Nonlinear Sciences

Self-similar curves arise naturally as the tension-free equilibrium states of conformally invariant bending energies. The simplest example is the Möbius invariant conformal arc-length on planar curves, dependent on the Frenet curvature κ through its first derivative with respect to arc-length. There are four conserved currents associated with this invariance: the tension and torque associated with Euclidean invariance, as well as scalar and vector currents reflecting invariance under scaling and special conformal transformations respectively. If the tension vanishes, all equilibrium states are self-similar: in the case of conformal arc-length, these are logarithmic spirals with no internal structure. More generally, the tension-free states are logarithmic spirals decorated with a repeating self-similar internal structure. Here it will be shown how the conservation laws can be used to construct these curves, while also endowing their geometry with a mechanical interpretation. The scaling current and the torque together provide a scale-invariant ode for the dimensionless variable κ ′ / κ 2 , which captures the internal structure of the spiral. For conformal arc-length it is constant. In tension-free states, the special conformal current vanishes. Its projections along orthogonal directions determine directly the distance from the spiral apex locally in terms of the curvature. The quadratic Casimir invariant of the Möbius group can be cast in terms of the four currents, none of which itself is invariant. For conformal arc-length, this is identified as the conformal curvature (the Schwarzian derivative of the Frenet curvature); it is constant along equilibrium curves.

In this paper, we examine the space of initial values for integrable lattice equations, which are lattice equations classified by Adler {\em et al} (2003), known as ABS equations. By considering the map which iterates the solution along particular directions on the lattice, we perform resolutions of singularities for several examples of ABS equations for the first time. Our geometric observations lead to new Miura transformations and reductions to ordinary difference equations.

We consider vector Non-linear Schrodinger Equation(NLSE) with balanced loss-gain(BLG), linear coupling(LC) and a general form of cubic nonlinearity. We use a non-unitary transformation to show that the system can be exactly mapped to the same equation without the BLG and LC, and with a modified time-modulated nonlinear interaction. The nonlinear term remains invariant, while BLG and LC are removed completely, for the special case of a pseudo-unitary transformation. The mapping is generic and may be used to construct exactly solvable autonomous as well as non-autonomous vector NLSE with BLG. We present an exactly solvable two-component vector NLSE with BLG which exhibits power-oscillation. An example of a vector NLSE with BLG and arbitrary even number of components is also presented.

The paper describes a number of simple but quite effective methods for constructing exact solutions of PDEs, that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat/diffusion equations, wave type equations, Klein--Gordon type equations, hydrodynamic boundary layer equations, Navier--Stokes equations, and some other PDEs. Apart from exact solutions to `ordinary' PDEs, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t) , these equations contain the same function at a past time, w=u(x,t?��? , where ?>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which in addition to the unknown u=u(x,t) , also contain the same functions with dilated or contracted arguments, w=u(px,qt) , where p and q are scaling parameters.

In this paper, an extended nonlinear Schrodinger equation with higher-order that includes fifth-order dispersion with matching higher-order nonlinear terms is investigated under zero boundary condition at infinity. Carrying out the spectral analysis, a kind of matrix Riemann-Hilbert problem is formulated on the real axis. Then on basis of the resulting matrix Riemann-Hilbert problem under restriction of no reflection, multiple soliton solutions of the extended nonlinear Schrodinger equation are generated explicitly.

An explicit solution formula for the matrix modified KdV equation is presented, which comprises the solutions given in Ref. 7 (S. Carillo, M. Lo Schiavo, and C. Schiebold. Matrix solitons solutions of the modified Korteweg-de Vries equation. In: Nonlinear Dynamics of Structures, Systems and Devices, edited by W. Lacarbonara, B. Balachandran, J. Ma, J. Tenreiro Machado, G. Stepan (Springer, Cham, 2020), pp. 75-83). In fact, the solutions in Ref.7 are part of a subclass studied in detail by the authors in a forthcoming publication. Here several solutions beyond this subclass are constructed and discussed with respect to qualitative properties.

We explore the connection between the equations describing Sisyphus dynamics and the generic Liénard type or Liénard equation from the viewpoint of branched Hamiltonians. The former provides the appropriate setting for classical time crystal being derivable from a higher order Lagrangian. However it appears the equations of Sisyphus dynamics have a close relation with the Liénard-II equation when expressed in terms of the `velocity' variable. Another interesting feature of the equations of Sisyphus dynamics is the appearance of velocity dependent "mass function" in contrast to the more commonly encountered position dependent mass. The consequences of such mass functions seem to have connections to cosmological time crystals .

We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless partial differential systems based on their contact Lax pairs and the related R -matrix theory for the Lie algebra of functions with respect to the contact bracket. We discuss various kinds of Lax representations for such systems, in particular, linear nonisospectral contact Lax pairs and nonlinear contact Lax pairs as well as the relations among the two. Finally, we present a large number of examples with finite and infinite number of dependent variables, as well as the reductions of these examples to lower-dimensional integrable dispersionless systems.

We introduce an equation defined on a multi-dimensional lattice, which can be considered as an extension to the coprimeness-preserving discrete KdV like equation in our previous paper. The equation is also interpreted as a higher-dimensional analogue of the Hietarinta-Viallet equation, which is famous for its singularity confining property while having an exponential degree growth. As the main theorem we prove the Laurent and the irreducibility properties of the equation in its "tau-function" form. From the theorem the coprimeness of the equation follows. In Appendix we review the coprimeness-preserving discrete KdV like equation whichis a base equation for our main system and prove the properties such as the coprimeness.

We propose a new integrable coupled dispersionless equation with self-consistent sources (CDESCS). We obtain the Lax pair and the equivalent generalized Heisenberg ferromagnet equation (GHFE), demonstrating its integrability. Specifically, we explore the geometry of these equations. Last, we consider the relation between the motion of curves/surfaces and the CDESCS and the GHFE.