Related papers: Integrability study of a four-dimensional eighth-o…
We study the $(3+1)$-dimensional eight-order nonlinear wave equation associated with the principal representation of the exceptional affine Lie algebra $E_6^{(1)}$, which was constructed by Kac and Wakimoto and stated that $N$-soliton…
The integrability of a four-dimensional sixth-order bilinear equation associated with the exceptional affine Lie algebra $D_4^{(1)}$ is studied by means of the singularity analysis. This equation is shown to pass the Painlev\'{e} test in…
An eighth-order equation in (3+1)-dimension is studied for its integrability. Its symmetry group is shown to be infinite-dimensional and is checked for Virasoro like structure. The equation is shown not to have Painlev$\acute{\rm e}$…
A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg - de Vries equation with a source. A Lax representation and a Backlund self-transformation are found…
We perform a detailed study on the integrability of the Benney-Lin and KdV-Kawahara equations by using the Lie symmetry analysis and the singularity analysis. We find that the equations under our consideration admit integrable…
Irreducible nonzero level modules with finite-dimensional weight spaces are studied for non-twisted affine Lie superalgebras. A complete classification is obtained for superalgebras A(m,n)^ and C(n)^. In other cases the classification…
First, we prove the Kac-Wakimoto conjecture on modular invariance of characters of exceptional affine W-algebras. In fact more generally we prove modular invariance of characters of all lisse W-algebras obtained through Hamiltonian…
In this paper we study the equation $$ w^{(4)} = 5 w" (w^2 - w') + 5 w (w')^2 - w^5 + (\lambda z + \alpha)w + \gamma, $$ which is one of the higher-order Painlev\'e equations (i.e., equations in the polynomial class having the Painlev\'e…
We show that the new third-order complex nonlinear wave equation, introduced recently by M\"{u}ller-Hoissen [arXiv:2202.04512], does not pass the Painlev\'{e} test for integrability. We find two reductions of this equation, one integrable…
We consider a system of equations for the description of nonlinear waves in a liquid with gas bubbles. Taking into account high order terms with respect to a small parameter, we derive a new nonlinear partial differential equation for the…
$K^2 S^2 T [5]$ recently derived a new 6$^{th}$-order wave equation $KdV6$: $(\partial^2_x + 8u_x \partial_x + 4u_{xx})(u_t + u_{xxx} + 6u_x^2) = 0$, found a linear problem and an auto-B${\ddot{\rm{a}}}$ckclund transformation for it, and…
In this paper, we investigate in detail a superintegrable extension of the singular harmonic oscillator whose wave functions can be expressed in terms of exceptional Jacobi polynomials. We show that this Hamiltonian admits a fourth-order…
A new class of infinite-dimensional Lie algebras given a name of Lax operator algebras, and the related unifying approach to finite-dimensional integrable systems with spectral parameter on a Riemann surface, such as Calogero--Moser and…
We prove that any classical affine W-algebra W(g,f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of Lax type equations. This is based on the theories of…
We derive a five-dimensional nonlinear first order matrix PDE which is a generalization of the completely integrable (2+1)-dimensional $N$-wave equation. Similar to the $\bar\partial$-problem, our algorithm is based on the linear integral…
We discuss Lie algebras of the Lie symmetry groups of two generically non-integrable equations in one temporal and two space dimensions arising in different contexts. The first is a generalization of the KP equation and contains 9 arbitrary…
A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlev\'e test for integrability well, and its 4$\times $4 Lax pair with two spectral parameters is found. The results…
Spectral method related to Lame equation with finite-gap potential is used to study the optical cascading equations. These equations are known not to be integrable by inverse scattering method. Due to "partial integrability" two-gap…
We prove that all classical affine W-algebras W(g,f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G_2,…
It is well known that the linear stability of solutions of partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the…