TY - GEN A1 - Kiselev, A. V. A1 - Wolf, Thomas T1 - Classification of integrable super-systems using the SsTools environment N2 - A classification problem is proposed for supersymmetric %scaling\/-\/in\-va\-ri\-ant evolutionary PDE that satisfy the assumptions of nonlinearity, nondegeneracy, and homogeneity. Four classes of nonlinear coupled boson\/-\/fermion systems are discovered under the weighting assumption $|f|=|b|=|D_t|=\oh$. The syntax of the \Reduce\ package \SsTools, which was used for intermediate computations, and the applicability of its procedures to the calculus of super\/-\/PDE are described. T3 - ZIB-Report - 10-17 KW - Integrable super-systems KW - symmetries KW - recursions KW - classifiaction Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11792 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas A1 - Efimovskaya, Olya V. T1 - Classification of integrable quadratic Hamiltonians on e(3) N2 - Linear Poisson brackets on e(3) typical of rigid body dynamics are considered. All quadratic Hamiltonians of Kowalevski type having additional first integral of fourth degree are found. Quantum analogs of these Hamiltonians are listed. T3 - ZIB-Report - 10-18 KW - Linear Poisson brackets on e(3) KW - rigid body dynamics KW - Kowalevski type Hamiltonians KW - first integrals Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11807 SN - 1438-0064 ER - TY - GEN A1 - Kiselev, Arthemy V. A1 - Wolf, Thomas T1 - On weakly non-local, nilpotent, and super-recursion operators for N=1 super-equations N2 - We consider nonlinear, scaling-invariant $N=1$ boson$+$fermion supersymmetric systems whose right-hand sides are homogeneous differential polynomials and satisfy some natural assumptions. We select the super-systems that admit infinitely many higher symmetries generated by recursion operators; we further restrict ourselves to the case when the dilaton dimensions of the bosonic and fermionic super-fields coincide and the weight of the time is half the weight of the spatial variable. We discover five systems that satisfy these assumptions; one system is transformed to the purely bosonic Burgers equation. We construct local, nilpotent, triangular, weakly non-local, and super-recursion operators for their symmetry algebras. T3 - ZIB-Report - 05-52 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8850 ER - TY - GEN A1 - Kiselev, Arthemy V. A1 - Wolf, Thomas T1 - Supersymmetric representations and integrable super-extensions of the Burgers and Boussinesq equations N2 - New evolutionary supersymmetric systems whose right-hand sides are homogeneous differential polynomials and which possess infinitely many higher symmetries are constructed. Their intrinsic geometry (symmetries, conservation laws, recursion operators, Hamiltonian structures, and exact solutions) is analyzed by using algebraic methods. A supersymmetric $N=1$ representation of the Burgers equation is obtained. An $N=2$ KdV-component system that reduces to the Burgers equation in the diagonal $N=1$ case $\theta^1=\theta^2$ is found; the $N=2$ Burgers equation admits and $N=2$ modified KdV symmetry. A one\/-\/parametric family of $N=0$ super\/-\/systems that exte nd the Burgers equation is described; we relate the systems within this family with the Burgers equation on associative algebras. A supersymmetric boson$+$fermion representation of the dispersionless Boussinesq equation is investigated. We solve this equation explicitly and construct its integrable deformation that generates two infinite sequences of the Hamiltonians. The Boussinesq equation with dispersion is embedded in a one-parametric family of two-component systems with dissipation. We finally construct a three-parametric supersymmetric system that incorporates the Boussinesq equation with dispersion and dissipation but never retracts to it for any values of the parameters. T3 - ZIB-Report - 05-53 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8869 ER - TY - JOUR A1 - Odesskii, Alexander A1 - Wolf, Thomas T1 - Compatible quadratic Poisson brackets related to a family of elliptic curves JF - Journal of Geometry and Physics Y1 - 2013 SN - 0393-0440 VL - 63 IS - 0 SP - 107 EP - 117 ER - TY - JOUR A1 - Anco, Stephen A1 - MacNaughton, Steven A. A1 - Wolf, Thomas T1 - Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions JF - Journal of Mathematical Physics Y1 - 2012 U6 - https://doi.org/10.1063/1.4711814 VL - 53 IS - 5 ER - TY - JOUR A1 - Wolf, Thomas A1 - Schrüfer, Eberhard A1 - Webster, Kenneth T1 - Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs JF - Programming and Computer Software Y1 - 2012 U6 - https://doi.org/10.1134/S0361768812020065 SN - 0361-7688 VL - 38 IS - 2 SP - 73 EP - 83 ER - TY - JOUR A1 - Euler, Marianna A1 - Euler, Norbert A1 - Wolf, Thomas T1 - The Two-Component Camassa-Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws JF - Journal of Nonlinear Mathematical Physics N2 - Recently, Holm and Ivanov, proposed and studied a class of multi-component generalizations of the Camassa–Holm equations [D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys A: Math. Theor. 43 (2010) 492001 (20pp)]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete set of first-order integrating factors for the systems in Cauchy–Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2). Y1 - 2013 U6 - https://doi.org/10.1142/S1402925112400025 VL - 19 IS - 1 SP - 13 EP - 22 ER - TY - JOUR A1 - Anco, Stephen A1 - Mohiuddin, Mohammad A1 - Wolf, Thomas T1 - Traveling waves and conservation laws for complex mKdV-type equations Y1 - 2012 U6 - https://doi.org/10.1016/j.amc.2012.06.061 VL - 219 IS - 2 SP - 679 EP - 698 ER - TY - JOUR A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction JF - SIGMA 7 N2 - A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this group-foliation reduction method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation. Y1 - 2011 U6 - https://doi.org/10.3842/SIGMA.2011.066 ER -