Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations

415 Pages·2019·4.049 MB·\415

by  Pham Loi Vu (Author)

Inverse Scattering
Problems and Their
Application to
Nonlinear Integrable
Equations

Inverse Scattering
Problems and Their
Application to
Nonlinear Integrable
Equations
Pham Loi Vu
Institute of Mechanics - Vietnam Academy of Science
and Technology
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Dedicated to the loving memory of
my late son, Pham Vu Nam (1975-2001), and
younger brother, Pham Quang Dien (1941-1999)

Contents
Acronyms xiii
Preface xv
Author xvii
Introduction xix
1 Inverse scattering problems for systems of first-order ODEs
on a half-line 1
1.1 The inverse scattering problem on a half-line with a potential
non-self-adjoint matrix . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 The representation of the solution of system (1.7). . . 4
1.1.2 The Jost solutions of system (1.7) . . . . . . . . . . . 8
1.1.3 The scattering function S(λ) and non-real eigenvalues 8
1.1.4 Connection between the analytic solution and Jost
solutions. . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.5 The scattering data . . . . . . . . . . . . . . . . . . . 20
1.1.6 Derivation of systems of fundamental equations . . . . 22
1.1.7 The estimates for the functions f(−x) and g(x) . . . . 24
1.1.8 The unique solvability of systems of fundamental
equations . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.1.9 The description of the scattering data . . . . . . . . . 28
1.2 The inverse scattering problem on a half-line with a potential
self-adjoint matrix . . . . . . . . . . . . . . . . . . . . . . . . 36
1.2.1 The unique solvability of the self-adjoint problem . . . 37
1.2.2 The Jost solutions of system (1.84) . . . . . . . . . . . 40
1.2.3 The scattering function and its properties . . . . . . . 42
1.2.4 The relation between the functions f(x,ξ), g(x,ξ) and
f(ξ), g(ξ). . . . . . . . . . . . . . . . . . . . . . . . . 45
1.2.5 The inverse scattering problem . . . . . . . . . . . . . 51
1.2.6 The complete description of the scattering function . . 54
2 Some problems for a system of nonlinear evolution equations
on a half-line 61
2.1 The IBVP for the system of NLEEs . . . . . . . . . . . . . . 63
vii
viii Contents
2.1.1 The Lax compatibility condition . . . . . . . . . . . . 63
2.1.2 The time-dependence of the scattering function . . . . 65
2.1.3 Evaluation of unknown BVs . . . . . . . . . . . . . . . 67
2.1.4 The time-dependence of the scattering data . . . . . . 69
2.1.5 The solution of the IBVP for the system of NLEEs (2.5) 70
2.1.6 The IBVP for the attractive NLS equation . . . . . . 74
2.2 Exact solutions of the system of NLEEs . . . . . . . . . . . . 76
2.2.1 Exact solutions of fundamental equations . . . . . . . 76
2.2.2 Thetime-dependenceofstandardizedmultipliersandan
exact solution of system (2.5) . . . . . . . . . . . . . . 78
2.2.3 An exact solution of the attractive NLS equation . . . 83
2.3 The Cauchy IVP problem for the repulsive NLS equation . . 85
3 Some problems for cubic nonlinear evolution equations on a
half-line 89
3.1 The direct and inverse scattering problem . . . . . . . . . . . 90
3.1.1 The representation of the solution of system (3.4). . . 90
3.1.2 The Jost solutions of system (3.4) . . . . . . . . . . . 92
3.1.3 The scattering function S(λ) and non-real eigenvalues 93
3.1.4 Connection between the analytic solution and
Jost solutions . . . . . . . . . . . . . . . . . . . . . . . 95
3.1.5 The scattering data . . . . . . . . . . . . . . . . . . . 98
3.1.6 The systems of fundamental equations . . . . . . . . . 100
3.1.7 The complete description of the scattering data . . . . 101
3.2 The IBVPs for the mKdV equations . . . . . . . . . . . . . . 102
3.2.1 The Lax compatibility condition . . . . . . . . . . . . 103
3.2.2 The time-dependence of the scattering function . . . . 104
3.2.3 Evaluation of unknown BVs . . . . . . . . . . . . . . . 106
3.2.4 The time-dependence of the scattering data . . . . . . 108
3.2.5 The solution of the IBVPs for mKdV equations . . . . 110
3.2.6 RelationbetweensolutionsofthemKdVandKdVequa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.3 Non-scattering potentials and exact solutions . . . . . . . . . 117
3.3.1 Exact solutions of systems of fundamental equations . 117
3.3.2 Thetime-dependenceofstandardizedmultipliersandan
exact solution of system (3.41) . . . . . . . . . . . . . 119
3.3.3 Exact solutions of equations mKdV and KdV . . . . . 122
3.4 The Cauchy problem for cubic nonlinear
equation (3.3) . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4 The Dirichlet IBVPs for sine and sinh-Gordon equations 129
4.1 The IBVP for the sG equation . . . . . . . . . . . . . . . . . 132
4.1.1 The Jost solutions . . . . . . . . . . . . . . . . . . . . 132
4.1.2 The Lax compatibility condition . . . . . . . . . . . . 134
4.1.3 Evaluation of unknown BVs . . . . . . . . . . . . . . . 135
Contents ix
4.1.4 The time-dependence of the scattering data . . . . . . 142
4.1.5 The IBVP (4.14)–(4.16) . . . . . . . . . . . . . . . . . 146
4.2 The IBVP for the shG equation . . . . . . . . . . . . . . . . 149
4.2.1 The self-adjoint problem associated with the shG
equation . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.2.2 The Lax compatibility condition . . . . . . . . . . . . 151
4.2.3 Evaluation of unknown BVs . . . . . . . . . . . . . . . 153
4.2.4 The time-dependence of the scattering function . . . . 158
4.2.5 The IBVP for the shG equation. . . . . . . . . . . . . 158
4.3 Exact soliton-solutions of the sG and shG equations . . . . . 162
5 Inverse scattering for integration of the continual system
of nonlinear interaction waves 167
5.1 The direct and ISP for a system of n first-order ODEs . . . . 169
5.1.1 The transition matrix S(λ) . . . . . . . . . . . . . . . 170
5.1.2 Representations of solutions of system (5.5) . . . . . . 170
5.1.3 The intermediate matrix S˜(λ) . . . . . . . . . . . . . . 181
5.1.4 The bilateral factorization of the transition
matrix S(λ) . . . . . . . . . . . . . . . . . . . . . . . . 182
5.1.5 The analytic and bilateral factorizations of S˜(λ) . . . 187
5.1.6 The inverse scattering problem . . . . . . . . . . . . . 190
5.2 The direct and ISP for the transport equation . . . . . . . . 195
5.2.1 The transition operator S(λ) . . . . . . . . . . . . . . 195
5.2.2 Volterra integral representations of solutions . . . . . 197
5.2.3 Bilateral Volterra factorization of the S-operator . . . 207
5.2.4 Analytic and bilateral Volterra factorizations of the in-
termediate operator S˜(λ) . . . . . . . . . . . . . . . . 212
5.2.5 The inverse scattering problem . . . . . . . . . . . . . 217
5.3 Integration of the continual system of nonlinear
interaction waves . . . . . . . . . . . . . . . . . . . . . . . . 227
5.3.1 The generalized Lax equation . . . . . . . . . . . . . . 227
5.3.2 The time-evolution of the operators F˜(λ;t) and G˜(λ;t) 232
5.3.3 The Cauchy problem for the continual system (5.213) 234
6 Some problems for the KdV equation and associated inverse
scattering 237
6.1 The direct and ISP . . . . . . . . . . . . . . . . . . . . . . . 239
6.1.1 The Jost solution and the analytic solution . . . . . . 240
6.1.2 The Parseval’s equality and the fundamental equation 244
6.1.3 The necessary conditions of the scattering data . . . . 247
6.1.4 The necessary and sufficient conditions of a
given data set . . . . . . . . . . . . . . . . . . . . . . . 249
6.2 The IBVP for the KdV equation . . . . . . . . . . . . . . . . 250
6.2.1 The Lax compatibility condition . . . . . . . . . . . . 251
6.2.2 The time-dependent Jost solution. . . . . . . . . . . . 254