compatible with the Lax pair are established.In Section 4 we study several classes of lump solutions and their dynamics and physical properties.Particular interest is addressed to the problem of determining the number of solitons that ensue from given analytic structure and to determine the motion of these entities.We find that typically lump's dynamics is a superposition of the "center of mass" motion, which proceeds in a uniform way and individual, lump depending, motion that behaves as ||  with  < 1. Interestingly for all cases analyzed the dynamics corresponds to a central configuration of a certain -body problem (a central configuration of the Euler-Lagrange -body problem corresponds to interacting point masses wherein for each body the acceleration vector is directed towards the center of mass and proportional to the distance to the center of mass) where they fall to the mass center.Particularly interesting is the case, similar to the "homothetic Lagrange" solution in the tree body problem, when pulses are located at the vertexes of an equilateral triangle which collapses to the center along a straight line as  1/3 .

Linear Problem
As we have already pointed out system (3) arises as the compatibility of a pair of operators; see [1,10,33,34] for genial ideas.Here we restrict our study to the class (, ) of potentials that are nonsingular and satisfy (2).Under these boundary conditions a convenient form of the Lax pair, depending on a complex spectral parameter , is given by the pair of operators: We assume the existence of an eigenfunction normalized to 1 as || → ∞.Then, (8) implies that () has the asymptotic expansion (with asymptotic coefficients  1 ,  2 , . ..): This expression permits recovering the physical amplitude || 2 .To determine the phase of the potential  consideration of the next order in the expansion is required.We find Thus, the potential is determined from

Discrete Spectrum and Relationships among Laurent Coefficients
Basic Assumption.The basic assumption of this work is the existence of a solution to ( 8) and ( 9), which has a finite or denumerable number of singularities {  ,  = 1, . . ., ∞}.We suppose that all singularities are higher-order poles and that () is analytic away from the singularities.Thus, around any pole  =  1 , () must have a local Laurent expansion: where are, respectively, the regular and singular parts of the eigenfunction at the pole and   =   (, , ), ]  = ]  (, , ) are the Laurent coefficients (LC).Notice that  Unlike what happens in the regular case, where the eigenfunction is fixed by the corresponding -problem, when singularities exist the inverse problem does not fix uniquely the singular part.Additional information relating different coefficients of the poles divisor is required.In this section we consider examples of meromorphic eigenfunctions related to nonsingular potentials and determine different relationships between coefficients of the poles divisor.We posit the existence of a linear relationship between coefficients of the Laurent expansion in the form where  is the pole multiplicity and   are certain functions to be determined.We say that the integer 1 +  is the index.

Simple Poles.
We shall first suppose that  ≡ (, , , ⋅) is an eigenfunction of ( 8), ( 9) with a meromorphic dependence on  and assume that  1 is a simple pole of .Let  1, 1 ≡  be the residue of () at  1 .Equation ( 14) reads Indeed, letting  →  1 in ( 8), (9) we obtain, at dominant orders, that  and ] ≡ ] 0 must satisfy the system of equations: 3.1.1.Simple Poles of Index One.We first consider the simplest linear relationship when Then, from the previous equations (, , ) must satisfy We can satisfy these equations by requiring  to solve the following system of constant differential equations: It follows that Therefore, eigenfunctions of (8) with the structure (16) can be constructed by requiring that  solves (17) and that the relationship holds for some complex constant .A similar relation was first established for KPI in [7].Whenever such a situation holds we say that the pole divisor is standard.We next study the possibility of having nonstandard pole divisors.

Simple Poles of Index 2.
We now show that the analytic structure (16) for eigenfunctions of (8) does not necessarily imply that (24) holds and different linear relationship between the LCs may hold; concretely we consider here the case when ] 1 , ] 0 , ] −1 ≡  are related as In this case we also need to supplement ( 17), (18) with higherorder equations: By insertion of ( 25) into (26) we obtain This equation will be satisfied if  is again given by ( 23) and  solves By integration where  and  are complex constants.Equation (30) gives yet another relationship between the three first LCs , ] 0 , and ] 1 of simple pole eigenfunctions compatible with the Lax pair.

Poles of Order
Two.We now show how the above ideas can be extended to cover the case when the eigenfunction has a pole of multiplicity 2 at some point  =  1 : that is,  = 2 in ( 14) and sing.() ≡ /( −  1 ) 2 + /( −  1 ), where, to avoid an awkward notation, we write  2, 1 ≡ ,  1, 1 ≡  and also ] 0, 1 ≡ ].By letting  →  1 in (8) we find that the main coefficients must satisfy   +   + 2 1   + 2   = 0, and also Again, the assumed analytic structure does not fix uniquely the way LCs are related.We next consider several possibilities.

Poles of Order
Two and Index Two.We first assume that the span ⟨, , ] 0 ⟩ of all linear combinations of ,  and ] 0 ≡ ] with coefficients dependent on , ,  is generated by just one of them; concretely we require Dim.⟨, , ]⟩ = 1; where , ℎ depend on , , .As previously mentioned, from the -equations we find  to be given by (23), while the remaining equations yield that ℎ must satisfy or, alternatively, ℎ  −  = 0, ℎ  = 1.It follows that ( 32) is compatible with the given analytic structure provided  is given by ( 23) and for some complex constants ,

Determination of Classes of Potentials
In this section we determine several classes solutions of (1) by considering proper election of meromorphic eigenfunctions and relationships between LCs of the form (15).

Potentials Corresponding to Simple Poles of Index One.
Suppose first that () is a meromorphic eigenfunction that has the representation ( 14) with a finite even number 2 < ∞ of simple poles.We make the following assumptions (denoted as condition C1): (1) Poles come in pairs   ,  + with corresponding residues   ,   : (2) At every pole   ,  = 1, . . ., 2 the pole divisor is standard and hence satisfies ( 24): (3) Suppose that  has just two poles  1 , − 1 .In this case from the previous assumptions and (24) we must have (45) Solving this system of linear equations and using (10) we find the potential where Δ is the determinant of the associated matrix: The field  given by (46) will be called the basic lump solution of (3).Obviously, if  1 ≡  1 +  1 ≡  + , where  ≡  1 ̸ = 0, the solution is regular on the entire plane and decreases rationally.
Consider a new Galilean frame (  ,   ) moving with velocities (, − 2 ): One finds also convenient to go to a frame of skew coordinates ,   where   , and   are defined in (48) and  ≡   −  is a shear transformation along the -axis, where  represents a slanting coefficient.In this frame the potential reads (50) Thus, in this inertial frame the solution is at rest for all time (objects moving uniformly with velocities (, − 2 ) in the unprimed frame remain at rest in the transformed primed frame) and strongly localized.However, the maxima structure is richer than what might have been expected.Inspection shows that critical points solve   =   /( 2 +  2 ),  = ±1.With  = 1 we find that critical points are (  ,   ) = (0, 0) and while for  = −1 one has These points are all candidates to maxima and minima; the number of them will vary depending on the values of the parameters.The parameter space is a two-dimensional plane deprived of the straight line  = 0. To describe the situation in the general case we restrict, with no loss of generality, to the first quadrant on the parameter space.The situation varies according to which of the regions do parameters belong.If (, ) is in the interior of  2 then two of the four points described above are maxima and the other two are minima.Point (  ,   ) = (0, 0) is a saddle point between the formers at which Particular cases are  = 0 (previously studied) and  = .In this case It has two symmetric maxima located at  = ± √ 2(1/4, /2) at which  ≡ || 2 − 1 = 1 while minima are to be found at the mirror images points at which  = −1.
We suppose that  has just two poles  1 , − 1 with the above properties.In this case one can prove that where Δ is the following definite-positive polynomial: Note how it depends on 3 complex parameters  1 ,  1 , .
Inspection of the solution shows that it corresponds to a coherent two pulsed structures.Each of the pulses has a rich internal structure with several maxima and minima in a similar disposition to that corresponding to the standard lump.
The dynamical behavior of this configuration is different and far more subtle than previously found.It turns out that lumps are not at rest in any inertial frame.
As indicated we consider a system of skew coordinates ,   defined in (48).We claim that the solutions to (61) Note that − ±∞ solves (60) if  ±∞ does; thus, the second pulse is the mirror image of the first respect to the origin, and it suffices to describe the motion of the first of them.We first consider the case  =  for which the dynamics and scattering process are easier to understand.In this case the moving frame (48) and solution are given by We then find from (60) that asymptotically the first of the pulses has coordinates  ±∞ () ≡ ( ±∞ (),  ±∞ ()), where  ± = ( √ 5 ∓ 1) 1/2 ,  ± = ±4 2 / ± , and Thus, in the moving reference frame the trajectories as  → ±∞ are straight lines with different slopes.Notice that initially ( → −∞) pulses are located in the second and fourth quadrants and will move to the first and third ones as  → ∞.For moderate times they collide head-on henceforth undergoing a scattering process.In Figures 1 and 2 we show the collision path and form of the pulses in solution (57) before and after scattering.The scattering angle Ω is easily found to be given by cos Note that cos Ω attains a maximum value 1 if (formally)  = 0 and decreases towards cos Ω = −1 as  → ∞.Thus, transparent scattering is obtained when  =  = 0 while perpendicular scattering corresponds to lumps with  2 = 1/2.
In the rest at frame the motion is the composition of a uniform motion (2, −4 3 ) and the slower one given by (63).Indeed when  =  both coordinate frames are related by a pure Galilean transformation and objects at rest in the primed frame pick a uniform motion with velocities (2, −4 3 ) in the original frame.Hence, in the stationary frame the path of the second pulse is no longer the mirror image of the first.

Potentials Corresponding to Double Poles with Index Three.
We now study potentials that correspond to eigenfunctions with double poles with index three.Concretely, assume the following: (1) Poles come in pairs   ,  + ,  = 1, . . .,  with  + = −  ≡   .Hence, (2) The multiplicity of every pole is two and the index three; namely, all pole divisors satisfy (36).
We consider the simplest such potential when  = 1.

4.3.1.
Pulses of Index Three.We consider in detail the physical properties and dynamics in the case  = 1.The potential is given by formula (46) where the tau function is Here  1 ≡  +  and ,  are complex constants, V,  are defined in (49), and The position and dynamics of the associated pulses are determined mainly by the constant  (alternatively by  ≡   +   ).Concretely, we have the following.

Proposition 1.
(1) If the constant  = 0 the solution is a multipeaked traveling wave of solitonic nature (i.e., stationary in the frame moving with the soliton).
(2) When  ̸ = 0 the solution is nonstationary.There are exactly three traveling pulses describing "homothetic motion."Namely, the motion in the frame at rest is a superposition of a "center of mass" uniform motion and an individual and slower motion proportional to  1/3 .
(3) In the "center of mass frame" pulses are located at the vertex of a triangle with center of gravity at the origin (see Figure 3 where this situation is shown).The time evolution contracts but does not deform the whole structure.As a result, pulses collapse onto the origin whereupon the configuration regains shape.No deflection angle appears.
To prove the claims we note that the solution reads neater with a skew transformation and a dilatation of coordinates defined by (48) and  ≡   −   ,  =   .Then dropping some irrelevant constants we have (see (42)) We simply consider here the case when  = 0, namely, when the parameters are chosen to satisfy V 2 (1 +  1 ) =  4  1 .
The -dependence of Δ drops out and hence the entire configuration (, , ) moves, with respect to the frame at rest, with constant velocity given by (V, −V 2 ).A plot of  shows that || 2 − 1 decreases to zero away from three regions.Thus, the configuration behaves like a multipeaked solution of solitonic nature.
Proof.The proposition is trivial if the number of pole singularities is finite so we shall assume an infinite number.Note then that (1) implies that they cannot accumulate anywhere on the finite plane.Thus, with no loss of generality we can assume that singularities pile up at infinity but not too quickly.

𝑘 1 )
sing.() is also termed the singularity principal part or the pole divisor.

Figure 1 :
Figure 1: The path of the two pulses for the cases incoming ( = −∞) and outgoing ( = ∞) corresponding to  = 0.3 with the scattering angle indicated.

Figure 2 :
Figure 2: Location and form of the two pulses: incoming ( = −10) and outgoing ( = 10) corresponding to the election of  =  = 1.The scattering process is clear.