TY - GEN A1 - Anco, Stephen A1 - Wolf, Thomas T1 - Some classifications of hyperbolic vector evolution equations N2 - Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several $O(N)$-invariant classes of hyperbolic equations $Utx=f(U,Ut,Ux)$ for an $N$-component vector $U(t,x)$ are considered. In each class we find all scaling-homogeneous equations admitting a higher symmetry of least possible scaling weight. Sigma model interpretations of these equations are presented. T3 - ZIB-Report - 05-08 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8429 ER - TY - GEN A1 - Anco, Stephen A1 - Bluman, George A1 - Wolf, Thomas T1 - Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Admitted Conservation Laws N2 - An algorithmic method using conservation law multipliers is introduced that yields necessary and sufficient conditions to find invertible mappings of a given nonlinear PDE to some linear PDE and to construct such a mapping when it exists. Previous methods yielded such conditions from admitted point or contact symmetries of the nonlinear PDE. Through examples, these two linearization approaches are contrasted. T3 - ZIB-Report - 08-11 KW - conservation laws KW - linearization KW - symmetries Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10652 SN - 1438-0064 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 - TY - JOUR A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions JF - Journal of Mathematical Analysis and Applications Y1 - 2011 U6 - https://doi.org/10.1016/j.jmaa.2011.01.073 VL - 379 IS - 2 SP - 748 EP - 763 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 - GEN A1 - Anco, Stephen A1 - Mohiuddin, Mohammad A1 - Wolf, Thomas T1 - Travelling waves and conservation laws for complex mKdV-type equations N2 - Travelling waves and conservation laws are studied for a wide class of $U(1)$-invariant complex mKdV equations containing the two known integrable generalizations of the ordinary (real) mKdV equation. The main results on travelling waves include deriving new complex solitary waves and kinks that generalize the well-known mKdV $\sech$ and $\tanh$ solutions. The main results on conservation laws consist of explicitly finding all 1st order conserved densities that yield phase-invariant counterparts of the well-known mKdV conserved densities for momentum, energy, and Galilean energy, and a new conserved density describing the angular twist of complex kink solutions. T3 - ZIB-Report - 12-34 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16124 SN - 1438-0064 ER - TY - GEN A1 - Anco, Stephen A1 - MacNaughton, Steven A. A1 - Wolf, Thomas T1 - Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions N2 - Symmetries and conservation laws are studied for two classes of physically and analytically interesting radial wave equations with power nonlinearities in multi-dimensions. The results consist of two main classifications: all symmetries of point type and all conservation laws of a general energy-momentum type are explicitly determined, including those such as dilations, inversions, similarity energies and conformal energies that exist only for special powers or dimensions. In particular, all variational cases (when a Lagrangian formulation exists) and non-variational cases (when no Lagrangian exists) for these wave equations are considered. As main results, the classification yields generalized energies and radial momenta in certain non-variational cases, which are shown to arise from a new type of Morawetz dilation identity that produces conservation laws for each of the two classes of wave equations in a different way than Noether's theorem. T3 - ZIB-Report - 12-33 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16111 SN - 1438-0064 ER - TY - GEN A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions N2 - A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in $n>1$ dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors. T3 - ZIB-Report - 11-26 KW - semilinear heat equation, similarity reduction, exact solutions, group foliation, symmetry Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13213 ER - TY - GEN A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Exact solutions of nonlinear partial differential equations by the method of group foliation reduction 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 is based on group foliation reduction and employs 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 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. T3 - ZIB-Report - 11-25 KW - semilinear heat equation, similarity reduction, exact solutions, group foliation, symmetry Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13171 ER -