We propose integrable discretizations of derivative nonlinear Schrödinger (DNLS) equations such as the Kaup–Newell equation, the Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reduc...

Recently, many nonlinear coupled evolution equations, such as the coupled Korteweg–de Vries (KdV) equation, the coupled Boussinesq equation, and the coupled mKdV equation, appear in scientific applications [1 – 13]. The coupled evolution equations attracted a considerable research work in the literature. The aims of these works have been the determination of soliton solutions and the proof of c...

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all L 2-based Sobolev spaces H s where local well-posedness is presently known, apart from the H 1 4 (R) endpoint for mKdV. The result for KdV relies on a new method for co...

By using solutions of an ordinary differential equation, an auxiliary equationmethod is described to seek exact solutions of variablecoefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for...

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The reverse space-time nonlocal complex modified Kortewewg–de Vries (mKdV) equation is investigated by using the consistent tanh expansion (CTE) method. According to CTE method, a nonauto-Bäcklund transformation theorem of mKdV obtained. interactions between one kink soliton and other different nonlinear excitations are constructed via theorem. By selecting cnoidal periodic waves, interaction w...

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...

کچ ی هد هکبش هکبش رگسح یاه رگسح زا یا هنیزه مک و دودمح یژرنا اب ،کچوک یاه دنتسه یا یم شخپ رظن دروم هقطنم رد هک هداد ات دنوش عجم طیمح زا ار ییاه شزادرپ یارب و هدونم یروآ یزکرم هرگ هب ١ دننک لاسرا . م نینچهم ی هب و هداد صیخشت طیمح رد ار یدادخر عوقو دنناوت دنناسرب یزکرم هرگ علاطا . هکبش هنوگنیا ریخا یالهاس رد هنیمز رد اه یاهدربراک فلتمخ یاه هدرک ادیپ یدایز دنا . زا دنترابع اهدربراک نیا هلجم زا...

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...