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Dark and antidark solitons of NNLS equation To obtain dark soliton solutions of NNLS equation, we choose plane wave solution as the seed solution, that is [MATH] and [MATH] [MATH] to the defocusing NNLS equation, that is Eqs. ( ) and ( ) with [MATH] . Substituting these solutions into the Lax pair equations ( 13 ) and ... |
[EQUATION] where [MATH] [MATH] [MATH] and [MATH] . Substituting these basic solutions into the first iterated DT formula ( 17 ) and simplifying the resultant expressions with [MATH] and [MATH] , we arrive at |
[EQUATION] where [EQUATION] and [MATH] [MATH] [MATH] [MATH] According to the definition of dark soliton, the soliton solution with real functions asymptotically approaches constant value. Therefore to identify dark soliton solution from ( 71 ) we must impose the conditions that [MATH] and [MATH] are to be real and [MAT... |
. In contrast to bright soliton, the dark soliton solution of NNLS equation exhibits stable propagation for [MATH] and [MATH] independently. The solution ( 71 ) consists of collision between two dark and/or antidark solitons. Interestingly when [MATH] exhibits dark soliton [MATH] exhibits either antidark or dark solito... |
Collision dynamics in two dark and antidark soliton solution To analyze the collision dynamics of dark/antidark solitons we name the two solitons in [MATH] as [MATH] and [MATH] and the two solitons in [MATH] as [MATH] and [MATH] . Each soliton in [MATH] can take two forms, namely dark and antidark soliton form. In othe... |
Collision Scenario 1: Figure 3(a) shows collision between two dark solitons of [MATH] while [MATH] exhibits collision between two antidark solitons for the same parameter values which is shown in Fig. 3(b). Figure 3(c) shows the plot of [MATH] , which gives two dark solitons collision. Unlike local NLS case here the tw... |
Collision Scenario 2: Figures 4(a) and 4(b) show contrasting structures compare to Figs. 3(a) and 3(b), that is [MATH] exhibits two antidark soliton collision and [MATH] gives two dark solitons collsion and Fig. 4(c) exhibits two dark solitons collision of [MATH] with a hump as in the previous case. |
Collision Scenario 3: Both [MATH] and [MATH] exhibit two dark solitons collision as shown in Figs. 5(a) and 5(b). In this case also [MATH] produces two dark solitons collision without any hump as shown in Fig. 5(c). |
Collision Scenario 4: Both [MATH] and [MATH] exhibit collision between two antidark solitons shown in Figs. 6(a) and 6(b). As we expected, [MATH] also exhibits two antidark solitons collision which is demonstrated in Fig. 6(c). |
Collision Scenario 5: Figures 7(a) and 7(b) illustrate the collsions between dark and antidark solitons in both the components [MATH] and [MATH] . But when we plot [MATH] , we get collision between two dark solitons, as shown in Fig. 7(c). |
Collision Scenario 6: [MATH] exhibits two dark solitons collision and [MATH] gives collision between one dark and one antidark solitons which are shown in Figs. 8(a) and 8(b). We then plot [MATH] for the same parameter values and obtain collision between two dark solitons as shown in Fig. 8(c). |
Collision Scenario 7: In contrast to the previous collision scenario here we get dark and antidark soliton collision in [MATH] and two dark soliton collision in [MATH] which are illustrated in Figs. 9(a) and 9(b). Here also we get two dark soliton collisions for [MATH] as demonstrated in Fig. 9(c). |
Collision Scenario 8: For the parameter values given in Fig. 10 we get two antidark soliton collision in [MATH] and dark and antidark soliton collision in [MATH] which are illustrated in Figs. 10(a) and 10(b). When we plot [MATH] for the same parameter values we get dark and antidark soliton collision that is displayed... |
Collision Scenario 9: In contrast to the collision scenario 8, here [MATH] exhibits collision between dark and antidark solitons, see Fig. 11(a), and [MATH] gives two antidark soliton collision, see Fig. 11(b). Plotting [MATH] gives dark and antidark collisions for the same parameter values which is given Fig. 11(c). |
To understand the above novel behaviours clearly we carry out appropriate asymptotic analysis in the following section. Asymptotic analysis for the defocussing case ( [MATH] |
The interpretation of the results in terms of actual motion of the soliton depends on the signs of the parameters [MATH] and [MATH] which appear in the expressions [MATH] [MATH] . To begin, let us assume that soliton 1 is in the vicinity of the line [MATH] . Now we change the frame co-moving with soliton 1 (coordinated... |
(i) Before Collision: Let us consider the limit [MATH] . In the vicinity of [MATH] , we have [MATH] , when [MATH] [MATH] . Substituting these asymptotic values in the solution ( 71 ) we obtain the following results: |
(a) Soliton 1: [EQUATION] and the squares of the absolute values are given by [EQUATION] From ( 74 ) one can find the amplitudes |
[EQUATION] and the phase as [EQUATION] While plotting the solution ( 73 ) we get dark soliton for [MATH] if [MATH] or antidark soliton for [MATH] if [MATH] . We obtain antidark soliton for [MATH] if [MATH] or dark soliton for [MATH] if [MATH] . So we get two different conditions for [MATH] and [MATH] to obtain dark and... |
(b) Soliton 2: In the same limit [MATH] , in the vicinity of [MATH] we have [MATH] , then we obtain [EQUATION] and the squares of their absolute values can be written as |
[EQUATION] From ( 78 ), the amplitudes can be written as [EQUATION] and the phase is given by [EQUATION] Now when we plot the solution ( 77 ), we obtain antidark soliton for [MATH] if [MATH] , and dark soliton for [MATH] if [MATH] . We get dark soliton for [MATH] if [MATH] or antidark soliton for [MATH] if [MATH] . Hen... |
(ii) After Collision Now we consider the other limit [MATH] . In the vicinity of [MATH] , we have [MATH] and we get the following results: |
(a) Soliton 1: [EQUATION] and the squares of their absolute values are given by [EQUATION] From ( 82 ), the amplitudes can be written as |
[EQUATION] and the phase is given by [EQUATION] (b) Soliton 2: In the same limit [MATH] , in the vicinity of [MATH] we have [MATH] , then we get |
[EQUATION] and the squares of their absolute values can be written as [EQUATION] From ( 86 ) the amplitudes can be written as [EQUATION] |
and the phase is given by [EQUATION] While plotting the solutions ( 81 ) and ( 85 ), we came across the same behaviour as in ( 73 ) and ( 77 ) respectively. The parametric regions for dark and antidark solitons in both [MATH] and [MATH] are listed in Table 1. From Eqs. ( 73 ) - ( 85 ), we can observe the following feat... |
To obtain symmetry preserving solution of two dark and antidark soliton solution one should consider [MATH] as complex variable such as [MATH] . By choosing [MATH] [MATH] [MATH] and making [MATH] , the solution ( 71 ) becomes |
[EQUATION] where [MATH] [MATH] [MATH] [MATH] . The solution ( 89 ) is the symmetry preserving solution since [MATH] can be obtained by taking complex conjugate of ( 89 ) and reversing the space variable in it. This solution coincides with the solution reported in Ref. |
[MATH] dark and antidark soliton solution Finally, Let us consider [MATH] set of basic solutions such that [EQUATION] where [MATH] [MATH] [MATH] and [MATH] . Substituting these basic solutions in the [MATH] th iterated DT formula ( 43 ), one can obtain [MATH] dark and antidark soliton solutions of NNLS equations ( ) an... |
Conclusions Using DT method we have constructed symmetry preserving and symmetry broken N-bright soliton solution for the NNLS equation of focussing type and given explicit one and two soliton solutions. To construct these solutions, we have considered appropriate eigenfunctions in the DT method. We have shown that due... |
Finally, we have indicated the structure of [MATH] -dark/antidark soliton solution formula and pointed out that this solution may have [MATH] combinations of collision. |
Acknowledgements NVP wishes to thank the University Grants Commission (UGC), Government of India, for providing financial support through Dr.D.S. Kothari Post Doctoral Fellowship Scheme. The work of MS forms part of a research project sponsored by Science and Engineering Research Board (SERB), DST, Government of India ... |
# Source: arxiv 1806.09392 # Title: Finding models through graph saturation # Sections: all # Downloaded: 2026-03-03T02:30:33.600621+00:00 |
Finding models through graph saturation Abstract We give a procedure that can be used to automatically satisfy invariants of a certain shape. These invariants may be written with the operations intersection, composition and converse over binary relations, and equality over these operations. We call these invariants sen... |
This is an accepted preprint, which will be published in the Journal of Logical and Algebraic Methods in Programming (JLAMP). Introduction |
The question ‘what models does a set of formulas [MATH] have’ has practical relevance, as it is an abstraction of an information system: We interpret the data set stored in an information system at a certain point in time as a model, and each invariant of the system corresponds to a formula in [MATH] This correspondenc... |
, that define an information system this way. Users of an information system try to change the data set continually. These changes might violate the constraints. While Ampersand responds to such violations by rejecting the change, it would be convenient to automatically add data items such that all constraints are sati... |
The question ‘does a set of formulas [MATH] have a model satisfying all formulas’ essentially asks whether [MATH] is free of contradictions. So far, we did not discuss the language in which we can write the formulas in [MATH] Several interesting problems arise when restricting the language in which we can write formula... |
Our interest in this language stems from experience in describing systems in Ampersand. All operations from relation-algebra are part of the Ampersand language. The operations considered here include only the most frequently used subset of those operations. Therefore, many of the formulas used in Ampersand will be sent... |
1.1 Approach We give a short summary of the basic algorithm presented here, so we can better relate our approach to other literature, describe our contributions, and give the outline of this paper. Italicised words in the next paragraph are defined later. |
The algorithm aims to determine whether there is a particular model for a set of sentences , say [MATH] , and is guaranteed to terminate if no such model exists. It proceeds to construct a (possibly infinite) model otherwise. The procedure has two phases: first, we translate the sentences in [MATH] into a set of graph ... |
entailment is the question whether a sentence [MATH] follows from a set of sentences [MATH] In an information system, a least consequence graph is a well suited to determine which data items to add: If conflict free, it corresponds to a graph that maintains the invariants. At the same, it only contains necessary conseq... |
1.2 Related Work We compare the work in this paper to existing work in two ways: work it is similar to in motivation, and work it is similar to in implementation from an abstract perspective. In motivation, our research is closely related to the Alcoa tool, which we’ll discuss first. In approach, our methods are relate... |
The Alcoa Tool. Our search for a reasoner for Ampersand is related to Alcoa , which is the analyzer for Alloy , a language based on Z |
Like Ampersand, the languages Z and Alloy are based on relations. Alloy is a simplification of Z: it reduces the supported operations to a set that is small yet powerful. This paper differs from Alloy in the expressivity of its operations, however: Alloy allows writing full first order formula’s plus the Kleene-star, m... |
In Alloy, a user may write assertions, which are formulas that the user believes follow from the specification. Alcoa tries to find counterexamples to those assertions, as well as a finite model for the entire specification. Unfortunately, several properties of the Alcoa tool hinder our purposes in Ampersand: Alcoa req... |
Description Logics. We can regard our procedure as a way to derive facts from previously stated facts: this is what happens in terms of sentences between subsequent graphs in the chain we create. So called description logics are languages used in conjunction with an engine, that gives a procedure to learn new facts fro... |
A set of derivation rules is consistent if it has a model. For a highly expressive description logic such as OWL DL, determining consistency is undecidable. Still, a rule engine for OWL DL will happily try to learn new facts until a model is found. Users of OWL DL typically need to ensure that the stated derivation rul... |
The description logic for which the language and implementation is closest to our language is the logic [MATH] and its extensions proposed by Baader et al |
Instead of using tableau-based procedures, as most description logics, it uses a saturation-based reasoner. Syntax of the derivation rules is limited to ensure termination of any saturation procedure: |
[MATH] allows statements about unary relations using top, bottom, individual elements called ‘nominal’, and conjunction. Statements about binary relations use a different syntax, that can be translated into sentences using composition, converse and the identity relation (but not necessarily vice-versa). By modeling [MA... |
In our work, we do not work under the assumption of termination: neither the user or the syntax guarantees it. This allows us to use a richer language than one that is syntactically guaranteed to terminate. Despite this lack of termination, we do ensure termination in case of conflicts: a conflict will be found if our ... |
Graph Rewriting. A central concept in graph rewriting is that a pushout can be used to apply a graph rule on a graph, as described by Wolfram Kahl |
The usual idea of such a pushout is that it models execution by removing a portion of the graph, and replacing it with the result of the execution step. Graph rewriting might then terminate when no rules can be applied anymore. Our approach diverges on this point: rather than execution, a step models learning a deducib... |
The term saturation is borrowed from the saturation procedure in resolution procedures, introduced by Robinson in 1965 His procedure solves an entailment problem over a certain language. As in his procedure, our procedure adds derivable facts iteratively. |
1.3 Contributions and Paper Outline We mentioned how this paper contributes by comparing it to related work: Compared to the work on [MATH] , our approach allows sentences in a richer language, and we present a translation to graph rules to separate the semantics from the core of the implementation. Compared to the wor... |
We also relate the contribution of this paper to a paper presenting Amperspiegel This earlier paper by the author conjectured that the problem whether no least consequence graph exists is undecidable. It also contains a procedure for finding such graphs, which it conjectures to be correct. We will show that the procedu... |
The procedure presented in this work is simpler than the one presented earlier. However, the latter can be obtained by applying optimizations to the former. We show correctness of the procedure, and show that the existence of a conflict free least consequence graph implies the existence of models for a set of sentences... |
The outline of this paper is as follows: we define the syntax and semantics of sentences in Section , and define the problems our procedure aims to solve: deciding consistency and entailment. Section then introduces the heart of the procedure by defining least consequence graphs and indicating how to obtain them throug... |
Background and Problem Statement As this paper primarily deals with directed labeled graphs, we choose to use these graphs for the semantics of sentences as well. There is no fundamental difference between this presentation and the usual binary relation based semantics usually presented as the canonical allegory (or as... |
Definition 1 (Graph, Empty, Finite) A directed labeled graph [MATH] is given by a set of labels [MATH] , a set of vertices [MATH] , and a set of edges [MATH] The set of all graphs with labels [MATH] is written as [MATH] We write graph when we mean a directed labeled graph. We say that a graph is finite if both its set ... |
Terms are built inductively from relation symbols [MATH] , combined with the operations [MATH] [MATH] , and [MATH] The operations stand for intersection, relational composition, and relational converse, respectively. The set of all terms over [MATH] is denoted as [MATH] We use the same letter [MATH] to indicate labels ... |
Definition 2 (Semantics) For a graph [MATH] , the semantics of a term [MATH] , written as [MATH] , is as in representable relation algebra: |
[EQUATION] A sentence is the proposition stating that two terms are equal: Definition 3 (Sentence, Holds) Given the terms [MATH] , the pair [MATH] is a sentence , written [MATH] We write [MATH] for a sentence of the shape [MATH] We say that a sentence holds in graph [MATH] if [MATH] , in which case we write: [MATH] If ... |
Lemma 1 Let [MATH] , and [MATH] [EQUATION] We deviate slightly from allegories: First, we are working in an untyped setting, or put differently: in an allegory with only a single object. In ‘typed allegories’, allegories with more than one object, relational composition is a partial operation. This deviation is not fun... |
Apart from the identity symbol , we also introduce bottom and top ( [MATH] and [MATH] ) as symbols in [MATH] In Definition we give the interpretation of these designated relation symbols, defining a graph as standard if it adheres to this interpretation. |
Definition 4 (Standard) We say that a set of labels [MATH] is standard with the (possibly empty) set of constant elements [MATH] if [MATH] and [MATH] We refer to elements in [MATH] simply as constants. Let [MATH] be a standard set of labels with the constants [MATH] A graph [MATH] is called standard if [MATH] , and: |
[EQUATION] This work looks at models for [MATH] , and investigates whether [MATH] entails [MATH] We can now give the definitions that necessary to make this precise. |
Definition 5 (Model, Consistent) Let [MATH] be a set of sentences over a standard set of labels [MATH] (with constants [MATH] ). We say that the graph [MATH] is a model for [MATH] if every sentence in [MATH] holds in [MATH] and [MATH] is standard. We say that [MATH] is consistent if such a graph exists. We may refer to... |
Definition 6 (Entails) Let [MATH] be a set of sentences over a standard set of labels [MATH] , and let [MATH] be a sentence over [MATH] We say that [MATH] is an instance of the entailment problem We say that [MATH] |
entails [MATH] if for all standard graphs G, [MATH] implies [MATH] Our use of ‘standard’ in these definitions is not a restriction: given a graph [MATH] over a language [MATH] with [MATH] , we can make it into a standard graph [MATH] over [MATH] , choosing the constants [MATH] , and adding the edges according to Defini... |
We prove a straightforward correspondence between the consistency problem and the entailment problem: Lemma 2 There is a standard graph [MATH] such that [MATH] if and only if [MATH] does not entail [MATH] |
Proof We first prove that if [MATH] entails [MATH] , then there is no standard graph with [MATH] A standard graph must have at least one vertex, say [MATH] Then [MATH] , and [MATH] , so [MATH] For the other direction: Suppose there is no standard graph with [MATH] , then entailment of any formula follows by definition.... |
We proceed with a small example of sentences, an entailment and a consistency problem. As an example, we make an administration of people and rooms. We use the label [MATH] to denote which room a person Inhabits, and [MATH] to denote which people are Roommates. We think of the labels in terms of their semantics: as bin... |
We ask ourselves if being a roommate is a transitive relation. That is, does [MATH] entail [MATH] or not? The answer is negative. A possible counter-example our procedure may produce is a graph [MATH] with: |
[EQUATION] In this example, [MATH] are people, and [MATH] are their rooms. While [MATH] and [MATH] are roommates of [MATH] [MATH] is not a roommate of [MATH] Note that person [MATH] has two rooms in this example. We may wish to forbid this: the sentence [MATH] expresses that [MATH] is univalent (if two rooms are inhabi... |
We elaborate on the same example for checking consistency, and add some constants to [MATH] Let [MATH] Let’s say we want Liz and Jon to be roommates, and ask ourselves if that’s possible. That is, we wish to solve the consistency problem for: |
[EQUATION] Our procedure then produces a graph like [MATH] with: [EQUATION] Without going into details on why, we remark that our procedure comes up with a new room, here called [MATH] , even with the Batcave and Room 11 available. Finally, if we require Liz and Jon to be in their rooms of their choice, the Batcave and... |
[EQUATION] Graph Rules and Consequence Graphs This section defines a least consequence graph, and gives conditions on a chain of graphs that ensure that its limit is a least consequence graph. When a graph is a least consequence graph, we can use it to answer both the entailment problem and the consistency problem. The... |
We introduce special notation for two basic operations on graphs: relabeling of vertices, and taking the union of two graphs. Suppose we have a function [MATH] , where [MATH] is the set of vertices of some graph. We can apply the function on the corresponding graph, written [MATH] |
[EQUATION] For taking the union of two graphs, we simply write [MATH] , defined as follows: [EQUATION] This leads to a natural definition of subgraph: |
Definition 7 (Subgraph) We say that [MATH] is a subgraph of [MATH] if [MATH] It follows that a subgraph of a finite graph is again finite. If [MATH] is a subgraph of [MATH] and [MATH] for some [MATH] , we write [MATH] |
In this article, we consider the set of labels [MATH] to be arbitrary but fixed. The relation ‘subgraph’ forms a complete lattice over [MATH] , which justifies the following definition: |
Definition 8 (Chain, Supremum) Given a set of labels [MATH] We say that [MATH] is a chain if for all [MATH] [MATH] is a subgraph of [MATH] The union of all graphs in a chain, written [MATH] , is called the supremum , defined as [MATH] with [MATH] |
The way we use graph rewriting is most closely related to the single-pushout rewriting found in the literature (e.g. ). In this approach, graph rules are related through a morphism that is, for instance, a partial function. Vertices in the left hand side of the rule not related to the right hand side get removed upon a... |
Definition 9 (Graph Rule) A pair of graphs [MATH] is called a graph rule if [MATH] is a subgraph of [MATH] , and [MATH] is finite. We say that a set [MATH] is a set of graph rules with labels |
[MATH] if each [MATH] is a graph rule, and [MATH] We proceed by giving an example of a graph rule, and do so visually. A graph can be drawn in the usual way. Figure 1(a) is an example of a graph with [MATH] A picture does not specify the set of labels [MATH] , only the set of edges and the set of vertices. An example o... |
We present a saturation procedure, so we need to capture when a graph is ‘saturated’. For this purpose, we define ‘maintained’, which indicates that a rule is applied sufficiently in a graph. For defining ‘maintained’, we first define graph embeddings: |
Definition 10 (Embedding) Let [MATH] If [MATH] , then [MATH] is an embedding of [MATH] in [MATH] In such case, we write [MATH] We say that [MATH] is embedded in [MATH] if such an [MATH] exists, written [MATH] It follows immediately that [MATH] |
We briefly explain our notations with the observation that embeddings form a category: its objects are graphs with labels [MATH] , and its arrows are embeddings. Although [MATH] , note that [MATH] is only the identity arrow if [MATH] which is why we avoid writing [MATH] |
Definition 11 (Maintained, (Least) Consequence Graph) A graph rule [MATH] with [MATH] is maintained in [MATH] if for every embedding [MATH] , there is an embedding [MATH] such that [MATH] for all [MATH] If for a set of graph rules [MATH] , each graph rule in [MATH] is maintained in [MATH] , we say that [MATH] is a cons... |
We use chains to find least consequence graphs. We look at two properties: ‘fairness’ and ‘weak pushout’, that help establish graphs to be a consequence graph and least, respectively. To get some intuition, and hopefully help dispel some overly optimistic conjectures, we look at some examples before defining these two ... |
We begin with an example of an embedding. Let [MATH] and [MATH] be graphs. Note that [MATH] is the graph rule drawn in Figure 1(b) We can embed [MATH] into the graph [MATH] as shown in Figure 1(a) A corresponding embedding is [MATH] with [MATH] for [MATH] There is also an embedding for [MATH] [MATH] with [MATH] for [MA... |
As an example of a consequence graph, let [MATH] with [MATH] as defined above, and let [MATH] , as drawn in Figure 1(d) Then [MATH] is a consequence graph maintaining [MATH] It is, however, not a least consequence graph of [MATH] maintaining [MATH] , since Figure 1(e) gives a consequence graph maintaining [MATH] in whi... |
The graph [MATH] defined above is an example of a least consequence graph of [MATH] maintaining [MATH] Graph [MATH] , consisting of two disjunctive copies of [MATH] , is a least consequence graph too, see Figure 1(f) If a least consequence graph is unique, it must be the empty graph. |
From the definition of maintained it follows that if [MATH] and [MATH] is maintained in [MATH] , then [MATH] is maintained in [MATH] too. Consequently, if [MATH] is maintained in a least consequence graph of [MATH] maintaining [MATH] , then [MATH] is maintained in every consequence graph maintaining [MATH] |
The following definition gives a sufficient condition to reach a consequence graph: Definition 12 (Fair Chain) Given a set of graph rules [MATH] and a chain [MATH] We say that [MATH] is a fair chain for [MATH] |
if for each graph rule [MATH] and for each embedding [MATH] there exists a [MATH] and an embedding [MATH] with [MATH] for all [MATH] in the set of vertices of [MATH] |
Lemma 3 If [MATH] is a fair chain for [MATH] [MATH] is a consequence graph maintaining [MATH] Proof By definition, [MATH] is a consequence graph if we can show that [MATH] is embedded in [MATH] for every [MATH] that is embedded in it. Take such an embedding [MATH] Then for each edge [MATH] of [MATH] there is an [MATH] ... |
We define weak pushout step as an upper limit to each step, to ensure that a consequence graph found as a supremum of a chain built out of these steps is also a least consequence graph. |
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