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is the object [MATH] , with comonoid structure given via the distributive law [MATH] , and monoid structure via the distributive law [MATH] Using a thick line for [MATH] , thin line for [MATH] , we depict the comultiplication and the multiplication of [MATH] by the following string diagrams |
[EQUATION] where all relevant arrows in [MATH] are uniquely determined by their source and target, so there is no need for labelling. |
Proposition 4.2 The semidirect product [MATH] is a bimonoid in [MATH] . If [MATH] and [MATH] are Hopf, with antipodes graphically represented by dots, then so is [MATH] , with the antipode given by diagram ( 18 ). |
[EQUATION] Proof. The defined (co)multiplication is already part of a (co)monoid structure. The compatibility of counit with unit, counit with multiplication and unit with comultiplication follows directly. What remains to show is the bimonoid axiom, which we have done using the manipulation of string diagrams shown in... |
In line ( 19 ), after rearrangement we used the compatibility of comultiplication of [MATH] with coaction of [MATH] on [MATH] , in the bottom right corner of the middle diagram. |
Going from line ( 19 ) to line ( 20 ) we used the bimonoid axiom for [MATH] on the top-left part of the diagram, followed by the (co)associativity for [MATH] . In the line ( 20 ) we used the braiding coelement axiom ( 13 ). When passing from line ( 20 ) to line ( 21 ) we used the (co)associativity for [MATH] , together... |
Passing from line ( 22 ) to line ( 23 ) uses the compatibility of comultiplication of [MATH] with coaction of [MATH] on [MATH] at three different places. In line ( 23 ) we used that the (co)multiplication of [MATH] is an [MATH] -comodule morphism. Finally, going from line ( 23 ) to line ( 24 ) uses the bimonoid axiom f... |
That ( 18 ) is indeed an antipode follows in a similar way. The strategy to show the “right inverse” axiom is to use the compatibility of [MATH] with [MATH] , and bimonoid axioms to get all multiplications to the top, and comultiplications to the bottom of the diagram, and then use the right inverse axiom for [MATH] mu... |
Proposition 4.3 The comparison functor [EQUATION] is strict monoidal and has a strict monoidal inverse [EQUATION] Proof. Using the dual of Beck’s monadicity theorem, we show that the forgetful functor |
[EQUATION] is comonadic. Since [MATH] is the composite of the two comonadic functors [MATH] and [MATH] , it has a right adjoint and reflects isomorphisms. The third criterion, not necessarily preserved by composition, is the existence and preservation of [MATH] -split equalizers. So, assume the parallel pair |
[EQUATION] in [MATH] has a split equalizer [MATH] in [MATH] . That is, there are maps [EQUATION] satisfying [EQUATION] Comonadicity of [MATH] implies that [MATH] is an [MATH] -comodule, with coaction |
[EQUATION] and that [MATH] is an equalizer in [MATH] , but not necessarily split. The proof involves the following identities (expressing the fact that [MATH] is an [MATH] -comodule morphism): |
[EQUATION] where the first equality follows from ( 32 ), the second from [MATH] and the fact that [MATH] and [MATH] are [MATH] -comodule morphisms, and the third comes from ( 31 ). Exactly the same equalities hold with [MATH] replaced by [MATH] , for the same reasons: |
[EQUATION] Now, the map [EQUATION] is an [MATH] -comodule morphism [EQUATION] compatible with counit and comultiplication on [MATH] , which follows from the compatibility of [MATH] with counit and comultiplication and equations ( 30 ) and ( 39 ). Therefore [MATH] is an object of [MATH] |
The arrow [MATH] is an [MATH] -comodule morphism, which follows directly from ( 39 ). To show that it equalizes [MATH] and [MATH] , take [MATH] to be an [MATH] -comodule and [MATH] an [MATH] -comodule morphism satisfying [MATH] In [MATH] [MATH] is the unique comparison map, since [MATH] is the equalizer of [MATH] and [... |
[EQUATION] completing the proof that [MATH] is comonadic. The comparison functor [MATH] is strict monoidal: the coaction for [EQUATION] |
is depicted as follows [EQUATION] while the coaction for [EQUATION] is depicted by [EQUATION] These are equal since [MATH] is compatible with comultiplication on [MATH] |
Example 4.1 When [MATH] , the multiplication of [MATH] is forced to be the diagonal map, the comultiplication gives [MATH] a monoid structure with identity denoted by [MATH] and the only possible cobraiding element is [MATH] An [MATH] -comodule bimonoid [MATH] is the same as a monoid morphism [MATH] , and [MATH] is pre... |
Birings In this section we consider two particular types of bimonoids in a braided monoidal additive category. The additivity condition is about existence of direct sums which, as absolute colimits, are preserved by all Ab -enriched functors, in particular tensoring. The naturality of the braiding implies it is compati... |
[EQUATION] which can be concisely written by specifying non-zero components [EQUATION] where concatenation is the tensor product and vertical empty space is the direct sum. From here we directly get the following lemma. |
Lemma 5.1 If an object [MATH] has as a symmetry morphism [EQUATION] then any decompositions of [MATH] into a sum [EQUATION] forces |
[EQUATION] since the components of [MATH] are isomorphisms this means that for [MATH] [EQUATION] 5.1 The grading Hopf ring Let [MATH] be a category which is, in addition, symmetric and has countable coproducts preserved by tensoring. Denote by [MATH] the copower of the object [MATH] by the set of integers [MATH] . In p... |
[EQUATION] The addition of integers gives [MATH] a group (Hopf monoid) structure in [MATH] , and induces a Hopf ring structure on [MATH] , given by |
[EQUATION] Tensoring with [MATH] gives a functor isomorphic to taking a copower by [MATH] [EQUATION] Since [MATH] is symmetric monoidal, the category |
[MATH] of [MATH] -comodules is monoidal, and becomes braided on using the braiding coelement [MATH] depicted by [EQUATION] Arrows [MATH] denote coproduct coprojections, and [MATH] satisfies the coelement axioms ( 13 )-( 15 ): |
[EQUATION] [EQUATION] [EQUATION] [MATH] inherits direct sums: if [MATH] and [MATH] are [MATH] -comodules, then [EQUATION] is a [MATH] -comodule as well, and the braiding induced from the braiding coelement [MATH] is compatible with direct sums. |
Example 5.1 When [MATH] , the biring [MATH] is the Laurent polynomial ring with integer coefficients. The coring structure is given by [MATH] . Then |
[EQUATION] is an equivalence of categories. Consider a [MATH] -comodule [EQUATION] [MATH] being a group homomorphism ensures that |
[EQUATION] which enable us to define abelian subgroups [EQUATION] while the compatibility with counit and comultiplication gives |
[EQUATION] which ensure that [EQUATION] Here [MATH] denotes the Kronecker delta. The braiding coelement ( 50 ) corresponds to the group homomorphism |
[EQUATION] and gives a braiding (symmetry, in fact) in GAb. 5.2 The differential Hopf ring Let [MATH] be a braided monoidal additive category. |
Proposition 5.1 An object [MATH] with braiding [MATH] induces a Hopf ring [MATH] , whose monoid structure [MATH] has non-zero components |
[EQUATION] the comonoid structure [MATH] has inverses of ( 54 ) as non-zero components, and the antipode is [EQUATION] Proof. The (co)associativity and (co)unit axioms follow from coherence for monoidal categories, after noting that a component is non-zero if and only if it contains either one [MATH] in its source and ... |
The compatibility of unit with counit and comultiplication is obvious. The bimonoid axiom [EQUATION] imposes that [EQUATION] equals |
[EQUATION] which follows from [MATH] , braiding coherences [MATH] [MATH] and [MATH] , and coherences for unit and associator. Finally, the Hopf axioms hold, for example the left inverse part gives |
[EQUATION] equals [EQUATION] 5.2.1 When [MATH] is a category of comodules Let [MATH] be a symmetric monoidal additive category, and [MATH] a biring there, with a braiding coelement [MATH] . Take [MATH] . Now [MATH] as an [MATH] -comodule is an object of [MATH] , together with a coaction [MATH] satisfying |
[EQUATION] where on the left we have the braiding in [MATH] and [MATH] is the symmetry in [MATH] From Proposition 5.1 , we have that [MATH] with the coaction |
[EQUATION] is a Hopf ring in [MATH] . Hence, by Proposition 4.2 , there is a semidirect product [MATH] , with the Hopf ring structure in [MATH] having components |
[EQUATION] [EQUATION] 5.2.2 When [MATH] Take [MATH] , and [MATH] . An [MATH] -comodule [MATH] can be thought of as a graded abelian group (see Example 5.1 ) with grades [MATH] . Condition ( 60 ) gives that for all [MATH] and [MATH] and [MATH] |
[EQUATION] The left hand side of the equality is an element of [MATH] component in the sum defining [MATH] , while the right hand side is an element of [MATH] . An argument similar to the one for Lemma 5.1 forces |
[EQUATION] Now, Lemma 5.1 further constrains the decomposition of individual [MATH] . We will consider those groups involved in decomposition of either finitely generated groups, namely [MATH] and [MATH] , or divisible groups |
, namely [MATH] and Prüfer groups [MATH] , where [MATH] is the [MATH] prime, and [MATH] . The tensor “multiplication table”, up to isomorphism, for these groups ( [MATH] is omitted) is given by |
[EQUATION] Each of these components has [MATH] which can be shown using the following pattern [EQUATION] and the fact that a pair [MATH] of elements determines [MATH] such that [MATH] and [MATH] for some integers [MATH] and [MATH] . In addition, when [MATH] we have [MATH] All this means that we can have: |
either only one copy of [MATH] in one of the odd degrees or no copies of [MATH] , at most one copy of [MATH] for each [MATH] and fixed [MATH] in odd degrees (except for [MATH] which can appear in an even degree) and arbitrary many copies of Prüfer groups, at arbitrary degrees. |
In the torsion-free-non-divisible part, we could ask for the following sufficient condition, slightly generalising the argument followed in ( 68 ). |
Lemma 5.2 If each two elements [MATH] and [MATH] of an abelian group [MATH] determine a set of elements [MATH] such that [MATH] for [MATH] and both [MATH] and [MATH] can be expressed as finite linear combination of elements from [MATH] , then [MATH] |
5.2.3 The Pareigis example Let [MATH] denote the degree of the differential, [MATH] , and denote its generator by [MATH] . By the argument above, [MATH] The biring [MATH] has underlying abelian group [MATH] . The (co)unit and (co)multiplication are determined using ( 62 ) and ( 63 ): |
[EQUATION] [EQUATION] [EQUATION] [EQUATION] To see what the antipode is, consider the general antipode diagram ( 18 ), and label the edges |
[EQUATION] where either [MATH] and [MATH] , or [MATH] . So we have [EQUATION] We get exactly the Pareigis Hopf ring [MATH] ) by setting |
[EQUATION] On the other hand, by setting [EQUATION] we get a modified Pareigis ring, [EQUATION] which corresponds to exchanging [MATH] and [MATH] , and gives, as comodules, chains like ( ). |
5.2.4 When [MATH] We can iterate the process by taking [MATH] , noting that the graded abelian group [MATH] , with [MATH] has a unique differential structure, and that [MATH] to obtain a second differential [MATH] on the chain [MATH] , satisfying |
[EQUATION] 5.2.5 When [MATH] Now we consider the grading monoid in [MATH] . Note that [MATH] for a particular abelian category [MATH] . Using the universal property of EM-objects in Ab-Cat we can conclude that [MATH] comodules in DGAb are equivalently chains in GAb. So comodules [MATH] turn each component [MATH] of the... |
Let [MATH] parametrise the two different coelements [MATH] can have: substitute [MATH] in ( 50 ) with [MATH] . Then, the tensor product |
[EQUATION] has braiding [EQUATION] For a differential [MATH] we can choose a graded chain with [MATH] and [MATH] for the other components (when [MATH] [MATH] gives two different directions, as in the previous example). Note that [MATH] in all cases. So we can consider [MATH] comodules to obtain a second differential: |
for [MATH] we get chains in DGAb, aka double complexes, with the second differential [MATH] satisfying [EQUATION] for [MATH] we get a second differential [MATH] satisfying [EQUATION] |
# Source: arxiv 1806.09359 # Title: On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation # Sections: all # Downloaded: 2026-03-02T08:55:55.553928+00:00 |
On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation |
Abstract We construct symmetry preserving and symmetry broken N-bright, dark and antidark soliton solutions of a nonlocal nonlinear Schrödinger equation. To obtain these solutions, we use appropriate eigenfunctions in Darboux transformation (DT) method. We present explicit one and two bright soliton solutions and show ... |
Introduction About five years ago, Ablowitz and Musslimani have proposed the following nonlocal nonlinear Schrödinger (NNLS) equation |
[EQUATION] where [MATH] is a slowly varying pulse envelope of the field, [MATH] and [MATH] represent space and time variables respectively and * denotes complex conjugation. The NNLS equation ( ) is invariant under the parity-time (PT) transformation. PT symmetric systems, which allow lossless propagation due to their ... |
. Equation ( ) attracted many researchers to study its physical and mathematical aspects intensively, see for example Refs. . The integrability of ( ) is proved by (i) the existence of a Lax pair, (ii) existence of infinite number of conservation laws and (iii) existence of N-soliton solutions |
. The initial value problem was studied by Ablowitz et al. . Breathers, dark, antidark soliton, algebraic soliton, higher order rational solutions, periodic and hyperbolic solutions of ( ) have been derived for this equation in Refs. |
. Discrete version of Eq. ( ) has also been proposed in . Recently, Stalin and two of the present authors have constructed more general bright soliton solutions for ( ) by developing a nonstandard bilinearization procedure |
. In this procedure, besides Eq. ( ) the authors have also considered the parity transformed complex conjugate equation of ( ), namely |
[EQUATION] since they have assumed [MATH] and [MATH] evolve independently. Since Eq. ( ) is nonlocal, to evaluate the dependent variable [MATH] at [MATH] , the other variable [MATH] has to be evaluated at [MATH] simultaneously. The authors have obtained more general one and two soliton solutions of Eqs. ( ) and ( ) by ... |
As far as the NNLS Eq. ( ) is concerned the symmetry broken and symmetry preserving solutions have been analyzed only for the bright soliton case. A natural question arises in this context is what happens to the dark soliton case. These soliton solutions for Eq. ( ) have already been reported in the literature |
. However, as we pointed out above, to bring out a more general dynamical evolution of dark soliton one should consider not only Eq. ( ) but also Eq. ( ) in the solution process. In this work, we intend to consider both the equations and construct a more general class of dark soliton solution. |
As a by-product of this work, we also extend Darboux transformation (DT) method suitable for this class of nonlocal equations. To make our studies a complete one, to begin with, we derive the bright soliton solution using the DT method by considering the nonlocal term [MATH] as a separate quantity. We then move on to c... |
By carrying out relevant asymptotic analysis of the two soliton solution we classify the parametric regions of dark and antidark solitons in both the components [MATH] and [MATH] . We then derive the four soliton solution from the second iteration of the DT method. Since the solution is cumbersome we only give plots of... |
The plan of the paper as follows. In Sec. II, we present the DT method to construct Nth iterated solution formula for obtaining N-bright, dark and antidark soliton solutions of Eqs. ( ) and ( ). We present explicit one and two bright soliton solutions and study the collision dynamics between two solitons in Sec. III. I... |
Darboux Transformation of NNLS equation In this section we recall the essential ingredients of the Darboux method to construct the desired solutions. The Lax pair of Eqs. ( ) and ( ) is given by, |
[EQUATION] where the block matrices [MATH] [MATH] [MATH] and [MATH] are given by [EQUATION] In the above [MATH] [MATH] [MATH] [MATH] [MATH] and [MATH] is isospectral parameter. The compatibility condition [MATH] leads to Eqs. ( ) and ( ), where the square bracket denotes the usual commutator. |
2.1 First Iteration of DT A Darboux transformation (DT) is a special gauge transformation [EQUATION] where [MATH] and [MATH] are old and new eigenfunctions of ( ), [MATH] is the DT matrix and [MATH] is a non-singular [MATH] matrix. The DT ( ) transforms the original Lax pair ( ) into a new Lax pair, |
[EQUATION] in which the matrices [MATH] [MATH] [MATH] and [MATH] assume the same forms as that of [MATH] [MATH] [MATH] and [MATH] except that the potentials [MATH] and [MATH] have now acquired new expressions, namely [MATH] and [MATH] in [MATH] and [MATH] . Substituting the transformation ( ) into ( ) and comparing the... |
[EQUATION] Plugging the expressions [MATH] [MATH] [MATH] [MATH] and [MATH] in Eq. ( ) and equating the coefficients of various powers of [MATH] on both sides, we get the following relations between old and new potentials, namely |
[EQUATION] The eigenvalue problem given in ( ) remains invariant under the transformation ( ) provided [MATH] satisfies all the Eqs. ( 8a )-( 8f ). We assume a general form for the matrix [MATH] , namely |
[EQUATION] Substituting the assumed form of [MATH] in Eq. ( 8d ) and equating the matrix elements on both sides, we find [EQUATION] |
To obtain two parameter family of symmetry preserving and symmetry broken solutions of NNLS equations ( ) and ( ) we consider [MATH] to be |
[EQUATION] where [MATH] is the solution of ( ) at [MATH] . The exact forms of [MATH] and [MATH] are given by, [EQUATION] where [MATH] is the solution of ( ) at [MATH] . Since we consider [MATH] as a separate quantity we assume [MATH] is the appropriate solution of ( ) at [MATH] , where [MATH] is an isospectral paramete... |
Next we shall prove that the above matrix [MATH] satisfies expressions ( 8a )-( 8c ) together with ( 8e ) and ( 8f ). If [MATH] is solution of eigenvalue equations ( ) then one can write them as |
[EQUATION] By considering the form of [MATH] as in Eq. ( 11 ) and rewriting the above Eqs. ( 13 ), we get [EQUATION] The above equations exactly match with the equations given in ( 8e ) and ( 8f ). Using the relation ( 8d ), together with the expressions given in ( 11 ), the Eqs. ( 8a ) - ( 8c ) are all satisfied. Thus... |
The first iterated DT is given by [MATH] (vide Eq.( )). If [MATH] is the solution of [MATH] at [MATH] then it should satisfy [EQUATION] |
Expressing Eq. ( 15 ) in matrix form and using Cramer’s rule we can determine the exact forms of [MATH] and [MATH] which are given by |
[EQUATION] From ( 16 ) it is evident that to determine [MATH] and [MATH] one should know the explicit expressions of [MATH] [MATH] [MATH] and [MATH] which are the solutions of the eigenvalue problem ( ). Solving ( ) with appropriate seed solution [MATH] and [MATH] , one can obtain the explicit expressions of [MATH] [MA... |
[EQUATION] Through the formula ( 17 ) we can generate symmetry preserving and symmetry broken one bright and two dark/antidark soliton solutions of ( ) and ( ). |
2.2 Second Iteration of DT Second iteration of DT can be written as [EQUATION] where [MATH] [MATH] [MATH] [MATH] If [MATH] [MATH] is solution of [MATH] at [MATH] then it should satisfy |
[EQUATION] where [MATH] and [MATH] are given by [EQUATION] The second iteration of DT provides us a new solution in the form [EQUATION] |
Expressing Eq. ( 19 ) in matrix elements and using Cramer’s rule we can find the exact forms of [MATH] and [MATH] as [EQUATION] Substituting ( 39 ) in ( 21 ) we can get the second iterated DT solution formula. Using this formula we can obtain two bright and dark soliton solutions of ( ) and ( ). |
2.3 [MATH] th Iteration of DT [MATH] th iteration of DT can be written as [EQUATION] If [MATH] [MATH] is solution of [MATH] at [MATH] , it should satisfy |
[EQUATION] In the above [MATH] [MATH] [MATH] [MATH] [MATH] and [MATH] [MATH] are given by [EQUATION] Nth iteration of DT leads us to a new solution of the form |
[EQUATION] Expressing Eq. ( 41 ) in matrix elements and using Cramer’s rule we can find exact forms of [MATH] and [MATH] as [EQUATION] |
where [MATH] [MATH] and [MATH] are given by [EQUATION] [EQUATION] [EQUATION] Substituting ( 44 ) in ( 43 ) one can get [MATH] th DT solution formula. Using this formula we can obtain symmetry preserving and symmetry broken [MATH] -bright and [MATH] -dark soliton solutions of ( ) and ( ). |
Bright soliton solutions of NNLS equation 3.1 One bright soliton solution In this subsection, we construct the one bright soliton solution of Eqs. ( ) and ( ). To construct them, we feed a vacuum solution, that is [MATH] as seed solution to the focusing NNLS equation ( [MATH] ). By solving the Lax pair equations ( ) wi... |
[EQUATION] where [MATH] . The solution ( 66 ) is the more general one soliton solution of ( ) and ( ). We call it as symmetry broken solution (except for specific parameter values given below) since [MATH] and [MATH] are independent and cannot deduce one from the other. By choosing [MATH] [MATH] [MATH] [MATH] [MATH] an... |
. One can also deduce symmetry preserving one soliton solution of NNLS equation from ( 66 ) by confining the parametric conditions in the form [MATH] [MATH] [MATH] [MATH] [MATH] and [MATH] , where [MATH] [MATH] [MATH] and [MATH] are real constants. As a result, we obtain |
[EQUATION] The above solution coincides with the one given in . One can get [MATH] from ( 67 ) by taking complex conjugate of it and inversing the space variable in it. The symmetry broken solution ( 66 ) for generic parametric conditions can also be written in terms of trigonometric functions as |
[EQUATION] The above solution ( 68 ) develops singularity at [MATH] [MATH] . This one bright soliton solution is plotted in Fig. 1 for the parameter values [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] . The absolute value of [MATH] plotted in Fig. 1(a) shows that the amplitude of the soliton decays at [MATH] in the [MATH]... |
3.2 Two bright soliton solution Now we derive the two bright soliton solution of Eqs. ( ) and ( ). For this, we have to solve the Lax pair equations ( ) at [MATH] [MATH] with the seed solutions [MATH] . By doing so, we obtain the basic solutions of the form [MATH] [MATH] [MATH] and [MATH] , where [MATH] and [MATH] [MAT... |
[EQUATION] [EQUATION] The two bright soliton solution ( 69 ) of NNLS equation is plotted in Fig.2. As in the one soliton case, two soliton also exhibits stable propagation only for the absolute value of [MATH] , which can be clearly seen in Figs. 2. This more general two soliton solution ( 69 ) also coincides with the ... |
If we consider [MATH] set of basic solutions [MATH] [MATH] [MATH] and [MATH] , where [MATH] and [MATH] [MATH] , we can substitute them in the Nth iterated DT formula and obtain the N-soliton solution of NNLS equation. |
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