Professor Jing Ping Wang
Professor of Applied Mathematics
REF Coordinator
About
Jing Ping serves on the School's Research and Innovation Committee and she is the School's REF Coordinator.
Research interests
- Integrability conditions and hidden structures of integrable systems
- Exact solutions for discrete and continuous nonlinear systems
Supervision
Nitin Serwa - Symbolic computation and integrable systems
Professional
Member of the EPSRC Peer Review College
Publications
Showing 50 of 52 total publications in the Kent Academic Repository. View all publications.
Article
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Hay, M., Hone, A., Novikov, V. and Wang, J. (2018). Remarks on certain two-component systems with peakon solutions. arXiv [online] [Online]. Available at: https://arxiv.org/pdf/1805.03323.pdf.We consider a Lax pair found by Xia, Qiao and Zhou for a family
of two-component analogues of the Camassa-Holm equation, including
an arbitrary function H, and show that this apparent freedom can be
removed via a combination of a reciprocal transformation and a gauge
transformation, which reduces the system to triangular form. The
resulting triangular system may or may not be integrable, depending
on the choice of H. In addition, we apply the formal series approach of
Dubrovin and Zhang to show that scalar equations of Camassa-Holm
type with homogeneous nonlinear terms of degree greater than three
are not integrable.
This article is dedicated to Darryl Holm on his 70th birthday. -
Carpentier, S., Mikhailov, A. and Wang, J. (2018). Rational recursion operators for integrable differential-difference equations. arXiv [Online] [Online]. Available at: https://arxiv.org/pdf/1805.09589.pdf.
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Hone, A., Novikov, V. and Wang, J. (2017). Generalizations of the short pulse equation. Letters in Mathematical Physics [Online]. Available at: https://doi.org/10.1007/s11005-017-1022-3.We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.
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Tian, K. and Wang, J. (2017). Symbolic Representation and Classification of N = 1 Supersymmetric Evolutionary Equations. Studies in Applied Mathematics [Online] 138:467-498. Available at: http:/dx.doi.org/10.1111/sapm.12163.We extend the symbolic representation to the ring of N = 1 supersymmetricdifferential polynomials, and demonstrate that operations on the ring, suchas the super derivative, Fr´echet derivative, and super commutator, can becarried out in the symbolic way. Using the symbolic representation, weclassify scalar ?-homogeneous N = 1 supersymmetric evolutionary equa-tions with nonzero linear term when ?>0 for arbitrary order and give acomprehensive description of all such integrable equations.
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Bury, R., Mikhailov, A. and Wang, J. (2017). Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system. Physica D: Nonlinear Phenomena [Online] 347:21-41. Available at: http://dx.doi.org/10.1016/j.physd.2017.01.003.In the paper we develop the dressing method for the solution of the two-dimensional periodic Volterra system with a period N. We derive soliton solutions of arbitrary rank k and give a full classification of rank 1 solutions. We have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial (zero) solution. The wave fronts are non-stationary and they propagate with a constant average velocity. The system also has soliton solutions similar to breathers, which resembles soliton webs in the KP theory. We associate the classification of soliton solutions with the Schubert decomposition of the Grassmannians View the MathML source and View the MathML source.
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Hone, A., Novikov, V. and Wang, J. (2017). Two-component generalizations of the Camassa-Holm equation. Nonlinearity [Online] 30:622-658. Available at: http://iopscience.iop.org/article/10.1088/1361-6544/aa5490/meta;jsessionid=0AADAAD96C412EF897587E993641D098.c2.iopscience.cld.iop.org.A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered.
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Mikhailov, A., Papamikos, G. and Wang, J. (2016). Dressing method for the vector sine-Gordon equation and its soliton interactions. Physica D: Nonlinear Phenomena [Online] 325:53-62. Available at: http://dx.doi.org/10.1016/j.physd.2016.01.010.In this paper, we develop the dressing method to study the exact solutions for the vector sine-Gordon equation. The explicit formulas for one kink and one breather are derived. The method can be used to construct multi-soliton solutions. Two soliton interactions are also studied. The formulas for position shift of the kink and position and phase shifts of the breather are given. These quantities only depend on the pole positions of the dressing matrices.
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Mikhailov, A., Papamikos, G. and Wang, J. (2016). Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere. Letters in Mathematical Physics [Online] 106:973-996. Available at: http://dx.doi.org/10.1007/s11005-016-0855-5.
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Wang, J. (2015). Representations of sl(2,?) in category O and master symmetries. Theoretical and Mathematical Physics [Online] 184:1078-1105. Available at: http://link.springer.com/article/10.1007/s11232-015-0319-6.We show that the indecomposable sl(2,?)-modules in the Bernstein-Gelfand-Gelfand category O naturally arise for homogeneous integrable nonlinear evolution systems. We then develop a new approach called the O scheme to construct master symmetries for such integrable systems. This method naturally allows computing the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For new integrable equations such as a Benjamin-Ono-type equation, a new integrable Davey-Stewartson-type equation, and two different versions of (2+1)-dimensional generalized Volterra chains, we generate their conserved densities using their master symmetries.
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Mikhailov, A., Papamikos, G. and Wang, J. (2014). Darboux transformation with dihedral reduction group. Journal of Mathematical Physics [Online] 55:113507. Available at: http://dx.doi.org/10.1063/1.4901224.We construct the Darboux transformation with Dihedral reduction group for the 2-dimensional generalisation of the periodic Volterra lattice. The resulting Bäcklund transformation can be viewed as a nonevolutionary integrable differential difference equation. We also find its generalised symmetry and the Lax representation for this symmetry. Using formal diagonalisation of the Darboux matrix, we obtain local conservation laws of the system.
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Khanizadeh, F., Mikhailov, A. and Wang, J. (2013). Darboux transformations and recursion operators for differential-difference equations. Theoretical and Mathematical Physics [Online] 177:1606-1654. Available at: http://dx.doi.org/10.1007/s11232-013-0124-z.We review two concepts directly related to the Lax representations of integrable systems: Darboux transformations and recursion operators. We present an extensive list of integrable differential-difference equations with their Hamiltonian structures, recursion operators, nontrivial generalized symmetries, and Darboux-Lax representations. The new results include multi-Hamiltonian structures and recursion operators for integrable Volterra-type equations and integrable discretizations of derivative nonlinear Schrödinger equations such as the Kaup-Newell, Chen-Lee-Liu, and Ablowitz-Ramani-Segur (Gerdjikov-Ivanov) lattices. We also compute the weakly nonlocal inverse recursion operators.
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Beffa, G. and Wang, J. (2013). Hamiltonian evolutions of twisted polygons in RP^n. Nonlinearity [Online] 26:2515-2551. Available at: http://dx.doi.org/10.1088/0951-7715/26/9/2515.In this paper we find a discrete moving frame and their associated invariants along projective polygons in RP^n , and we use them to describe invariant evolutions of projective N-gons. We then apply a reduction process to obtain a natural Hamiltonian structure on the space of projective invariants for polygons, establishing a close relationship between the projective N-gon invariant evolutions and the Hamiltonian evolutions on the invariants of the flow. We prove that any Hamiltonian evolution is induced on invariants by an invariant evolution of N-gons—what we call a projective realization—and both evolutions are connected explicitly in a very simple way. Finally, we provide a completely integrable evolution (the Boussinesq lattice related to the lattice W3-algebra), its projective realization in RP2 and its Hamiltonian pencil. We generalize both structures to n-dimensions and we prove that they are Poisson, defining explicitly the n-dimensional generalization of the planar evolution (a discretization of the Wn-algebra). We prove that the generalization is completely integrable, and we also give its projective realization, which turns out to be very simple.
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Mansfield, E., Marà BeffaG. and Wang, J. (2013). Discrete Moving Frames and Discrete Integrable Systems. Foundations of Computational Mathematics [Online] 13:545-582. Available at: http://dx.doi.org/10.1007/s10208-013-9153-0.
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Wang, J. (2012). Recursion Operator of the Narita–Itoh–Bogoyavlensky Lattice. Studies in Applied Mathematics [Online] 129:309-327. Available at: http://dx.doi.org/10.1111/j.1467-9590.2012.00556.x.We construct a recursion operator for the family of Narita-Itoh-Bogoyavlensky infinite lattice equations using its Lax presentation and present their mastersymmetries and bi-Hamiltonian structures. We show that this highly nonlocal recursion operator generates infinitely many local symmetries.
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Mikhailov, A. and Wang, J. (2011). A new recursion operator for Adler’s equation in the Viallet form. Physics Letters A [Online] 375:3960-3963. Available at: http://dx.doi.org/10.1016/j.physleta.2011.09.018.For Adler?s equation in the Viallet form and Yamilov?s discretisation of the Krichever–Novikov equation we present new recursion and Hamiltonian operators. This new recursion operator and the recursion operator found in [A.V. Mikhailov, et al., Theor. Math. Phys. 167 (2011) 421, arXiv:1004.5346] satisfy the spectral curve associated with the equation.
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Mikhailov, A., Wang, J. and Xenitidis, P. (2011). Cosymmetries and Nijenhuis recursion operators for difference equations. Nonlinearity [Online] 24:2079-2097. Available at: http://dx.doi.org/10.1088/0951-7715/24/7/009.In this paper we discuss the concept of cosymmetries and co-recursion operators for difference equations and present a co-recursion operator for the Viallet equation. We also discover a new type of factorization for the recursion operators of difference equations. This factorization enables us to give an elegant proof that the pseudo-difference operator R presented in Mikhailov et al 2011 Theor. Math. Phys. 167 421-43 is a recursion operator for the Viallet equation. Moreover, we show that the operator R is Nijenhuis and thus generates infinitely many commuting local symmetries. The recursion operator R and its factorization into Hamiltonian and symplectic operators have natural applications to Yamilov's discretization of the Krichever-Novikov equation.
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Mikhailov, A., Wang, J. and Xenitidis, P. (2011). Recursion operators, conservation laws and integrability conditions for difference equations. Theoretical and Mathematical Physics [Online] 167:421-443. Available at: http://dx.doi.org/10.1007/s11232-011-0033-y.We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.
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Wang, J. (2010). The Hunter-Saxton equation: remarkable structures of symmetries and conserved densities. Nonlinearity 23:2009-2028.
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Wang, J. (2009). Lenard scheme for two-dimensional periodic Volterra chain. Journal of Mathematical Physics [Online] 50:23506. Available at: http://dx.doi.org/10.1063/1.3054921.We prove that for compatible weakly nonlocal Hamiltonian and symplectic operators, hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating Nijenhuis operators are weakly nonlocal or not. We construct a recursion operator of the two-dimensional periodic Volterra chain from its Lax representation and prove that it is a Nijenhuis operator. Furthermore we show that this system is a (generalized) bi-Hamiltonian system. Rather surprisingly, the product of its weakly nonlocal Hamiltonian and symplectic operators gives rise to the square of the recursion operator.
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Wang, J. (2009). Extension of integrable equations. Journal of Physics A: Mathematical and Theoretical [Online] 42:362004. Available at: http://dx.doi.org/10.1088/1751-8113/42/36/362004.In this communication, we show that there is general construction to produce non-evolutionary integrable equations from a given integrable evolutionary equation. To support the main theorem, a few examples are explicitly given.
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Hone, A. and Wang, J. (2008). Integrable peakon equations with cubic nonlinearity. Journal of Physics A: Mathematical and Theoretical [Online] 41. Available at: http://dx.doi.org/10.1088/1751-8113/41/37/372002.We present a new integrable partial differential equation found by Vladimir Novikov. Like the Camassa-Holm and Degasperis-Procesi equations, this new equation admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic. We give a matrix Lax pair for V Novikov's equation, and show how it is related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. Infinitely many conserved quantities are found, as well as a bi-Hamiltonian structure. The latter is used to obtain the Hamiltonian form of the finite-dimensional system for the interaction of N peakons, and the two-body dynamics (N = 2) is explicitly integrated. Finally, all of this is compared with some analogous results for another cubic peakon equation derived by Zhijun Qiao.
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Novikov, V. and Wang, J. (2007). Symmetry Structure of Integrable Nonevolutionary Equations. Studies in Applied Mathematics [Online] 119:393-428. Available at: http://dx.doi.org/10.1111/j.1467-9590.2007.00390.x.We study a class of evolutionary partial differential systems with two components related to second order (in time) nonevolutionary equations of odd order in spatial variable. We develop the formal diagonalization method in symbolic representation, which enables us to derive an explicit set of necessary conditions of existence of higher symmetries. Using these conditions we globally classify all such homogeneous integrable systems, i.e., systems which possess a hierarchy of infinitely many higher symmetries.
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Mikhailov, A., Novikov, V. and Wang, J. (2007). On classification of integrable nonevolutionary equations. Studies in Applied Mathematics [Online] 118:419-457. Available at: http://www.blackwell-synergy.com/doi/abs/10.1111/j.1467-9590.2007.00376.x.We study partial differential equations of second order (in time) that possess a hierarchy of infinitely many higher symmetries. The famous Boussinesq equation is a member of this class after the extension of the differential polynomial ring. We develop the perturbative symmetry approach in symbolic representation. Applying it, we classify the homogeneous integrable equations of fourth and sixth order (in the space derivative) equations, as well as we have found three new tenth-order integrable equations. To prove the integrability we provide the corresponding bi-Hamiltonian structures and recursion operators.
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Wang, J. (2006). On the structure of (2+1)-dimensional commutative and noncommutative integrable equations. Journal of Mathematical Physics [Online] 47:113508. Available at: http://dx.doi.org/10.1063/1.2375032.We develop the symbolic representation method to derive the hierarchies of (2+1)-dimensional integrable equations from the scalar Lax operators and to study their properties globally. The method applies to both commutative and noncommutative cases in the sense that the dependent variable takes its values in C or a noncommutative associative algebra. We prove that these hierarchies are indeed quasi-local in the commutative case as conjectured by Mikhailov and Yamilov [J. Phys. A 31, 6707 (1998)]. We propose a ring extension in the noncommutative case based on the symbolic representation. As examples, we give noncommutative versions of Kadomtsev-Petviashvili (KP), modified Kadomtsev-Petviashvili (mKP), and Boussinesq equations.
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Sanders, J. and Wang, J. (2006). Integrable systems in n-dimensional conformal geometry. Journal of Difference Equations and Applications [Online] 12:983-995. Available at: http://dx.doi.org/10.1080/10236190600986784.In this paper we show that if we write down the structure equations for the flow of the parallel frame of a curve embedded in a flat n-dimensional conformal manifold, this leads to two compatible Hamiltonian operators. The corresponding integrable scalar-vector equation is where is the standard Euclidean inner product of the vector with itself. These results are similar to those we obtained in the Riemannian case, implying that the method employed is well suited for the analysis of the connection between geometry and integrability.
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Mikhailov, A., Novikov, V. and Wang, J. (2005). Partially integrable nonlinear equations with one higher symmetry. Journal of Physics A: Mathematical and General [Online] 38:L337-L341. Available at: http://dx.doi.org/10.1088/0305-4470/38/20/L02.In this letter, we present a family of second order in time nonlinear partial differential equations, which have only one higher symmetry. These equations are not integrable, but have a solution depending on one arbitrary function.
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Sanders, J. and Wang, J. (2004). On the Integrability of Systems of second order Evolution Equations with two Components. Journal of Differential Equations [Online] 203:1-27. Available at: http://dx.doi.org/10.1016/j.jde.2004.04.010.This paper is devoted to classifying second order evolution equations with two components. Combining the symbolic method and number theory, we give the complete list of such homogeneous polynomial symmetry-integrable systems with non-zero diagonal linear terms. The technique is applicable for more general systems.
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Hone, A. and Wang, J. (2003). Prolongation algebras and Hamiltonian operators for peakon equations. Inverse Problems [Online] 19:129-145. Available at: http://dx.doi.org/10.1088/0266-5611/19/1/307.We consider a family of non-evolutionary partial differential equations, labelled by a single parameter b, all of which admit multi-peakon solutions. For the two special integrable cases, namely the Camassa-Holm and Degasperis-Procesi equations (b = 2 and 3), we explain how their spectral problems have reciprocal links to Lax pairs for negative flows, in the Korteweg-de Vries and Kaup-Kupershmidt hierarchies respectively. An analogous construction is presented in the case of the Sawada-Kotera hierarchy, leading to a new zero-curvature representation for the integrable Vakhnenko equation. We show how the two special peakon equations are isolated via the Wahlquist-Estabrook prolongation algebra method. Using the trivector technique of Olver, we provide a proof of the Jacobi identity for the non-local Hamiltonian structures of the whole peakon family. Within this class of Hamiltonian operators (also labelled by b), we present a uniqueness theorem which picks out the special cases b = 2, 3.
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Wang, J. and Sanders, J. (2003). Integrable systems in n-dimensional Riemannian geometry. Moscow Mathematical Journal 3:1369-1393.
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Wang, J. (2002). A List of 1 + 1 Dimensional Integrable Equations and Their Properties. Journal of Nonlinear Mathematical Physics [Online] 9 - Su:213-233. Available at: http://dx.doi.org/10.2991/jnmp.2002.9.s1.18.This paper contains a list of known integrable systems. It gives their recursion-, Hamiltonian-, symplectic- and cosymplectic operator, roots of their symmetries and their scaling symmetry.
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Olver, P., Sanders, J. and Wang, J. (2002). Ghost symmetries. Journal of Nonlinear Mathematical Physics [Online] 9:164-172. Available at: http://dx.doi.org/10.2991/jnmp.2002.9.s1.14.We introduce the notion of a ghost characteristic for nonlocal differential equations. Ghosts are essential for maintaining the validity of the Jacobi identity for the charateristics of nonlocal vector fields.
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Sanders, J. and Wang, J. (2002). On a family of operators and their Lie algebras. Journal of Lie Theory 12:503-514.An infinite family of differential operators is constructed. Each of these operators defines a Lie bracket and the operator is a homomorphism from the new Lie algebra to the standard Lie algebra. An interesting feature of these operators is that they factorize into first order operators with integer coefficients. This generalizes recent results of Zhiber and Sokolov.
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Beffa, G., Sanders, J. and Wang, J. (2002). On integrable systems in 3-dimensional Riemannian geometry. Journal of Nonlinear Science [Online] 12:143-167. Available at: http://dx.doi.org/10.1007/s00332-001-0472-y.In this paper we introduce a new infinite-dimensional pencil of Hamiltonian structures. These Poisson tensors appear naturally as the ones governing the evolution of the curvatures of certain flows of curves in 3-dimensional Riemannian manifolds with constant curvature. The curves themselves are evolving following arclength-preserving geometric evolutions for which the variation of the curve is an invariant combination of the tangent, normal, and binormal vectors. Under very natural conditions, the evolution of the curvatures will be Hamiltonian and, in some instances, bi-Hamiltonian and completely integrable.
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Sanders, J. and Wang, J. (2001). Integrable Systems and their Recursion Operators. Nonlinear Analysis: Theory, Methods & Applications [Online] 47:5213-5240. Available at: http://dx.doi.org/10.1016/S0362-546X(01)00630-7.In this paper we discuss the structure of recursion operators. we show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and co symmetries. this enables us to compute recursion operators more systematically. Under certain conditions (which hold for all examples known to us)Nijenhuis are well defined i.e, they give rise to hierarchies of infinitely many commuting symmetries of the operator. Moreover, the non local part of a Nijenhuis operator contains the candidates of roots and coroots.
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Sanders, J. and Wang, J. (2001). On recursion operators. Physica D: Nonlinear Phenomena [Online] 149:1-10. Available at: http://dx.doi.org/10.1016/S0167-2789(00)00188-3.We observe that application of a recursion operator of the Burgers equation does not produce the expected symmetries. This is explained by the incorrect assumption that D-x(-1), D-x = 1. We then proceed to give a method to compute the symmetries using the recursion operator as a first-order approximation. (C) 2001 Elsevier Science B.V. All rights reserved.
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Beukers, F., Sanders, J. and Wang, J. (2001). On integrability of systems of evolution equations. Journal of Differential Equations [Online] 172:396-408. Available at: http://dx.doi.org/10.1006/jdeq.2000.3859.We prove the conjecture, formulated in [BSW98], that almost all systems in the family[formula]have at most finitely many symmetries by using number theory. We list the nine exceptional cases when the systems do have infinitively many symmetries. For such systems, we give the recursive operators to generate their symmetries. We treat both 1the commutative and the noncommutative (or quantum) cases. This is the first example of a class of equations where such a classification has been possible.
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Sanders, J. and Wang, J. (2000). On the Integrability of Non-Polynomial Scalar Evolution Equations. Journal of Differential Equations [Online] 166:132-150. Available at: http://dx.doi.org/10.1006/jdeq.2000.3782.We show the existence of infinitely many symmetries for ?-homogeneous equations when ?=0. If the equation has one generalized symmetry, we prove that it has infinitely many and these can be produced by recursion operators. Identifying equations under homogeneous transformations, we find that the only integrable equations in this class are the Potential Burgers, Potential Modified Korteweg–de Vries, and Potential Kupershmidt Equations. We can draw some conclusions from these results for the case ?=?1 which, although theoretically incomplete, seem to cover the known integrable systems for this case.
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Olver, P. and Wang, J. (2000). Classification of Integrable One-Component Systems on Associative Algebras. Proceedings of the London Mathematical Society [Online] 81:566-586. Available at: http://dx.doi.org/10.1112/S0024611500012582.This paper is devoted to the complete classification of integrable one-component evolution equations whose field variable takes its values in an associative algebra. The proof that the list of non-commutative integrable homogeneous evolution equations is complete relies on the symbolic method. Each equation in the list has infinitely many local symmetries and these can be generated by its recursion (recursive) operator or master symmetry
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Sanders, J. and Wang, J. (1998). On the integrability of homogeneous scalar evolution equations. Journal of Differential Equations [Online] 147:410-434. Available at: http://dx.doi.org/10.1006/jdeq.1998.3452.We determine the existence of (infinitely many) symmetries for equations of the form
u(t) = u(k) + f(u, ..., u(k-1))
when they are lambda-homogeneous (with respect to the scaling u(k) lambda + k) with lambda > 0. Algorithms are given to determine whether a system has a symmetry (also independent of t and x). If it has one generalized symmetry, we prove it has infinitely many and these can be found using recursion operators or master symmetries. The method of proof uses the symbolic method and results From diophantine approximation theory. We list the 10 integrable hierarchies. The methods can in principle be applied to the lambda less than or equal to 0 cast. as we illustrate ibr one example with lambda = 0, which seems to be new. In principle they can also be used for systems of evolution equations, but so far this has only been demonstrated for one class of examples. -
Sanders, J. and Wang, J. (1998). Combining Maple and Form to decide on integrability questions. Computer Physics Communications [Online] 115:1-13. Available at: http://dx.doi.org/10.1016/S0010-4655(98)00122-2.We consider the existence problem of (infinitely many) symmetries for equations of the form u(t) = u(k) + F(u,..., u(k-1)) when they are lambda-homogeneous (with respect to the scaling u(k) bar right arrow mu(lambda+k)u(k)). This involves fairly large calculations which are carried out in a mixture of Form and Maple. We give an introduction to this mixed language style of programming, indicating how the choice of language will be determined by bottlenecks in the computation. We give some fairly complete results for the lambda = 0 case, which leads to more difficult computer algebra problems involving differential ideals. Finally we show some results extending the methods to cosymmetries.
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Beukers, F., Sanders, J. and Wang, J. (1998). One symmetry does not imply integrability. Journal of Differential Equations [Online] 146:251-260. Available at: http://dx.doi.org/10.1006/jdeq.1998.3426.We show that Bakirov's counter-example (which had been checked by computer algebra methods up to order 53) to the conjecture that one nontrivial symmetry of an evolution equation implies infinitely many is indeed a counter-example. To prove this we use thesymbolic methodof Gel'fand–Dikii andp-adic analysis. We also formulate a conjecture to the effect that almost all equations in the family considered by Bakirov have at most finitely many symmetries. This conjecture depends on the solution of a diophantine problem, which we explicitly state.
Book section
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Sanders, J. and Wang, J. (2009). Number Theory and the Symmetry Classification of Integrable Systems. In: Mikhailov, A. V. ed. Integrability. USA: Princeton University Press.
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Mikhailov, A., Novikov, V. and Wang, J. (2008). Symbolic representation and classification of integrable systems. In: MacCallum, M. A. and Mikhailov, A. V. eds. Algebraic Theory of Differential Equations. Cambridge: Cambridge university press, pp. 156-216.
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Hereman, W., Sanders, J., Sayers, J. and Wang, J. (2005). Symbolic Computation of Polynomial Conserved Densities, Generalized Symmetries, and Recursion Operators for Nonlinear Differential-Difference Equations. In: Group Theory and Numerical Analysis. Amer. Math. Soc., pp. 133-148. Available at: http://www.ams.org/bookstore?fn=20&arg1=crmpseries&item=CRMP-39.Algorithms for the symbolic computation of polynomial conserved densities, fluxes, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations are presented. In the algorithms we use discrete versions of the Frechet and variational derivatives and the Euler and homotopy operators.
The algorithms are illustrated for prototypical nonlinear polynomial lattices, including the Kac-van Moerbeke (Volterra) and Toda lattices. Results are shown for the modified Volterra and Ablowitz-Ladik lattices. -
Olver, P., Sanders, J. and Wang, J. (2001). Classification of symmetry-integrable evolution equations. In: Alan, C., Pavel, W., Colin, R., Robert, M. and Decio, L. eds. Bäcklund and Darboux Transformations. The Geometry of Solitons (Halifax, NS, 1999). Providence, RI: American Mathematical Society, pp. 363-372. Available at: http://bookshop.blackwell.co.uk/jsp/id/Backlund_and_Darboux_Transformations/9780821828038.This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Backlund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory. The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Backlund transformations of Painleve equations, to the dressing method and Backlund and Darboux transformations, and to the classical geometry of Backlund transformations and their applications to soliton theory.The second part contains original contributions that represent new developments in the theory and applications of these transformations. Both the introductory lectures and the original talks were presented at an International Workshop that took place in Halifax, Nova Scotia (Canada). This volume covers virtually all recent developments in the theory and applications of Backlund and Darboux transformations.
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Sanders, J. and Wang, J. (2001). On Integrability of Evolution Equations and Representation Theory. In: Joshua, L. and Thierry, R. eds. The Geometrical Study of Differential Equations. America: American Mathematical Society, pp. 85-99.This volume contains papers based on some of the talks given at the NSF-CBMS conference on 'The Geometrical Study of Differential Equations' held at Howard University (Washington, DC). The collected papers present important recent developments in this area, including the treatment of nontransversal group actions in the theory of group invariant solutions of PDEs, a method for obtaining discrete symmetries of differential equations, the establishment of a group-invariant version of the variational complex based on a general moving frame construction, the introduction of a new variational complex for the calculus of difference equations and an original structural investigation of Lie-Backlund transformations.The book opens with a modern and illuminating overview of Lie's line-sphere correspondence and concludes with several interesting open problems arising from symmetry analysis of PDEs. It offers a rich source of inspiration for new or established researchers in the field. This book can serve nicely as a companion volume to "Selected Topics in the Geometrical Study of Differential Equations", by Niky Kamran, in the "AMS" series, "CBMS Regional Conference Series in Mathematics
Conference or workshop item
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Wang, J. (2003). Generalized Hasimoto Transformation and vector Sine-Gordon equation. In: Abenda, S., Gaeta, G. and Walcher, S. eds. Symmetry and Perturbation Theory, SPT 2002. World Scientific, pp. 277-285.
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Sanders, J. and Wang, J. (2000). The symbolic method and cosymmetry integrability of evolution equations. In: Fiedler, B., Gröger, K. and Sprekels, J. eds. the International Conference on Differential Equations. Singapore: world scientific, pp. 824-831.
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Sanders, J. and Wang, J. (1998). On the classification of integrable systems. In: Desanto, J. ed. Fourth International Conference on Mathematical and Numerical Aspects of Wave Propagation. Philadelphia: SIAM, pp. 393-397.
Thesis
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Ashcroft, J. (2017). Topological Solitons and Their Dynamics.Topological solitons are particle-like solutions of nonlinear field equations with important applications in physics. This thesis presents four research projects concerning topological solitons and their dynamics. We investigate solitons in (1+1)- and (2+1)-dimensions, and develop numerical methods to obtain static solutions and simulate soliton scattering.
We first study kink collisions in a model with two scalar fields in the presence of false vacua. We find a variety of scattering outcomes depending on the initial velocity and vacuum structure. Kinks can either repel, form a true or false domain wall, annihilate, or collide and escape to infinity. These behaviours occur in alternating windows of initial velocity. When the kinks escape to infinity, there are a number of oscillations or ``bounces" before the kinks escape, and this bounce number is conserved in each of the windows.
In the second project we design new baby Skyrme models that do not require a potential term to allow topological soliton solutions. We raise the Skyrme and sigma terms to fractional powers, which enables us to evade Derrick's theorem. We calculate topological energy bounds for our models and numerically obtain minimal energy solutions for solitons of charge 1, 2, and 3. For each charge, the minimal energy solution is a ring.
The last two projects concern vortices in the Ginzburg-Landau model. In the first of these, we numerically investigate the scattering of multi-vortex rings. When two 2-vortex rings collide, there are two distinct scattering outcomes. In both cases, one pair of vortices will scatter at right angles and escape along the $y$-axis. The remaining two vortices will either form a bound state or escape along an axis after colliding a number of times.
Finally, we study vortices scattering with magnetic impurities of the form $\sigma(r)=ce^{-dr^2}$. An impurity will attract or repel a vortex depending on the coupling constant $\lambda$ and the parameters $c$ and $d$. We scatter critically coupled vortices with two different impurities and explore the relationship between the scattering angle and impact parameter. We also find that a 2-vortex ring will break up in a head-on collision with an impurity.
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