| 1. |
- Euler, Marianna, et al.
(författare)
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Linearizable hierarchies of evolution equations in (1+1) dimensions
- 2003
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Ingår i: Studies in applied mathematics (Cambridge). - : Wiley. - 0022-2526 .- 1467-9590. ; 111:3, s. 315-337
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Tidskriftsartikel (refereegranskat)abstract
- In our article [1], "A tree of linearisable second-order evolution equations by generalised hodograph transformations" we present a class of linearizable (C-integrable) second-order evolution equations in (1+1) dimensions, using a generalized hodograph transformation. We report here the complete set of recursion operators for this class and present the resulting linearizable (C-integrable) hierarchies in (1+1) dimensions. The autonomous class of linearizable hierarchies are extended further by considering the equations in potential form followed by the pure hodograph transformation.
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| 2. |
- Lenells, Jonatan, 1981-
(författare)
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Exactly solvable model for nonlinear pulse propagation in optical fibers
- 2009
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Ingår i: Studies in applied mathematics (Cambridge). - : Wiley. - 0022-2526 .- 1467-9590. ; 123:2, s. 215-232
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Tidskriftsartikel (refereegranskat)abstract
- The nonlinear Schrödinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation, which was first derived by means of bi-Hamiltonian methods in 1. The purpose of the present paper is threefold: (a) We show how this generalized NLS equation arises as a model for nonlinear pulse propagation in monomode optical fibers when certain higher-order nonlinear effects are taken into account; (b) We show that the equation is equivalent, up to a simple change of variables, to the first negative member of the integrable hierarchy associated with the derivative NLS equation; (c) We analyze traveling-wave solutions.
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| 3. |
- Marciniak, Krzysztof, 1968-, et al.
(författare)
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On Reciprocal Equivalence of Stäckel Systems
- 2012
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Ingår i: Studies in applied mathematics (Cambridge). - : Wiley-Blackwell. - 0022-2526 .- 1467-9590. ; 129:1, s. 26-50
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we ivestigate Stäckel transforms between different classes of parameter-dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel-related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.
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| 4. |
- Marciniak, Krzysztof, et al.
(författare)
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Separable systems of coordinates for triangular Newton equations q¨i = Mi(q1,..., qi)
- 2007
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Ingår i: Studies in applied mathematics (Cambridge). - : Wiley. - 0022-2526 .- 1467-9590. ; 118:1, s. 45-84
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Tidskriftsartikel (refereegranskat)abstract
- Triangular form of Newton equations is a strong property. Together with the existence of a single quadratic with respect to velocities integral of motion, it usally implies existence of further n - 1 integrals that are also quadratic. These integrals make the triangular system separable in new type of coordinates. The separation coordinates are built of quadric surfaces that are nonorthogonal and noconfocal and can intersect along lower dimensional singular manifolds. We present here separability theory for n-dimensional triangular systems and analyze the structure of separation coordinates in two and three dimensions. © 2007 by the Massachusetts Institute of Technology.
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| 5. |
- Petersson, Niclas, et al.
(författare)
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Recursion operators for a class of integrable third-order evolution equations
- 2004
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Ingår i: Studies in applied mathematics (Cambridge). - : Wiley. - 0022-2526 .- 1467-9590. ; 112:2, s. 201-225
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Tidskriftsartikel (refereegranskat)abstract
- We consider ut=uuxxx+n(u)uxuxx+m(u)u3x+r(u)uxx+p(u)u2x+q(u)ux+s(u) with α= 0 and α= 3, for those functional forms of m, n, p, q, r, s for which the equation is integrable in the sense of an infinite number of Lie–Bäcklund symmetries. Recursion operators which are x- and t-independent that generate these infinite sets of (local) symmetries are obtained for the equations. A combination of potential forms, hodograph transformations, and x-generalized hodograph transformations are applied to the obtained equations.
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| 6. |
- Sachdev, P L, et al.
(författare)
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Large-time asymptotics for periodic solutions of the modified burgers equation
- 2005
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Ingår i: Studies in applied mathematics (Cambridge). - : Wiley. - 0022-2526 .- 1467-9590. ; 114:3, s. 307A-323
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we construct large-time asymptotic solution of the modified Burgers equation with sinusoidal initial conditions by using a balancing argument. These asymptotics are validated by a careful numerical study.
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| 7. |
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| 8. |
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| 9. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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On the convergence of a linearly implicit finite element method for the nonlinear Schrödinger equation
- 2024
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Ingår i: Studies in Applied Mathematics. - 1467-9590 .- 0022-2526. ; In press
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Tidskriftsartikel (refereegranskat)abstract
- We consider a model initial- and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second-order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal-order error estimates in the ?2 and the ?1 norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a courant-friedrichs-lewy (CFL) condition between the space mesh size and the time step sizes.
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| 10. |
- Bergström, Lars
(författare)
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Initial algebraic growth of small angular dependent disturbances in pipe Poiseuille flow
- 1992
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Ingår i: Studies in applied mathematics (Cambridge). - : Wiley. - 0022-2526 .- 1467-9590. ; 87:1, s. 61-79
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Tidskriftsartikel (refereegranskat)abstract
- The evolution of small, angular dependent velocity disturbances in laminar pipe flow is studied. In particular, streamwise independent perturbations are considered. To fully describe the flow field, two equations are required, one for the radial and the other for the streamwise velocity perturbation. Whereas the former is homogeneous, the latter has the radial velocity component as a forcing term. The normal modes of the system are determined and analytical solutions for eigenfunctions, damping rates and phase velocities are calculated. Then, the initial value problem is treated and the time development of the disturbances is determined.
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