相关论文: Solutions to the complex Korteweg-de Vries equatio…

We consider the modified Korteweg-de Vries equation. Given a self-similar solution, and a subcritical perturbation of any size, we prove that there exists a unique solution to the equation which behaves at blow-up time as the self-similar…

For the quintic, mass critical generalized Korteweg-de Vries equation, for any $\nu \in (\frac{1}{2}, 1)$, we prove the existence of solutions in the energy space that blow up in finite time $T>0$ with the blow-up rate $\|\partial_x…

A new approach to double-sub equation method is introduced to construct novel solutions for the nonlinear partial differential equations. It is applied to the Korteweg-de Vries (KdV) equation and yields new complexiton solutions of both the…

Complexiton solutions (or complexitons for short) are exact solutions newly introduced to integrable equations. Starting with the solution classification for a linear differential equation, the Korteweg-de Vries equation and the Toda…

The real, nonsingular elliptic solutions of the Korteweg-deVries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This…

The Korteweg-de Vries equation is a fundamental nonlinear equation that describes solitons with constant velocity. On the contrary, here we show that this equation also presents accelerated wavepacket solutions. This behavior is achieved by…

We formulate the $N$ soliton solution of the Wadati-Konno-Ichikawa equation that is determined by purely algebraic equations. Derivation is based on the matrix Riemann-Hilbert problem. We give examples of one soliton solution that include…

Certain explicit solutions to the Korteweg-de Vries equation in the first quadrant of the $xt$-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational…

The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws…

This paper deals with blow-up for the complex-valued semilinear wave equation with power nonlinearity in dimension 1. Up to a rotation of the solution in the complex plane, we show that near a characteristic blow-up point, the solution…

Two new methods of numerical integration of Cauchy problems for ODEs with blow-up solutions are described. The first method is based on applying a differential transformation, where the first derivative (given in the original equation) is…

We consider the mass-critical generalized Korteweg--de Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this…

This is a continuation of Ref.[1](arXiv:nlin.SI/0603008). In the present paper we review solutions to the modified Korteweg-de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation…

We provide a detailed numerical study of various issues pertaining to the dynamics of the Burgers equation perturbed by a weak dispersive term: blow-up in finite time versus global existence, nature of the blow-up, existence for "long"…

We provide explicit criteria for blow-up solutions of autonomous ordinary differential equations. Ideas are based on the quasi-homogeneous desingularization (blowing-up) of singularities and compactifications of phase spaces, which suitably…

A proper bilinear form is proposed for the N=1 supersymmetric modified Korteweg-de Vries equation. The bilinear B\"{a}cklund transformation of this system is constructed. As applications, some solutions are presented for it.

This paper constructs the $N$-fold Darboux transformation (DT) for the vector complex modified Korteweg-de Vries (vcmKdV) equation and presents its determinant representation. Utilizing the DT and multi-fold eigenvalue degeneracy, we derive…

The soliton resolution for the focusing modified Korteweg-de vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation…

In this paper we develop two different types of criteria for the finite time blow-up solutions to the combined nonlinear Schr\"odinger equation in 1D. The first one is a negative energy criterion developed for triple combined nonlinearity…

In this paper, we study coupled complex modified Korteweg-de Vries (ccmKdV) equation by combining the Hirota's method and the Kadomtsev-Petviashvili (KP) reduction method. First, we show that the bilinear form of the ccmKdV equation under…