scholarly journals Initial–boundary problems for the vector modified Korteweg–de Vries equation via Fokas unified transform method

2016 ◽  
Vol 440 (2) ◽  
pp. 578-596 ◽  
Author(s):  
Huan Liu ◽  
Xianguo Geng
Keyword(s):  
Vries Equation ◽  
Initial Boundary ◽  
Transform Method ◽  
Boundary Problems ◽  
De Vries ◽  
Unified Transform Method

Asymptotics of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane

2020 ◽  
pp. 1-46
Author(s):  
Nan Liu ◽  
Boling Guo
Keyword(s):  
Large Time ◽  
Vries Equation ◽  
Large Time Behavior ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Time Behavior ◽  
Transform Method ◽  
Quarter Plane ◽  
De Vries ◽  
Fifth Order

The large-time behavior of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane is established. Our approach uses the unified transform method of Fokas and the nonlinear steepest descent method of Deift and Zhou.

Generalised Dirichlet to Neumann maps for linear dispersive equations on half-line

2017 ◽  
Vol 164 (2) ◽  
pp. 297-324
Author(s):  
ATHANASSIOS S. FOKAS ◽  
ZIPENG WANG
Keyword(s):  
Initial Boundary ◽  
Half Line ◽  
Unknown Boundary ◽  
Dispersive Equations ◽  
Transform Method ◽  
Boundary Problems ◽  
Linear Evolution ◽  
Explicit Representations ◽  
Essential Set

AbstractA large class of initial-boundary problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit representations are presented for the generalised Dirichlet to Neumann maps. Namely, the determination of the unknown boundary values when an essential set of initial and boundary data is given.

scholarly journals The unified transform method for the Sasa–Satsuma equation on the half-line

2013 ◽  
Vol 469 (2159) ◽  
pp. 20130068 ◽  
Author(s):  
Jian Xu ◽  
Engui Fan
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Half Line ◽  
Transform Method ◽  
Global Relation ◽  
Unified Transform Method ◽  
Initial Boundary Value ◽  
Dirichlet To Neumann Map

We implement the unified transform method to the initial-boundary value (IBV) problem of the Sasa–Satsuma equation on the half line. In addition to presenting the basic Riemann–Hilbert formalism, which linearizes this IBV problem, we also analyse the associated general Dirichlet to Neumann map using the so-called global relation.

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