We establish a new a priori bound for L-bounded sequences of solutions to the mKdV equations on the torus. This first enables us to construct weak solutions in L for this equation and to check that the ”solutions” constructed by Kappeler and Topalov in the defocusing case satisfy the equation in some weak sense. In a second time, we prove that the solution-map associated with the mKdV and the K...

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces b H s (R) defined by the norm ‖v0‖ b Hr s (R) := ‖〈ξ〉 b v0‖Lr′ ξ , 〈ξ〉 = (1 + ξ) 1 2 , 1 r + 1 r = 1. Local well-posedness for the jth equation is shown in the parameter range 2 ≥ r > 1, s ≥ 2j−1 2r . The proof uses an appropriate variant of the Fourier restriction norm method. A coun...

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of di↵usion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-di↵usion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A r...

In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in some half-strip (0 < x < ∞) × [0, T ], or in the quarter plane (0 < x < ∞) × (0 < t < ∞). We suppose that this solution has a C∞ initial function vanishing as ...

We derive a zero-curvature formalism for a combined sine-Gordon (sG) and modi-ed Korteweg{de Vries (mKdV) equation which yields a local sGmKdV hierarchy. In complete analogy to other completely integrable hierarchies of soliton equations, such as the KdV, AKNS, and Toda hierarchies, the sGmKdV hierarchy is recursively constructed by means of a fundamental polynomial formalism involving a spectr...

We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) RmKdV mKdV), are connected through a transformation that combines both Miura transformations. Lax pairs straightforwardly obtained by means aforementioned have also identified classical Lie symmetries RmKdV. Non-trivial similari...

An explicit construction of solutions of the modified Korteweg-de Vries equation given a solution of the (ordinary) Korteweg-de Vries equation is provided. Our theory is based on commutation methods (i.e., N = 1 supersymmetry) underlying Miura's transformation that links solutions of the two evolution equations. In connection with the extensively studied Korteweg-de Vries (KdV-) equation its co...

We show that a number of nonlocal nonlinear equations, including the Ablowitz–Musslimani and Yang variant Schrödinger (NLS) equation, modified Korteweg de Vries (mKdV) Hirota admit novel kinklike pulselike superposed periodic solutions. Furthermore, we mKdV equation also admits (hyperbolic) kink–antikink solution. In addition, while NLS complex parity-time reversal-invariant kink pulse solution...

Recent advances in the numerical solution of Riemann–Hilbert problems allow for the implementation of a Cauchy initial value problem solver for the Korteweg–de Vries equation (KdV) and the defocusing modified Korteweg–de Vries equation (mKdV), without any boundary approximation. Borrowing ideas from the method of nonlinear steepest descent, this method is demonstrated to be asymptotically accur...

New solutions of some important partial differential equations are obtained using the first integral method. The efficiency of the method is demonstrated by applying it for the linear Klein-Gordan equation, MKdV equation, Burgers’ equation in two and three dimensions.