Boston University. University Professors Program Senior theses.

Doctor of Philosophy (PhD). / The term `Lax pair' refers to linear systems (of various types) that are related to nonlinear equations through a compatibility condition. If a nonlinear equation possesses a Lax pair, then the Lax pair may be used to gather information about the behaviour of the solutions to the nonlinear equation. Conserved quantities, asymptotics and even explicit solutions to the nonlinear equation, amongst other information, can be calculated using a Lax pair. Importantly, the existence of a Lax pair is a signature of integrability of the associated nonlinear equation. While Lax pairs were originally devised in the context of continuous equations, Lax pairs for discrete integrable systems have risen to prominence over the last three decades or so and this thesis focuses entirely on discrete equations. Famous continuous systems such as the Korteweg de Vries equation and the Painleve equations all have integrable discrete analogues, which retrieve the original systems in the continuous limit. Links between the different types of integrable systems are well known, such as reductions from partial difference equations to ordinary difference equations. Infinite hierarchies of integrable equations can be constructed where each equation is related to adjacent members of the hierarchy and the order of the equations can be increased arbitrarily. After a literature review, the original material in this thesis is instigated by a completeness study that finds all possible Lax pairs of a certain type, including one for the lattice modified Korteweg de Vries equation. The lattice modified Korteweg de Vries equation is subsequently reduced to several q-discrete Painleve equations, and the reductions are used to form Lax pairs for those equations. The series of reductions suggests the presence of a hierarchy of equations, where each equation is obtained by applying a recursion relation to an earlier member of the hierarchy, this is confirmed using expansions within the Lax pairs for the q-Painleve equations. Lastly, some explorations are included into fake Lax pairs, as well as sets of equivalent nonlinear equations with similar Lax pairs.

Doctor of Philosophy (PhD). / The term `Lax pair' refers to linear systems (of various types) that are related to nonlinear equations through a compatibility condition. If a nonlinear equation possesses a Lax pair, then the Lax pair may be used to gather information about the behaviour of the solutions to the nonlinear equation. Conserved quantities, asymptotics and even explicit solutions to the nonlinear equation, amongst other information, can be calculated using a Lax pair. Importantly, the existence of a Lax pair is a signature of integrability of the associated nonlinear equation. While Lax pairs were originally devised in the context of continuous equations, Lax pairs for discrete integrable systems have risen to prominence over the last three decades or so and this thesis focuses entirely on discrete equations. Famous continuous systems such as the Korteweg de Vries equation and the Painleve equations all have integrable discrete analogues, which retrieve the original systems in the continuous limit. Links between the different types of integrable systems are well known, such as reductions from partial difference equations to ordinary difference equations. Infinite hierarchies of integrable equations can be constructed where each equation is related to adjacent members of the hierarchy and the order of the equations can be increased arbitrarily. After a literature review, the original material in this thesis is instigated by a completeness study that finds all possible Lax pairs of a certain type, including one for the lattice modified Korteweg de Vries equation. The lattice modified Korteweg de Vries equation is subsequently reduced to several q-discrete Painleve equations, and the reductions are used to form Lax pairs for those equations. The series of reductions suggests the presence of a hierarchy of equations, where each equation is obtained by applying a recursion relation to an earlier member of the hierarchy, this is confirmed using expansions within the Lax pairs for the q-Painleve equations. Lastly, some explorations are included into fake Lax pairs, as well as sets of equivalent nonlinear equations with similar Lax pairs.

Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 5 uppsatser.

Lax är en av de mest frekvent förekommande matfiskarna i våra hem. Den största delen av den lax som når hem och restaurang är odlad. Det är allt vanligare med sjukdomar hos lax som odlas. Kocken ställs inför svåra frågor gällande den odlade fiskens kvalité. En viktig aspekt är att ha kunskap om de eventuella sjukdomar som förekommer i vår samtids fiskodling. Syftet med detta arbete är att undersöka den i vardagens måltider förekommande råvaran odlad lax utifrån hur sjukdomar påverkar fiskens kvalité. Den information som ges i de fem vetenskapliga artiklar arbetet är baserat på presenteras i avsnittet resultat. Det framgår att en stressad och sjuk fisk tappar avsevärt i kvalité och ger en sämre avkastning. I diskussionen jämförs olika författares resultat och det framgår här att bättre foder, smittokontroll och högre levnadsstandard för laxen ger en bättre produkt ur kvalitetssynpunkt. Slutsatserna som dras är att vi tillagar en fisk med sämre kvalité än nödvändigt och att vi för denna betalar ett högre pris än nödvändigt. Litteraturstudien bygger på fem vetenskapliga artiklar från olika databaser, litteraturen redovisas i resultatet och databassökningarna i sökmatriser. Sammanfattningar av de fem artiklarna återfinns i artikelmatriser i slutet av arbetet. / B-uppsatser

This dissertation deals with a class of nonlinear wave equations of the type discovered by R. Camassa and D. D. Holm which includes the Camassa-Holm, the Degasperis-Procesi, and the two component Camassa-Holm equations. All these equations admit certain non-smooth soliton-like solutions, called peakons as well as other non-smooth solutions like cuspons. We apply the techniques of the theory of distributions of L. Schwartz to study these solutions. In particular, every non-smooth traveling wave which is a distributional solution of the two component Camassa-Holm equation is a distributional solution of the Camassa-Holm equation if the set of points where the height of the wave equals its speed, is of measure zero. This includes peakon or cuspon traveling wave solutions.<p>We also develop a suitable modification of the classical Lax pair formalism to deal with singular solutions. We show that the Lax pair formalism can be extended to a distributional weak Lax pair which is appropriate for dealing with the peakon solutions of the Camassa-Holm equation.

This dissertation deals with a class of nonlinear wave equations of the type discovered by R. Camassa and D. D. Holm which includes the Camassa-Holm, the Degasperis-Procesi, and the two component Camassa-Holm equations. All these equations admit certain non-smooth soliton-like solutions, called peakons as well as other non-smooth solutions like cuspons. We apply the techniques of the theory of distributions of L. Schwartz to study these solutions. In particular, every non-smooth traveling wave which is a distributional solution of the two component Camassa-Holm equation is a distributional solution of the Camassa-Holm equation if the set of points where the height of the wave equals its speed, is of measure zero. This includes peakon or cuspon traveling wave solutions.<p>We also develop a suitable modification of the classical Lax pair formalism to deal with singular solutions. We show that the Lax pair formalism can be extended to a distributional weak Lax pair which is appropriate for dealing with the peakon solutions of the Camassa-Holm equation.

Det atlantiska laxbeståndet i Östersjön är idag hotat, vilket är ett resultat av antropogen påverkan, och det har i praktiken inte funnits någonförvaltningsplan för laxen i Östersjön på nästan 10 år. Ett förslag gällande en flerårig förvaltningsplan för Östersjölaxen lades fram av EUkommissionen2011, vilken syftade till att bidra till ett hållbart nyttjande av laxen, och att den genetiska integriteten och variationen bevaras.Huvudsyftet med denna studie är att undersöka utvalda remissvar som framtagits efter förslaget. Remissvaren är officiella utlåtanden från berördamyndigheter och icke-statliga organisationer, och som har framförts till regeringen. Innehållsanalysen kommer att användas i syfte att nå enfördjupning samt identifiera återkommande mönster och teman utifrån remissvaren och de två utförda intervjuerna. Analysen visar att mångaaktörer påvisar en generellt sett positiv inställning till förslaget, men meningsskiljaktigheter och kritik av föreslagna åtgärder och tillvägagångssättkaraktäriserar flera av remissvaren. Detta märks tydligt under förslagen gällande utfasning av utsättningar och smoltproduktionsmålen, där mångaåtgärds- och tillvägagångsförslag är mycket tidskrävande, och förändringar sker över längre tidsperioder. Det är därför av stor vikt attförändringsprocesser påbörjas långt innan man riskerar att nå en punkt där skadan blir oåterkallelig. / The population of the Atlantic salmon in the Baltic Sea is threatened by substantial anthropogenic activities, and in practice, an action plan hasbeen missing for almost a decade. A proposal for a multi annual action for the Baltic salmon stock was presented by the EU-commission 2011,this plan aimed to aid attempts of reaching a sustainable exploitation of the salmon stock, and to sustain the genetic integrity and diversity of thespecies. The main purpose of this thesis is to examine selected statements of opinion in regards of mentioned proposal, these statements areofficial recommendations carried out by concerned authorities and NGO’s, and have been delivered to the Swedish government. Content analysiswill be used on the statements of opinion and the two interviews. The analysis shows that various actors are generally positive towards theproposal of a multiannual action plan, but disagreements and critique of proposed approaches tend to be dominant in several statements. This isvery clear in terms of the phasing out of the salmon releases and the smolt production goals, which the proposal deals with. A large number ofproposed measures and associated processes are very time consuming, and thus, changes in matters like these happen over extended periods oftime. It is therefore of great importance that actions are put in motion long before the risks of reaching irreversible damages becomes evident.

Braunschweig, Techn. Univ., Diss., 2002. / Computerdatei im Fernzugriff.

Braunschweig, Techn. Univ., Diss., 2002. / Computerdatei im Fernzugriff.