Soliton-generating τ-functions revisited

Within the framework of the Inverse-Scattering formalism and the Hirota algorithm, soliton solutions of evolution equations are images of {\tau}-functions. Typically, the latter are expressed in terms of exponentials, the arguments of which are linear in the coordinates. Consequently, often, {\tau}-functions are unbounded in space and time. However, they are not unique. Exploitation of their non-uniqueness uncovers physically interesting possibilities: 1) One can construct equivalent {\tau}-functions, which generate the same traditional (Inverse-Scattering/Hirota)) soliton solutions, yet allow for the extension of the family of soliton solutions to a wider, parametric family, in which the traditional solutions are a subset. The parameters are shifts in individual soliton trajectories. 2) When two wave numbers in a multi-soliton solution are made to coincide, the reduction of the solution to one with a lower number of solitons is qualitatively different for solutions that are within the traditional subset and those that are outside this subset. 3) One can construct {\tau}-functions that are bounded in space and time, in terms of which soliton solutions become images of localized sources.


I. INTRODUCTION
Many well-known nonlinear evolution equations provide approximate descriptions of phenomena in physical systems. For example, the Korteweg-deVries (KdV) equation describes the propagation of waves in (1 + 1) dimensions on the surface of a shallow water layer, 1,2 along a Fermi-Pasta-Ulam chain, 3 and of ion acoustic waves in plasma physics; 4,5 the Kadomtsev-Petviashvili II (KP II) equation describes the propagation of waves in (1 + 2) dimensions on the surface of a shallow water layer. 6 In the inverse-scattering [7][8][9][10][11][12][13][14][15] and Hirota approaches, [16][17][18][19][20][21] soliton solutions of integrable evolution equations are transforms of τ-functions. Typically, the latter are expressed in terms of exponentials, the arguments of which are linear in the coordinates. Often, the τ-functions are extended and unbounded in space and time, whereas their images, the soliton solutions, are bounded and localized structures.
The goal of this paper is to obtain a representation of soliton solutions as images of bounded and spatially localized sources. This is attained though exploitation of the non-uniqueness of the τ-functions in order to construct τ-functions, which are equivalent to the ones traditionally used in the inverse-scattering/Hirota formulation. Namely, they generate the same soliton solutions. Using The non-uniqueness of τ-functions has been exploited in, probably, tens if not hundreds of papers. In the case of the KdV equation, the resulting equivalent τ-function arrived at in this paper was first presented and exploited in the study of the tropical limit of the KdV equation. 22 The structure of the equivalent τ-functions has another desirable attribute: It elucidates in a simple manner the role of the arbitrary shifts in the positions of soliton trajectories in space and time in controlling the limit of multi-soliton solutions when two wave numbers are made to coincide. Specifically, whether an N-soliton solution is reduced to a solution with (N 1) or (N 2) solitons.
In the traditional forms of τ-functions, this is not as transparent. Examples are discussed in the cases of the KdV and KP II equations.
The case of the modified KdV equation is particularly interesting. It is well known that the traditional form of the τ-function forces the two-soliton solution to be reduced to a single-soliton solution when the two wave numbers, from which the solution is constructed, are made to coincide. However, exploiting an equivalent τ-function, it is shown that the solution is reduced to a δ-function along a line in the x-t plane for a wide range of choices of the arbitrary shifts in soliton trajectories. Furthermore, depending on the properties of these shifts, a three-soliton solution may be reduced to a two-or a single-soliton solution.
The general case of the KdV equation is discussed in the Appendix and Secs. II A and II B. Examples are discussed in Secs. II C-II F. The case of the KP II equation is discussed in Sec. III. The case of the modified KdV equation is discussed in Sec. IV.
Throughout the paper, any quantity associated with the usual formulation within the framework of the inverse-scattering/Hirota formalism will be called a "traditional" quantity.

A. Motivation
The soliton solutions of the KdV equation, are constructed in terms of a τ-function through 10-17 The traditional form of τ(t,x) for an N-soliton solution is 10-17 Here and in the following, the subscript T denotes the traditional (inverse-scattering/Hirota) form.
In view of Eqs. (2) and (3), it is clear that the soliton solutions, which are bounded and spatially localized, are images of unbounded and spatially extended structures.
In addition, the role of the arbitrary shifts, δ i , in determining the properties of the solution, is not transparent. For specificity, let us discuss the two-soliton solution, for which Eq. (3) becomes Consider the case of constant shifts, δ 1 and δ 2 . Trivially, in the limit when the wave numbers, k 1 and k 2 , coincide, the two-soliton solution tends to a single-soliton one. However, this is not the only possibility. As δ 1 , and δ 2 are arbitrary, assign them the following wave number dependence: Equation (6) then becomes For any value of γ, the resulting solution tends to zero when k 2 → k 1 . While reaching this conclusion in the two-soliton case is relatively simple, deciphering the roles of the arbitrary shifts in the behavior of solutions with N > 2 solitons becomes far less transparent.
In the remainder of this paper, it will be shown that both issues discussed above can be taken care of through the non-uniqueness of τ(t,x): Multiplying τ(t,x) by any term of the form e g(t) x + f (t) yields an equivalent τ-function that generates the same solutions. An equivalent τ-function will be presented, in terms of which one defines In Eq. (9), the subscript E stands for "equivalent." S(t,x) will be shown to be bounded and localized in the x-t plane so that the solution of Eq. (1), expressed as is now an image of a localized source. The same choice of equivalent τ-function also elucidates the role of the free shifts in soliton solutions in the changes in multi-soliton solutions when wave numbers are made to coincide.

Search for a localized source
In order to express u(t,x) in terms of a bounded and localized source as in Eqs. (9) and (10), one needs to find an appropriate equivalent τ-function. To this end, consider an equivalent τ-function τ E (t,x) is a sum of 2 N terms, each a product of N exponentials. For S(t,x) of Eq. (9) to be bounded and localized in the x-t plane, one must ensure that τ E (t,x) is unbounded asymptotically in all directions in the plane. Namely, in any direction in the plane, some exponential terms in τ E (t,x) must become unbounded asymptotically. This can be achieved by choosing the coefficients, µ i , so that the sum of the exponents of all the 2 N terms in Eq. (11) vanishes. Then, if in a domain in the plane some exponential terms in τ E (t,x) vanish asymptotically, others become exponentially unbounded.
The sum of all the exponents is For this sum to vanish, the independence of θ i requires The resulting equivalent τ-function is (see the Appendix for the details of the transformation from Under Eqs. (9), (10), and (14), the solution is an image of a bounded, spatially localized source. The source for a single-soliton solution is a soliton (see Sec. II C). The source for multi-soliton solutions is a hump, concentrated around the soliton interaction region (see Secs. II D-II F). Note that δ i , the constant shifts in the traditional expression of Eq. (4), are replaced by wave number dependent shifts, ∆ i k . The traditional values of ∆ i k are determined by the transformation from τ T to τ E . Examples are presented in Secs. II D-II F.

Peculiar properties of N-soliton solutions
An N-soliton solution, constructed from τ T of Eq. (3), is reduced to an (N 1)-soliton solution when two wave numbers are made to coincide, provided all shifts, δ i , are constant.
If δ i are wave-number dependent, the limit of the solution may be different. This characteristic is buried in the traditional form, Eq. (3), but is not transparent. Using the equivalent τ-function, τ E of Eq. (14), the role played by ∆ i k , the shifts in soliton trajectories, is elucidated directly.
The singular wave-number dependence of the traditional ∆ i k (obtained in the transformation from τ T to τ E ; see the Appendix and Secs. II C-II F) leads to the reduction of an N-soliton solution to an (N 1)-soliton one. This can be seen directly from inspecting Eq. (14) with the aid of the Appendix. Examples of the two-, three-, and four-solitons are provided in Secs. II D-II F.
If ∆ i k lack this singular wave number dependence, then when k 2 → k 1 , the number of terms in τ E of Eq. (14) is reduced from 2 N1 to 2 N3 , corresponding to an (N 2)-soliton solution!
In Secs. II D-II F, it is shown that, when the ∆ i k are independent of the wave numbers, the limits of the two-, three-, and four-soliton solutions are, respectively, zero, one-, and two-soliton solutions.
With N ≥ 4 solitons, there are more possibilities. If some ∆ i k are constant and others have the singular structure of the traditional case, then when some wave number pairs coincide, the solution is reduced to (N 2) solitons, while for other pairs, it is reduced to (N 1) solitons.

C. Single-soliton solution
This trivial case is discussed only so as to show the emerging pattern. The traditional τ-function, where δ is a constant shift, generates the single-soliton solution Multiplying the expression in Eq. (15) by e −(kx+4k 3 t+δ) /2 yields an equivalent τ-function, which generates the same single-soliton solution through Eq. (2), Now define the source function of Eq. (9)

Limit of k 2 → k 1
The structure of the traditional τ T of Eq. (17) forces the solution to be reduced to a single-soliton in the limit when δ i are constants. Constructing the solution through τ E of Eqs. (23), the limit depends on the value of α. If α has the traditional singular wave number dependence of Eq. (22), then the limit of one soliton is attained. If α does not have this singular nature, the limit vanishes,

E. Three-soliton solution
The traditional form of the three-soliton τ-function (wave numbers (26)

Soliton trajectory shifts as solution parameters
It does not take much to show that α σ 1 σ 2 σ 3 obey the constraint Thus, only three of α σ 1 σ 2 σ 3 are linearly independent. This constraint is identical in shape to the constraint obeyed by the four θ σ 1 σ 2 σ 3 , Equation (32) is a trivial consequence of Eq. (29). This suggests a similar decomposition for α σ 1 σ 2 σ 3 , Equation (28) can be re-written in a form that exhibits the role of the shifts in soliton trajectories in solution properties, withθ

Localized source
With τ E of Eq. (34), the source, S(t,x), defined by Eq. (9), is bounded and localized in the x-t plane. Hence, the three-soliton solution becomes the image of a localized source. Figure 3 shows a three-soliton solution, and Fig. 4 shows its source, S(t,x).

Limit of k 2 → k 1
When k 2 → k 1 , Eq. (26) becomes a traditional two-soliton τ-function if the δ i are constants; the three-soliton solution is reduced to a two-soliton one. With τ E of Eq. (28) or (34), the role of the shifts of soliton trajectories in the x-t plane becomes clear. Using τ E of Eq. (28), the limit depends on α σ 1 σ 2 σ 3 . In the traditional case, the two-soliton limit is obtained owing to the singular nature of α σ 1 σ 2 σ 3 of Eq. (30). However, when α σ 1 σ 2 σ 3 do not obey Eq. (30), the limit is different, J. Math. Phys. 59, 122701 (2018) Equation (37) is a τ-function that generates a single KdV-soliton solution, with wave number k 3 .

F. Four-soliton solution
The traditional τ-function for four-soliton solution (wave numbers (38)

Equivalent τ-function
Following the procedure delineated in the Appendix, one obtains an equivalent τ-function , the source of the three-KdV-soliton solution. The parameters are the same as in Fig. 3. J. Math. Phys. 59, 122701 (2018)

Soliton trajectory shifts as solution parameters
Following the procedure delineated in the Appendix, the structure of τ T of Eq. (38) dictates the expressions for α σ in the traditional case. Furthermore, α σ obey the following constraints: As in the case of the three-soliton solution, the constraints of Eq. (41) are identical in shape to four constraints obeyed by θ σ . The latter are a trivial consequence of Eq. (40). This allows, again, for the construction of the eight linearly dependent shifts in terms of four independent shifts and for re-writing of Eq. (39) in the form of Eq. (14). Again, the parameters, on which the four-soliton solution depends, have been formulated as shifts of soliton trajectories in the x-t plane.

Localized source
Using Eq. (39) in the definition, Eq. (9), of S(t,x), the four-soliton solution becomes the image of a source, a hump that is localized in the x-t plane.

Limit of coinciding wave numbers
The structure of τ T of Eq. (38) ensures that when k 2 → k 1 , if the δ i are constants, the solution is reduced to a three-soliton solution (wave numbers k 1 , k 3 , and k 4 ). If one next considers the limit of k 4 → k 3 , then the three-soliton solution is reduced to a two-soliton solution (wave numbers k 1 and k 3 ). If the solution is constructed from τ E of Eq. (39), the limit depends on α σ . If the latter assume the singular wave-number dependent values dictated by the structure of τ T , then the above-mentioned limits are reached. If, however, α σ do not have the singular behavior, the limit may be different.
If all α σ are constants, the limit, k 2 → k 1 , of the solution is a two-soliton solution (wave numbers k 3 and k 4 ). Imposing, in addition, k 4 → k 3 , this two-soliton solution is reduced to zero.

III. THE KADOMTSEV-PETVIASHVILI II EQUATION
The line-soliton solutions of the Kadomtsev-Petviashvili II (KP II) equation, are constructed as follows: 20,21 The traditional τ-function is given by In Eqs. (45) and (46)

A. Generating a localized source
To generate through Eq. (9) a bounded source, S(t,x,y), that is localized in the (1 + 2)-dimensional space, the equivalent τ-function, τ E (t,x,y), must become unbounded asymptotically in any direction in space. To achieve this, let us replace τ T of Eq. (44) by τ E of Eq. (49) is a sum of exponentials. The generic form of the exponential terms in τ E is For S(t,x,y) to be bounded, at least some of the exponential terms must become unbounded asymptotically in any direction in the (1 + 2)-dimensional space. This can be achieved by requiring that the sum of the exponents in all the terms in τ E vanishes. Then, if in any direction, some exponential terms in τ E (t,x,y) vanish asymptotically, others become exponentially unbounded. In that sum, each θ i (t,x,y) is multiplied by The first term in Eq. (52) counts the number of times each θ i (t,x,y) appears with a positive sign, and the second term counts the number of times it appears with a negative sign. As all the θ i (t,x,y) are independent, the vanishing of the sum requires that the coefficient of each θ i (t,x,y) vanishes,
Using τ E in Eq.
Equation (60) Figures 5 and 6 show, respectively, a three-soliton solution and its localized source. For constant ξ M (i), this solution tends to a single soliton solution if two of the wave numbers are made to coincide. For example, in the limit of k 2 → k 1 , it tends to a single soliton constructed out of k 1 and k 3 . To make the limit vanish, one must make two ξ M (i) vanish in that limit, making it an uninteresting option.

C. Four wave numbers: (4,2) four-soliton solution
In (M,N) solutions with N > 1, the numerical coefficients depend on the wave numbers [see Eq. (45)], allowing for a dependence of the characteristics of the solutions on the wave numbers. An   FIG. 6. S(t,x,y) [Eq. (9) In Eq. (62), (t,x,y) have been omitted from θ i for the sake of brevity. The ξ's obey Eq. (49).
(65) (The notation is as in the case of the KdV equation.)

Limit of k 2 → k 1
Consider now the limit when two wave numbers coincide, say, k 2 → k 1 . From τ T of Eq. (62), one deduces that if the shifts, ξ i , are independent of the wave numbers, then the (4,2) solution is reduced to a (3,2) solution, which is a three-soliton solution (Y -shaped) with wave numbers k 1 , k 3 , and k 4 . Constructing the solution through τ E of Eq. (63), this limit is reached if α σ are assigned the required singular expressions of Eq. (65). If they have other values, then the k 2 → k 1 limit may be different. For constant α σ , τ E of Eq. (63) tends to which generates a single-soliton solution with wave numbers k 3 and k 4 ! FIG. 10. S(t,x,y) [Eq. (9)], the source of the (4,2)-KP II-soliton solution. The parameters are the same as in Fig. 9.

Localized source
Using Eq. (62) in Eq. (9) yields a source, S(t,x,y), which is bounded in the (1 + 2)-dimensional space and is localized in the x-y plane at any time. The (4,2) solution is the image of this localized source under Eq. (3) . Figures 9 and 10 present a (4,2) solution and its source, S(t,x,y), respectively.

IV. THE MODIFIED Kdv EQUATION
The soliton solutions of the modified KdV (mKdV) equation, are constructed through a transformation of a different structure 18 Owing to the fact that the connection between the solution and the τ-function is not through a logarithmic transformation, a simple procedure of the type described in Secs. II and III has not been found. However, the Miura transformation connecting the solutions of the KdV and mKdV equations 24 ensures that equivalent τ-functions can be found. Rather than embarking upon a full analysis, the case of the two-and three-soliton solutions is presented as examples.
The traditional two-soliton τ-function is given by τ T of Eq. (69) is unbounded in the vicinity of a line in the x-t plane. This singular behavior is of no concern, as it is remedied by the transformation in Eq. (68). An equivalent τ-function is obtained by multiplying the top and the bottom of Eq. (69) by (k 1 + k 2 ) 2 e − 1 2 (θ 1 +θ 2 ) .
The result leads to The structure of τ T of Eq. (69) dictates the wave-number dependence of α to be As evident from Eq. (69), in the traditional case, the two soliton solution is reduced to a singlesoliton solution when k 2 → k 1 , provided the shifts, δ i , are constant. Using Eq. (71), this limit is a consequence of the singular nature of the traditional value of α, given by Eq. (72). The limit is different if α does not have that singular structure. For example, for constant α, the leading singular term in Eq. (71) is where the final sign depends on which side of the line is. Hence, in the limit, the arctanh in Eq. (67) jumps between π/2 and +π/2. Consequently, the limit of the two-soliton solution is a zero-width single soliton, J. Math. Phys. 59, 122701 (2018) The traditional three-soliton τ-function is given by In the limit of k 2 → k 1 , Eq. (76) yields a τ-function for a two-soliton solution with wave numbers k 3 and k 1 . To see other possibilities, it pays to consider the following equivalent τ-function: The traditional values for the shifts are If the shifts obtain these traditional values, then their singular behavior ensures that the solution tends to a two-soliton solution with wave numbers k 1 and k 3 in the limit of k 2 → k 1 . If, however, the shifts do not have the singular nature dictated by Eqs. (78)-(80), then the solution tends to a single soliton with wave number k 3 !

V. CONCLUDING COMMENTS
In this paper, it has been shown that the well-known non-uniqueness of the τ-functions, from which soliton solutions of nonlinear evolution equations are constructed, allows for the association of soliton solutions with bounded and spatially localized sources. This has been achieved through the construction of τ-functions that are equivalent to the form used traditionally. The physics gain is the fact that all the information regarding the structure if a multi-soliton solution is contained in a source that is localized in the soliton interaction region.
A byproduct of this study is the observation that the equivalent τ-functions allow for a simple exposition of the role of shifts in the positions of soliton trajectories in determining solution characteristics. In particular, the limit into which an N-soliton solution is reduced when two wave numbers are made to coincide has been elucidated. In the case of the KdV equation, this allows for seeing in a simple manner when an N-soliton solution degenerates into a traditional (N 1)-soliton solution and when into an (N 2)-soliton solution. In the case of the KP II equation, the discussion of the (4,2) four-soliton solution shows when it degenerates into a three-soliton solution and when to a single soliton. In the case of the mKdV equation, depending on the shift, a two-soliton solution is reduced to a single-soliton solution or may degenerate into a δ-function.
Clearly, the analysis presented here can be applied to other KdV-like equations, such as the bi-directional KdV equation 25 and the Sawada-Kotera 26 equation.

APPENDIX: CONSTRUCTION OF EQUIVALENT τ-FUNCTION FOR N-SOLITON SOLUTION OF KdV EQUATION
One first multiplies the traditional τ-function, The 2 N exponential terms in the result, are split into 2 N1 pairs of terms. The simplest pair is the one in which all θ i have the same signs. It is obtained from the sum of the following two terms in τ T : When these are multiplied by the factor of Eq. (A2), the sum of the two terms becomes This term can be re-written as where sinh α ++···+ The pair of the next level of complication is that in which one of the θ i has a negative sign. Take as an example the case that this is θ N . It is obtained from the sum of the following two terms in τ T : When these are multiplied by the factor of Eq. (A2), the sum of the two terms becomes This term can be re-written as where sinh α + + · · · + N−1times The next level of complication is of pairs, in which two of the θ i have a negative sign. Take as an example the case that these are θ N and θ N1 . It is obtained from the sum of the following two terms in τ T : When these are multiplied by the factor of Eq. (A2), their sum becomes This term can be re-written as In each term, the multiplicative wave number dependent coefficients appear only for indices i and j, for which This pattern recurs in all other terms. Hence, the general term in τ E has the form N j=1 N l=j+1 σ j =σ i k j − k l k j + k l cos(θ σ + α σ ), where σ = {σ 1 , σ 2 , . . . , σ N }, σ 1 = +1, σ i ≥2 = ±1 .
Next, note that the same denominator appears in the definitions of all α σ . Thus, in the traditional (inverse-scattering/Hirota) construction, the shifts in the position of soliton trajectories in the x-t plane all have a singular dependence on the wave numbers whenever any two of them coincide.
Finally, in the traditional construction, the exponents, θ i , may include arbitrary shifts, δ i , The cumulative contribution of these shifts in any term, In τ E , the 2 N1 α σ constitute additional shifts in the locations of soliton trajectories in the x-t plane. Hence, they should be in a similar form This ensures translation invariance along the trajectory of each soliton, once sufficiently far away from all other solitons. This allows for re-writing the equivalent τ-function, τ E , in the form The shifts, ∆ i k , may contain the wave number dependent contributions required in the traditional case but, as discussed in the main body of the paper, may assume any values.