Related papers: Solutions to the complex Korteweg-de Vries equatio…
We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final…
We consider the modified Korteweg-de Vries equation (mKdV) and prove that given any sum P of solitons and breathers of (mKdV) (with distinct velocities), there exists a solution p of (mKdV) such that p(t) -- P(t) $\rightarrow$ 0 when t…
A new matrix modified Korteweg-de Vries (mmKdV) equation with a $p\times q$ complex-valued potential matrix function is first studied via Riemann-Hilbert approach, which can be reduced to the well-known coupled modified Korteweg-de Vries…
The shape-preserving and shape-altering collisions of dark solitons are investigated in the complex modified Korteweg-de Vries equation. The obtained dark soliton solutions are classified into two distinct types, referred to as type-I and…
In this paper, we investigate the blow-up phenomenon of the $H^2$ norm of solutions to the inhomogeneous biharmonic Schrodinger equation in two distinct scenarios. First, we consider the case of negative energy, analyzing separately the…
The modification of simplest equation method to look for exact solutions of nonlinear partial differential equations is presented. Using this method we obtain exact solutions of generalized Korteweg-de Vries equation with cubic source and…
For a singular Liouville equation, it is plausible that a non-simple blowup phenomenon occurs around a quantized singular pole. The presence of complex blowup profiles of bubbling solutions presents substantial challenges in applications.…
We construct examples of blowup from smooth data for complex-valued solutions to linear uniformly parabolic equations in dimension $n \geq 2$, which are exactly as irregular as parabolic energy estimates allow.
We present a numerical study of solutions to the generalized Kadomtsev-Petviashvili equations with critical and supercritical nonlinearity for localized initial data with a single minimum and single maximum. In the cases with blow-up, we…
We study positive blowing-up solutions of the system: $$u_{t}-\delta\Delta u=v^p,\,\,\, v_{t}-\Delta v=u^{q},$$ as well as of some more general systems. For any $p,\,q>1$, we prove single-point blow-up for any radially decreasing, positive…
We discuss integrable extensions of real nonlinear wave equations with multi-soliton solutions, to their bicomplex, quaternionic, coquaternionic and octonionic versions. In particular, we investigate these variants for the local and…
Following our previous paper in the radial case, we consider blow-up type II solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the…
We show that the Helmholtz equation describing the propagation of transverse electric waves in a Kerr slab with a complex linear permittivity and a complex Kerr coefficient admits blow-up solutions provided that the real part of the Kerr…
In this note we try to understand the blow-up of solutions to Nakao's problem by using nonlinear ordinary differential inequalities.
Quaternion-valued solutions to the non-commutative KdV equation are produced using determinants. The solutions produced in this way are (breather) soliton solutions, rational solutions, spatially periodic solutions and hybrids of these…
Blowing-up solutions of Klein-Gordon equations with gauge variant semilinear terms are considered in Friedmann-Lema\'itre-Robertson-Walker spacetimes. Effects of spatial expansion or contraction on the solutions are studied through the…
Taking the example of Koretweg--de Vries equation, it is shown that soliton solutions need not always be the consequence of the trade-off between the nonlinear terms and the dispersive term in the nonlinear differential equation. Even the…
This work studies nonnegative solutions for the Cauchy, Neumann, and Dirichlet problems of a logistic type reaction-diffusion equation. The finite time blowup results for nonnegative solutions under various restrictions on the coefficients…
In this work we consider the focusing, energy-critical wave equation in 3D radial case. It has been verified that any global or type II blow-up solution decomposes into a superposition of several decoupled grounds states, a free wave and a…
In this paper we present a vectorial Darboux transformation, in terms of ordinary determinants, for the supersymmetric extension of the Korteweg-de Vries equation proposed by Manin and Radul. It is shown how this transformation reduces to…