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The application of the Gardner method for generation of conservation laws to all the ABS equations is considered. It is shown that all the necessary information for the application of the Gardner method, namely B\"acklund transformations…
Recently, Quesne on the one side and on the other side Bougie, Gangopadhyaya and Mallow, have discovered new translational shape invariant potentials not present in previous classifications. By using ordinary B\"acklund--Darboux…
We find a B\"acklund transformation between 4-dimensional Mart\'{\i}nez Alonso -- Shabat and Ferapontov -- Khusnutdinova equations.
We extend the recent improvement of Roth's theorem on three term arithmetic progressions by Sanders to obtain similar results for the problem of locating non-trivial solutions to translation invariant linear equations in many variables in…
The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence…
In this article, we provide a modified argument for proving conditional stability for inverse problems of determining spatially varying functions in evolution equations by Carleman estimates. Our method needs not any cut-off procedures and…
We show that there exist some intimate connections between three unconventional Schr\"odinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space, respectively.…
Symmetry invariant local interaction of a many body system leads to global constraints. We obtain explicit forms of the global macroscopic condition assuring that at the microscopic level the evolution respects the overall symmetry.
We present a B\"acklund transformation (a discrete symmetry transformation) for the self-duality equations for supersymmetric gauge theories in N-extended super-Minkowski space ${\cal M}^{4|4N}$ for an arbitrary semisimple gauge group. For…
For the supersymmetric KdV equation, a proper Darboux transformation is presented. This Darboux transformation leads to the B\"{a}cklund transformation found early by Liu and Xie \cite{liu2}. The Darboux transformation and the related…
A two-parameters family of Backlund transformations for the classical elliptic Gaudin model is constructed. The maps are explicit, symplectic, preserve the same integrals as for the continuous flows and are a time discretization of each of…
We introduce a method of approximate nonclassical Lie-B\"acklund symmetries for partial differential equations with a small parameter and discuss applications of this method to finding of approximate solutions both integrable and…
A defining characteristic of the Kadomstev-Petviashvili (KP) model equation is that the well-posedness results are subject to the restriction that at all transverse positions, the mass $\int u \,dx = \text{constant independent of $y$}.$ In…
We develop a variational approach in order to study the qualitative properties of non-autonomous parabolic equations. Based on the method of product integrals, we discuss long-time behavior, invariance properties, and ultracontractivity of…
In this paper we derive two examples of fully-nonlinear symmetry-integrable evolution equations with algebraic nonlinearities, namely one class of 3rd-order equations and a 5th-order equation. To achieve this we study the equations'…
Using the Balitsky-Kovchegov (BK) equation as an explicit example, we show that nonlinear QCD evolution leads to an instability in the propagation toward the infrared of the gluon transverse momentum distribution, if one starts with a state…
The Hirota bilinear difference equation is generalized to discrete space of arbitrary dimension. Solutions to the nonlinear difference equations can be obtained via B\"acklund transformation of the corresponding linear problems.
We consider second-order evolution equations in an abstract setting with intermittently delayed/ not-delayed damping. We give sufficient conditions for asymptotic and exponential stability, improving and generalising our previous results…
Utilizing an appropriate ansatz to the wave function, we reproduce the exact bound-state solutions of the radial Schrodinger equation to various exactly solvable sextic anharmonic oscillator and confining perturbed Coulomb models in…
We introduce twisted permutation-equivariant GW-invariants, and compute them in terms of untwisted ones. The computation is based on Grothendieck-like RR formula corresponding to Adams' operations from K-theory to itself, and the result can…